Analysis on some powered integral inequalities with retarded argument and application

Abstract

We investigate a certain class of nonlinear Gronwall–Bellman–Pachpatte type of integral inequalities involving retarded term and nonlinear integrals. Our results derive some new inequalities as well as give an upper bound estimation of the unknown function. The significance of these inequalities originate from the truth that it is far applicable in specific situations in which different available inequalities do not apply legitimately. Our consequences additionally can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is also illustrated to show the validity of our established theorems.

Keywords:

• Retarded integral inequalities
• explicit bounds
• non-decreasing
• power

Mathematics Subject Classifications 2010:

• 26D10
• 34A34

1. Introduction

Integral inequalities possess an exceptionally advantageous position in the improvement of the hypothesis of ordinary differential and integral equations. In this account, fundamental integral inequality by Gronwall [1] and Bellman [2] contributes a significant part in examining the stability and asymptotic behaviour in order to find the solutions of integral equations. In the ongoing years, Gronwall–Bellman type inequalities have been enormously improved by the affirmation of their importance and inherent value in numerous branches of the applied sciences. These inequalities with their linear and nonlinear generalizations have earned much consideration in the works of Abdeldaim et al. [3], Bainov et al. [4], Bihari [5], El-Owaidy et al. [6], Jiang et al. [7], Kim [8], Liu et al. [9], Ngoc et al. [10], Oguntuase [11], Pachpatte [12], Pachpatte [13], Wang et al. [14], Zhou et al. [15].

Furthermore, El-Owaidy et al. [16] introduced a new essential inequality of the type (1) $\begin{array}{rl}r\left(u\right)& \le r_0+{\int }_0^{u}l\left(s\right)\\ & \times [r^{\varrho }\left(s\right)+{\int }_0^{s}t\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma ]\mathrm{d}s,u\in [0,\mathrm{\infty }),\end{array}$(1) in which the integrand function incorporates the power and makes it tougher to estimate the unknown function where $0\le \varrho <1$. Nevertheless, many real life issues that have been formulated for differential equations in the past were sometimes based on initial value problems. As a matter of fact, include a critical memory impact that can be interpreted in a more sophisticated version of differential equation that integrates delayed or retarded arguments. In these circumstances, we need to speak about some retarded nonlinear integral inequalities. In this record, Agarwal et al. [17] discovered the integral inequality as (2) $\begin{array}{rl}r\left(u\right)& \le x\left(u\right)\\ & +\sum _{i=1}^n{\int }_{{\omega }_i\left(u_0\right)}^{{\omega }_i\left(u\right)}{l}_i(u,s){\rho }_i\left(r\right(s\left)\right)\mathrm{d}s,u_0\le u<u_1,\end{array}$(2) where ${\rho }_i$, $i=1,\dots ,n$ are positive, continuous and non-decreasing functions on $[0,\mathrm{\infty })$ and ${\rho }_1\propto {\rho }_2\cdots \propto {\rho }_n$ with n terms on the right integral of inequality. The concept of recursion is expected to get the estimation. Very recently, Abdeldaim et al. [18] proved the retarded inequality of another kind (3) $\begin{array}{rl}r\left(u\right)& \le r_0+{\int }_0^{\omega \left(u\right)}[l\left(s\right)r\left(s\right)+q\left(s\right)]\mathrm{d}s\\ & +{\int }_0^{\omega \left(u\right)}l\left(s\right)\left({\int }_0^{s}e\left(\varsigma \right)r\left(\phi \right)\mathrm{d}\varsigma \right)\mathrm{d}s,u\in [0,\mathrm{\infty }),\end{array}$(3) such that $\omega \left(u\right)\le u$ and $\omega \left(0\right)=0$ and multiplication of two functions exist on the right of the third term, that's a unique form of inequality (2(2) $\begin{array}{rl}r\left(u\right)& \le x\left(u\right)\\ & +\sum _{i=1}^n{\int }_{{\omega }_i\left(u_0\right)}^{{\omega }_i\left(u\right)}{l}_i(u,s){\rho }_i\left(r\right(s\left)\right)\mathrm{d}s,u_0\le u<u_1,\end{array}$(2) ). Moreover, El-Deeb et.al [19] investigated the retarded Volterra–Fredholm type integral inequality (4) $\begin{array}{rl}{r}^{\varrho }\left(u\right)& \le c\left(u\right)+{\int }_a^{\omega \left(u\right)}l\left(s\right)r\left(s\right)\mathrm{d}s\\ & +{\int }_a^{b}e\left(s\right)r^{\varrho }\left(s\right)\mathrm{d}s,u\in [a,b],\end{array}$(4) where $\omega \left(u\right)\le u$, $\omega \left(a\right)=a$ and $\varrho \ge 1$ is a constant. Other than the results referenced over, a number of examiners have observed many beneficial integral inequalities, mainly stimulated by their applications in exclusive parts of the differential equations (see Boudeliou [20], Ferreira et al. [21], Khan et al. [22], Lipovan [23], Belmor et al. [24], Lipovan [25], Wang et al. [26], Ul-Haq et al. [27], Xu et al. [28]).

However, in some circumstances, various classes of powered delay integral and differential equations, it is expected to inspect new delay inequalities so as to survive and accomplish the ideal objectives. Therefore, in this article, we explore certain new generalizations of retarded integral inequalities with power for previous well-known results, which can be utilized as helpful apparatuses to demonstrate the classifications of integral inequalities and integral equations. Lastly, we give one example to describe the benefit of our results.

2. Results and discussion

Given notations will be followed throughout this text for the ease of reading: ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$, $\mathbb{R}$ denotes the set of real numbers, $C(N,S)$ be the class of all continuous functions defined on set N with range in the set S and

(H1)	The function $r\left(u\right)$ is a nonnegative and continuous on $[u_0,\mathrm{\infty })$.
(H2)	$\omega \left(u\right)$ be a continuous, differentiable and increasing function on $[u_0,\mathrm{\infty })$ with $\omega \left(u\right)\le u$, $\omega \left(u_0\right)=u_0$.
(H3)	Let $m\left(u\right)$, $j\left(u\right)$, $l\left(u\right)$, $n\left(u\right)\in C\left(\right[u_0,\mathrm{\infty }),{\mathbb{R}}_{+})$ and $m\left(u\right)$ be a non-decreasing function.
(H4)	$\mu ,\phi \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ be non-decreasing function and $\mu \left(r\right)>0$, $\phi \left(r\right)>0$ for r>0.

Presently, a principle lemma which will be utilized later in verification of the exhibited paper is given below:

Lemma 2.1

[29]

Let $z\ge 0$, $\varphi \ge \eta \ge 0$ and $\varphi \ne 0$, then (5) $\begin{array}{r}{z}^{\frac{\eta }{\varphi }}\le \frac{\eta }{\varphi }z+\frac{\varphi -\eta }{\varphi }.\end{array}$(5)

Proof.

inequality (5(5) $\begin{array}{r}{z}^{\frac{\eta }{\varphi }}\le \frac{\eta }{\varphi }z+\frac{\varphi -\eta }{\varphi }.\end{array}$(5) ) is valid for $\eta =0$, but if $\eta >0$ and $\delta =\eta /\varphi$, then $\delta =1$ by [29], we have $z^{\frac{\eta }{\varphi }}\le \frac{\eta }{\varphi }{L}^{\left(\eta -\varphi \right)/\varphi }z+\frac{\varphi -\eta }{\varphi }{L}^{\eta /\varphi },$ for any L>0. Let L = 1, we get (5(5) $\begin{array}{r}{z}^{\frac{\eta }{\varphi }}\le \frac{\eta }{\varphi }z+\frac{\varphi -\eta }{\varphi }.\end{array}$(5) ).

Theorem 2.2

If the conditions (H1)–(H3) and (6) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\\ & \times (r^2\left(s\right)+{\int }_{u_0}^{s}n\left(\varsigma \right)r^2\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\mathrm{\forall }u\in {\mathbb{R}}_{+},\end{array}$(6) hold. Then (7) $\begin{array}{r}r\left(u\right)\le \frac{1}{\exp (-m_1(u\left)\right)-{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s},\end{array}$(7) provided with (8) $\begin{array}{r}{m}_1\left(u\right)=\ln \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mathrm{d}s.\end{array}$(8)

Proof.

Denote (9) $\begin{array}{rl}P\left(u\right)& =m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(r^2\left(s\right)+{\int }_{u_0}^{s}n\left(\varsigma \right)r^2\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(9) $P\left(u\right)$ is non-decreasing function, so (6(6) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\\ & \times (r^2\left(s\right)+{\int }_{u_0}^{s}n\left(\varsigma \right)r^2\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\mathrm{\forall }u\in {\mathbb{R}}_{+},\end{array}$(6) ) restates as (10) $\begin{array}{r}r\left(u\right)\le P\left(u\right),P\left(u_0\right)=m\left(u_0\right),\end{array}$(10) differentiating (9(9) $\begin{array}{rl}P\left(u\right)& =m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(r^2\left(s\right)+{\int }_{u_0}^{s}n\left(\varsigma \right)r^2\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(9) ) and employ (10(10) $\begin{array}{r}r\left(u\right)\le P\left(u\right),P\left(u_0\right)=m\left(u_0\right),\end{array}$(10) ), we get (11) $\begin{array}{rl}{P}^{\mathrm{\prime }}\left(u\right)& =m^{\mathrm{\prime }}\left(u\right)+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)r\left(\omega \right(u\left)\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(r^2\left(\omega \right(u\left)\right)+{\int }_{u_0}^{u}n\left(s\right)r^2\left(s\right)\mathrm{d}s)\\ & \le m^{\mathrm{\prime }}\left(u\right)+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(P^2\left(\omega \right(u\left)\right)+{\int }_{u_0}^{u}n\left(s\right)P^2\left(s\right)\mathrm{d}s),\end{array}$(11) (11(11) $\begin{array}{rl}{P}^{\mathrm{\prime }}\left(u\right)& =m^{\mathrm{\prime }}\left(u\right)+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)r\left(\omega \right(u\left)\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(r^2\left(\omega \right(u\left)\right)+{\int }_{u_0}^{u}n\left(s\right)r^2\left(s\right)\mathrm{d}s)\\ & \le m^{\mathrm{\prime }}\left(u\right)+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(P^2\left(\omega \right(u\left)\right)+{\int }_{u_0}^{u}n\left(s\right)P^2\left(s\right)\mathrm{d}s),\end{array}$(11) ) becomes (12) $\begin{array}{rl}\frac{P^{\mathrm{\prime }}\left(u\right)}{P\left(u\right)}& \le \frac{m^{\mathrm{\prime }}\left(u\right)}{P\left(u\right)}+\frac{j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)}{P\left(u\right)}\\ & +\frac{l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P^2\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)}{P\left(u\right)}\\ & \le \frac{m^{\mathrm{\prime }}\left(u\right)}{m\left(u\right)}+\frac{j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)}{P\left(\omega \right(u\left)\right)}\\ & +\frac{l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P^2\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)}{P\left(\omega \right(u\left)\right)}\\ & =\frac{m^{\mathrm{\prime }}\left(u\right)}{m\left(u\right)}+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)P\left(\omega \right(u\left)\right).\end{array}$(12) Integrate the both sides of (12(12) $\begin{array}{rl}\frac{P^{\mathrm{\prime }}\left(u\right)}{P\left(u\right)}& \le \frac{m^{\mathrm{\prime }}\left(u\right)}{P\left(u\right)}+\frac{j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)}{P\left(u\right)}\\ & +\frac{l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P^2\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)}{P\left(u\right)}\\ & \le \frac{m^{\mathrm{\prime }}\left(u\right)}{m\left(u\right)}+\frac{j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)}{P\left(\omega \right(u\left)\right)}\\ & +\frac{l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)P^2\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)}{P\left(\omega \right(u\left)\right)}\\ & =\frac{m^{\mathrm{\prime }}\left(u\right)}{m\left(u\right)}+j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\\ & +l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)(1+{\int }_{u_0}^{s}n\left(s\right)\mathrm{d}s)P\left(\omega \right(u\left)\right).\end{array}$(12) ) from $u_0$ to u, we have (13) $\begin{array}{r}\ln \left(P\right(u\left)\right)\le m_1\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )P\left(s\right)\mathrm{d}s,\end{array}$(13) where $m_1\left(u\right)$ be as defined in (8(8) $\begin{array}{r}{m}_1\left(u\right)=\ln \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mathrm{d}s.\end{array}$(8) ). Consider (14) $\begin{array}{r}\ln \left(P\right(u\left)\right)\le L\left(u\right),P\left(u\right)\le \exp \left(L\right(u\left)\right),\end{array}$(14) and (15) $\begin{array}{r}L\left(u\right)=m_1\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )P\left(s\right)\mathrm{d}s.\end{array}$(15) Clearly $m_1\left(u\right)>0$ in (15(15) $\begin{array}{r}L\left(u\right)=m_1\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )P\left(s\right)\mathrm{d}s.\end{array}$(15) ), so $L\left(u\right)>0$ and $m\left(u\right)>1$ from (8(8) $\begin{array}{r}{m}_1\left(u\right)=\ln \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mathrm{d}s.\end{array}$(8) ), taking derivative (15(15) $\begin{array}{r}L\left(u\right)=m_1\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )P\left(s\right)\mathrm{d}s.\end{array}$(15) ) and from (14(14) $\begin{array}{r}\ln \left(P\right(u\left)\right)\le L\left(u\right),P\left(u\right)\le \exp \left(L\right(u\left)\right),\end{array}$(14) ), we deduce (16) $\begin{array}{rl}{L}^{\mathrm{\prime }}\left(u\right)& =m_1^{\mathrm{\prime }}\left(u\right)+l\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s){\omega }^{\mathrm{\prime }}\left(u\right)P\left(\omega \right(u\left)\right)\\ & \le m_1^{\mathrm{\prime }}\left(u\right)+l\left(\omega \right(u\left)\right)\\ & \times (1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s){\omega }^{\mathrm{\prime }}\left(u\right)\exp \left(L\right(\omega \left(u\right)\left)\right),\end{array}$(16) or, equivalently, (17) $\begin{array}{rl}\frac{L^{\mathrm{\prime }}\left(u\right)}{\exp \left(L\right(u\left)\right)}& \le \frac{m_1^{\mathrm{\prime }}\left(u\right)}{\exp \left(m_1\right(u\left)\right)}\\ & +\frac{\begin{array}{c}l\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s)\\ {\omega }^{\mathrm{\prime }}\left(u\right)\exp \left(L\right(\omega \left(u\right)\left)\right)\end{array}}{\exp \left(L\right(\omega \left(u\right)\left)\right)}\\ & =\frac{m_1^{\mathrm{\prime }}\left(u\right)}{\exp \left(m_1\right(u\left)\right)}+l\left(\omega \right(u\left)\right)\\ & \times (1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s){\omega }^{\mathrm{\prime }}\left(u\right),\end{array}$(17) by integrating (17(17) $\begin{array}{rl}\frac{L^{\mathrm{\prime }}\left(u\right)}{\exp \left(L\right(u\left)\right)}& \le \frac{m_1^{\mathrm{\prime }}\left(u\right)}{\exp \left(m_1\right(u\left)\right)}\\ & +\frac{\begin{array}{c}l\left(\omega \right(u\left)\right)(1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s)\\ {\omega }^{\mathrm{\prime }}\left(u\right)\exp \left(L\right(\omega \left(u\right)\left)\right)\end{array}}{\exp \left(L\right(\omega \left(u\right)\left)\right)}\\ & =\frac{m_1^{\mathrm{\prime }}\left(u\right)}{\exp \left(m_1\right(u\left)\right)}+l\left(\omega \right(u\left)\right)\\ & \times (1+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s){\omega }^{\mathrm{\prime }}\left(u\right),\end{array}$(17) ) from $u_0$ to u, it is noticed that (18) $\begin{array}{r}L\left(u\right)\le \ln \left(\frac{1}{\begin{array}{c}\exp (-m_1(u\left)\right)-{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\\ (1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\end{array}}\right),\end{array}$(18) the required estimation in (7(7) $\begin{array}{r}r\left(u\right)\le \frac{1}{\exp (-m_1(u\left)\right)-{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)(1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s},\end{array}$(7) ) can be obtained by combining (10(10) $\begin{array}{r}r\left(u\right)\le P\left(u\right),P\left(u_0\right)=m\left(u_0\right),\end{array}$(10) ), (14(14) $\begin{array}{r}\ln \left(P\right(u\left)\right)\le L\left(u\right),P\left(u\right)\le \exp \left(L\right(u\left)\right),\end{array}$(14) ) and (18(18) $\begin{array}{r}L\left(u\right)\le \ln \left(\frac{1}{\begin{array}{c}\exp (-m_1(u\left)\right)-{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\\ (1+{\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\end{array}}\right),\end{array}$(18) ).

Theorem 2.3

Suppose that (H1), (H2), (H3) be satisfied. Moreover (19) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)(r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(19) for all $u\in {\mathbb{R}}_{+}$ such that $\varrho \in (0,1]$ is a positive constant. Then (20) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)\\ & +\frac{m_2\left(u\right)\exp ({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)}{1-m_2\left(u\right){\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)\exp ({\int }_{u_0}^s\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s},\end{array}$(20) where (21) $\begin{array}{rl}{m}_2\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)m\left(s\right)([\varrho (m\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma )\\ & +(1-\varrho )+{\int }_{u_0}^{s}n\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma ])\mathrm{d}s,\end{array}$(21) (22) $\begin{array}{rl}\mathrm{{\rm Y}}\left(u\right)& =j\left(u\right)[2\varrho m\left(u\right)(1+{\int }_{u_0}^{u}l\left(s\right)\mathrm{d}s)\\ & +(1-\varrho )+2m\left(u\right){\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s],\end{array}$(22) (23) $\begin{array}{rl}\chi \left(u\right)& =j\left(u\right)[\varrho (1+{\int }_{u_0}^{u}l\left(s\right)\mathrm{d}s)+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s].\end{array}$(23)

Proof.

Let (24) $\begin{array}{rl}Y\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)(r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(24) therefore, (19(19) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)(r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(19) ) reaches to (25) $\begin{array}{r}r\left(u\right)\le m\left(u\right)+Y\left(u\right),Y\left(u_0\right)=0,\end{array}$(25) by applying Lemma 2.1 and (25(25) $\begin{array}{r}r\left(u\right)\le m\left(u\right)+Y\left(u\right),Y\left(u_0\right)=0,\end{array}$(25) ) into (24(24) $\begin{array}{rl}Y\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)(r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\end{array}$(24) ), it follows that (26) $\begin{array}{rl}Y\left(u\right)& \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\\ & \times [\varrho (r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )+(1-\varrho )]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [\varrho (m\left(s\right)+Y\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma )\\ & +(1-\varrho )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [{\int }_{u_0}^{s}n\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma ]\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)m\left(s\right)(\varrho (m\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)m\left(s\right)\\ & +j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\\ & \times [j\left(s\right)(2\varrho m\left(s\right)(1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)+2j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varrho \right)d\varrho )]Y\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)(\varrho (1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +j\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )]Y^2\left(s\right)\mathrm{d}s\\ & \le m_2\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)Y\left(s\right)\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)Y^2\left(s\right)\mathrm{d}s,\end{array}$(26) where $m_2\left(u\right)$, $\mathrm{{\rm Y}}\left(u\right)$ and $\chi \left(u\right)$ are quoted in (21(21) $\begin{array}{rl}{m}_2\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)m\left(s\right)([\varrho (m\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma )\\ & +(1-\varrho )+{\int }_{u_0}^{s}n\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma ])\mathrm{d}s,\end{array}$(21) ), (22(22) $\begin{array}{rl}\mathrm{{\rm Y}}\left(u\right)& =j\left(u\right)[2\varrho m\left(u\right)(1+{\int }_{u_0}^{u}l\left(s\right)\mathrm{d}s)\\ & +(1-\varrho )+2m\left(u\right){\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s],\end{array}$(22) ) and (23(23) $\begin{array}{rl}\chi \left(u\right)& =j\left(u\right)[\varrho (1+{\int }_{u_0}^{u}l\left(s\right)\mathrm{d}s)+{\int }_{u_0}^{u}n\left(s\right)\mathrm{d}s].\end{array}$(23) ) respectively. For an arbitrary $u\in [u_0,U^{\star }]$, fixed $U^{\star }\in [u_0,\mathrm{\infty })$ and taking $Y_1\left(u\right)$ by the right hand side of (26(26) $\begin{array}{rl}Y\left(u\right)& \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\\ & \times [\varrho (r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )+(1-\varrho )]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [\varrho (m\left(s\right)+Y\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma )\\ & +(1-\varrho )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [{\int }_{u_0}^{s}n\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma ]\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)m\left(s\right)(\varrho (m\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)m\left(s\right)\\ & +j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\\ & \times [j\left(s\right)(2\varrho m\left(s\right)(1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)+2j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varrho \right)d\varrho )]Y\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)(\varrho (1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +j\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )]Y^2\left(s\right)\mathrm{d}s\\ & \le m_2\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)Y\left(s\right)\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)Y^2\left(s\right)\mathrm{d}s,\end{array}$(26) ), we deduce (27) $\begin{array}{rl}{Y}_1\left(u\right)& =m_2\left(U^{\star }\right)+{\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)Y\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)Y^2\left(s\right)\mathrm{d}s,\end{array}$(27) (26(26) $\begin{array}{rl}Y\left(u\right)& \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\\ & \times [\varrho (r\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )+(1-\varrho )]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)r\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [\varrho (m\left(s\right)+Y\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma )\\ & +(1-\varrho )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)[m\left(s\right)+Y\left(s\right)]\\ & \times [{\int }_{u_0}^{s}n\left(\varsigma \right)(m\left(\varsigma \right)+Y\left(\varsigma \right))\mathrm{d}\varsigma ]\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)m\left(s\right)(\varrho (m\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)m\left(s\right)\\ & +j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)m\left(\varsigma \right)\mathrm{d}\varsigma )]\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\\ & \times [j\left(s\right)(2\varrho m\left(s\right)(1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +(1-\varrho )j\left(s\right)+2j\left(s\right)m\left(s\right)({\int }_{u_0}^{s}n\left(\varrho \right)d\varrho )]Y\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}[j\left(s\right)(\varrho (1+{\int }_{u_0}^{s}l\left(\varsigma \right)\mathrm{d}\varsigma ))\\ & +j\left(s\right)({\int }_{u_0}^{s}n\left(\varsigma \right)\mathrm{d}\varsigma )]Y^2\left(s\right)\mathrm{d}s\\ & \le m_2\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)Y\left(s\right)\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)Y^2\left(s\right)\mathrm{d}s,\end{array}$(26) ) can be carried out as (28) $\begin{array}{rl}Y\left(u\right)& \le Y_1\left(u\right),Y\left(\omega \right(u\left)\right)\le Y_1\left(\omega \right(u\left)\right)\le Y_1\left(u\right),\\ Y_1\left(u_0\right)& =m_2\left(U^{\star }\right),\end{array}$(28) since $Y_1\left(u\right)$ is non-dereasing, hence we can write (27(27) $\begin{array}{rl}{Y}_1\left(u\right)& =m_2\left(U^{\star }\right)+{\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)Y\left(s\right)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)Y^2\left(s\right)\mathrm{d}s,\end{array}$(27) ) by using (28(28) $\begin{array}{rl}Y\left(u\right)& \le Y_1\left(u\right),Y\left(\omega \right(u\left)\right)\le Y_1\left(\omega \right(u\left)\right)\le Y_1\left(u\right),\\ Y_1\left(u_0\right)& =m_2\left(U^{\star }\right),\end{array}$(28) ) as the following (29) $\begin{array}{r}{Y}_1^{\mathrm{\prime }}\left(u\right)\le \mathrm{{\rm Y}}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Y_1\left(u\right)+\chi \left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Y_1^2\left(u\right),\end{array}$(29) take $Z\left(u\right)=Y_1^{-1}\left(u\right)$, then $Z^{\mathrm{\prime }}\left(u\right)=-Y_1^{-2}\left(u\right)Y_1^{\mathrm{\prime }}\left(u\right)$, therefore (29(29) $\begin{array}{r}{Y}_1^{\mathrm{\prime }}\left(u\right)\le \mathrm{{\rm Y}}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Y_1\left(u\right)+\chi \left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Y_1^2\left(u\right),\end{array}$(29) ) yields (30) $\begin{array}{r}{Z}^{\mathrm{\prime }}\left(u\right)+\mathrm{{\rm Y}}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Z\left(u\right)\ge -\chi \left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right).\end{array}$(30) Multiplying both sides of (30(30) $\begin{array}{r}{Z}^{\mathrm{\prime }}\left(u\right)+\mathrm{{\rm Y}}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)Z\left(u\right)\ge -\chi \left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right).\end{array}$(30) ) by $\exp \left({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\right(s\left)\mathrm{d}s\right)$, integrating the resultant inequality and applying $Z\left(u_0\right)=m_2^{-1}\left(U^{\star }\right)$, we attain (31) $\begin{array}{r}Z\left(u\right)\ge \frac{\begin{array}{c}1-m_2\left(U^{\star }\right){\int }_{u_0}^u\chi \left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\\ \exp ({\int }_{u_0}^{\omega \left(s\right)}\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\end{array}}{m_2\left(U^{\star }\right)\exp ({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)},\end{array}$(31) substituting $Z\left(u\right)=Y_1^{-1}\left(u\right)$ in (31(31) $\begin{array}{r}Z\left(u\right)\ge \frac{\begin{array}{c}1-m_2\left(U^{\star }\right){\int }_{u_0}^u\chi \left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\\ \exp ({\int }_{u_0}^{\omega \left(s\right)}\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s\end{array}}{m_2\left(U^{\star }\right)\exp ({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)},\end{array}$(31) ) and from (28(28) $\begin{array}{rl}Y\left(u\right)& \le Y_1\left(u\right),Y\left(\omega \right(u\left)\right)\le Y_1\left(\omega \right(u\left)\right)\le Y_1\left(u\right),\\ Y_1\left(u_0\right)& =m_2\left(U^{\star }\right),\end{array}$(28) ), we get (32) $\begin{array}{r}Y\left(u\right)\le \frac{m_2\left(U^{\star }\right)\exp ({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)}{1-m_2\left(U^{\star }\right){\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)\exp ({\int }_{u_0}^s\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s},\end{array}$(32) so (25(25) $\begin{array}{r}r\left(u\right)\le m\left(u\right)+Y\left(u\right),Y\left(u_0\right)=0,\end{array}$(25) ) leads to (33) $\begin{array}{rl}r\left(U^{\star }\right)& \le m\left(U^{\star }\right)\\ & +\frac{m_2\left(U^{\star }\right)\exp ({\int }_{u_0}^{\omega \left(U^{\star }\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)}{1-m_2\left(U^{\star }\right){\int }_{u_0}^{\omega \left(U^{\star }\right)}\chi \left(\right(s)\exp ({\int }_{u_0}^s\mathrm{{\rm Y}}(\varsigma )\mathrm{d}\varsigma )ds},\end{array}$(33) the arbitrariness of $U^{\star }$ in (33(33) $\begin{array}{rl}r\left(U^{\star }\right)& \le m\left(U^{\star }\right)\\ & +\frac{m_2\left(U^{\star }\right)\exp ({\int }_{u_0}^{\omega \left(U^{\star }\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)}{1-m_2\left(U^{\star }\right){\int }_{u_0}^{\omega \left(U^{\star }\right)}\chi \left(\right(s)\exp ({\int }_{u_0}^s\mathrm{{\rm Y}}(\varsigma )\mathrm{d}\varsigma )ds},\end{array}$(33) ) gives the desired bound in (20(20) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)\\ & +\frac{m_2\left(u\right)\exp ({\int }_{u_0}^{\omega \left(u\right)}\mathrm{{\rm Y}}\left(s\right)\mathrm{d}s)}{1-m_2\left(u\right){\int }_{u_0}^{\omega \left(u\right)}\chi \left(s\right)\exp ({\int }_{u_0}^s\mathrm{{\rm Y}}\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s},\end{array}$(20) ). This completes the proof.

Remark 2.1

If $m\left(u\right)=c$, $\omega \left(u\right)\le u$, $n\left(u\right)=0$, then Theorem 2.3 converts to Theorem 3.2 due to Abdeldaim et al. [3].

Remark 2.2

By taking $m\left(u\right)=u_0$, $\varrho =1$, $r\left(u\right)\le \phi \left(u\right(t\left)\right)$,$n\left(u\right)=g\left(t\right)$ and $g\left(u\right)=f\left(t\right)$, then Theorem 2.3 will be a slight variation of Theorem 2.2 with $q\left(t\right)=0$ given in Abdeldaim et al. [18].

Another interesting and slightly different version of the nonlinear integral inequality may be stated as follows.

Corollary 2.4

If (H1), (H2), (H5) $m\left(u\right)$, $j\left(u\right)$, $l\left(u\right)\in C\left(\right[u_0,\mathrm{\infty }),{\mathbb{R}}_{+})$ be fulfilled and (34) $\begin{array}{rl}{r}^{\varrho }\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)r\left(s\right)\\ & \times ({\int }_{u_0}^{s}l\left(\varsigma \right)r^{\varrho }\left(\varsigma \right)\mathrm{d}\varsigma )\mathrm{d}s,\mathrm{\forall }u\in {\mathbb{R}}_{+}.\end{array}$(34) with $\varrho >0$. Then (35) $\begin{array}{r}r\left(u\right)\le m^{\frac{1}{\varrho }}\left(u\right)(1+\frac{m^{-1}\left(u\right)}{\exp (-m_4(u\left)\right)-{\int }_{u_0}^{\omega \left(u\right)}{\chi }_1\left(s\right)\mathrm{d}s}),\end{array}$(35) where $m_4\left(u\right)=\ln \left(m_3\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}{\mathrm{{\rm Y}}}_1\left(s\right)\mathrm{d}s$, $m_3\left(u\right)=j\left(u\right)\left(\right(m\left(u\right)+\varrho -1\left)/\varrho \right)\left({\int }_{u_0}^{u}l\right(s\left)m\right(s\left)\mathrm{d}s\right)$, ${\chi }_1\left(u\right)=\left(1/\varrho \right)j\left(u\right)\left({\int }_{u_0}^{t}l\right(s\left)\mathrm{d}s\right)$ and ${\mathrm{{\rm Y}}}_1\left(u\right)=\left(1/\varrho \right)j\left(u\right)\left(2m\right(u)+\varrho -1)\left({\int }_{u_0}^{u}l\right(s\left)\mathrm{d}s\right)$.

Proof.

The proof of Corollary 2.4 procee ds in much a comparable way as in the evidence of Theorem 2.2 with a few alterations. We leave the information to the reader.

Theorem 2.5

The assumptions (H1), (H2), (H4), (H5) be satisfied. Also (36) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mu \left(r\right(s\left)\right)\\ & \times (r^{\lambda }\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)\phi \left(r\right(\varsigma \left)\right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s,\mathrm{\forall }u\in {\mathbb{R}}_{+},\end{array}$(36) with $\varrho >0$, $\lambda \ge 1$ are constants and $\varrho +\lambda >1$. Then (37) $\begin{array}{r}r\left(u\right)\le {\xi }^{-1}(\xi \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s),\end{array}$(37) ∀$u\in [u_0,U^{\star }]$, provided with (38) $\begin{array}{rl}Q\left(u\right)& =({\mathrm{\Omega }}^{-1}[{\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(u\right))\\ & +{\int }_{u_0}^{u}l\left(s\right)\mathrm{d}s)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j(s)\mathrm{d}s)]{)}^{\varrho }\end{array}$(38) (39) $\begin{array}{rl}\mathrm{\Omega }\left(z\right)& ={\int }_1^z\frac{\mathrm{d}s}{\phi \left(s\right)},;z>0,\end{array}$(39) (40) $\begin{array}{rl}\mathrm{\Lambda }\left(z\right)& ={\int }_1^z\frac{\phi \left({\mathrm{\Omega }}^{-1}\right(s\left)\right)\mathrm{d}s}{\left(\mu \right({\mathrm{\Omega }}^{-1}\left(s\right)\left)\right)\left({\mathrm{\Omega }}^{-1}\right(s){)}^{(\varrho +\lambda -1)}},;z>0,\end{array}$(40) (41) $\begin{array}{rl}\xi \left(z\right)& ={\int }_1^z\frac{\mathrm{d}s}{\mu \left(s\right)},;z>0,\end{array}$(41) ${\mathrm{\Omega }}^{-1}$, ${\mathrm{\Lambda }}^{-1}$, ${\xi }^{-1}$ are the inverses of Ω, Λ, ξ and $U^{\star }$ is chosen such that (42) $\begin{array}{rl}& \mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(u\right))+{\int }_{u_0}^{u}l(s\left)\mathrm{d}s\right)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mathrm{d}s\\ & \le {\int }_1^z\frac{\phi \left({\mathrm{\Omega }}^{-1}\right(s\left)\right)\mathrm{d}s}{\left(\mu \right({\mathrm{\Omega }}^{-1}\left(s\right)\left)\right)\left({\mathrm{\Omega }}^{-1}\right(s){)}^{(\varrho +\lambda -1)}},\end{array}$(42) (43) $\begin{array}{rl}& \xi \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s\le \frac{\mathrm{d}s}{\mu \left(s\right)},\end{array}$(43) (44) $\begin{array}{rl}& {\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(u\right)\left)+{\int }_{u_0}^{u}l\right(s\left)\mathrm{d}s\right)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mathrm{d}s\le \frac{\mathrm{d}s}{\phi \left(s\right)}.\end{array}$(44)

Proof.

Fixing $U^{\star }\in [u_0,\mathrm{\infty })$ for an arbitrary $u\in [u_0,U^{\star }]$ and letting (45) $\begin{array}{rl}V\left(u\right)& =m\left(U^{\star }\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mu \left(r\right(s\left)\right)\\ & \times (r^{\lambda }\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)\phi \left(r\right(\varsigma \left)\right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s,\end{array}$(45) (36(36) $\begin{array}{rl}r\left(u\right)& \le m\left(u\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mu \left(r\right(s\left)\right)\\ & \times (r^{\lambda }\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)\phi \left(r\right(\varsigma \left)\right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s,\mathrm{\forall }u\in {\mathbb{R}}_{+},\end{array}$(36) ), (45(45) $\begin{array}{rl}V\left(u\right)& =m\left(U^{\star }\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mu \left(r\right(s\left)\right)\\ & \times (r^{\lambda }\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)\phi \left(r\right(\varsigma \left)\right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s,\end{array}$(45) ) imply that (46) $\begin{array}{r}r\left(u\right)\le V\left(u\right),V\left(u_0\right)=m\left(U^{\star }\right),\end{array}$(46) since $V\left(u\right)$ is non-decreasing, then (45(45) $\begin{array}{rl}V\left(u\right)& =m\left(U^{\star }\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)\mu \left(r\right(s\left)\right)\\ & \times (r^{\lambda }\left(s\right)+{\int }_{u_0}^{s}l\left(\varsigma \right)\phi \left(r\right(\varsigma \left)\right)\mathrm{d}\varsigma {)}^{\varrho }\mathrm{d}s,\end{array}$(45) ) equals to (47) $\begin{array}{rl}{V}^{\mathrm{\prime }}\left(u\right)& =j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(r\right(\omega \left(u\right)\left)\right)\\ & \times (r^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(r\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)\\ & \times (V^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(V\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)W^{\varrho }\left(\omega \right(u\left)\right),\end{array}$(47) where (48) $\begin{array}{r}W\left(\omega \right(u\left)\right)=V^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(V\right(s\left)\right)\mathrm{d}s,\end{array}$(48) and (49) $\begin{array}{r}W\left(\omega \right(u_0\left)\right)=m^{\lambda }\left(U^{\star }\right),V\left(\omega \right(u\left)\right)\le W\left(\omega \right(u\left)\right),\end{array}$(49) by the definition of $W\left(\omega \right(u\left)\right)$, utilizing (47(47) $\begin{array}{rl}{V}^{\mathrm{\prime }}\left(u\right)& =j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(r\right(\omega \left(u\right)\left)\right)\\ & \times (r^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(r\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)\\ & \times (V^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(V\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)W^{\varrho }\left(\omega \right(u\left)\right),\end{array}$(47) ), (49(49) $\begin{array}{r}W\left(\omega \right(u_0\left)\right)=m^{\lambda }\left(U^{\star }\right),V\left(\omega \right(u\left)\right)\le W\left(\omega \right(u\left)\right),\end{array}$(49) ) and $W\left(\omega \right(u\left)\right)>0$, we have (50) $\begin{array}{rl}& W^{\mathrm{\prime }}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\\ & =\lambda V^{\lambda -1}{V}^{\mathrm{\prime }}\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)+l\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\phi \left(V\right(\omega \left(u\right)\left)\right)\end{array}$(50) so that (51) $\begin{array}{rl}{W}^{\mathrm{\prime }}\left(\omega \right(u\left)\right)& \le \lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(W\right(\omega \left(u\right)\left)\right)W^{\varrho +\lambda -1}\left(\omega \right(u\left)\right)\\ & +l\left(\omega \right(u\left)\right)\phi \left(W\right(\omega \left(u\right)\left)\right)\end{array}$(51) (52) $\begin{array}{rl}& \frac{W^{\mathrm{\prime }}\left(\omega \right(u\left)\right)}{\phi \left(W\right(\omega \left(u\right)\left)\right)}\le \frac{\begin{array}{c}\lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \\ \left(W\right(\omega \left(u\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(u\left)\right)\end{array}}{\phi \left(W\right(\omega \left(u\right)\left)\right)}+l\left(\omega \right(u\left)\right),\end{array}$(52) integrate (52(52) $\begin{array}{rl}& \frac{W^{\mathrm{\prime }}\left(\omega \right(u\left)\right)}{\phi \left(W\right(\omega \left(u\right)\left)\right)}\le \frac{\begin{array}{c}\lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \\ \left(W\right(\omega \left(u\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(u\left)\right)\end{array}}{\phi \left(W\right(\omega \left(u\right)\left)\right)}+l\left(\omega \right(u\left)\right),\end{array}$(52) ), using (39(39) $\begin{array}{rl}\mathrm{\Omega }\left(z\right)& ={\int }_1^z\frac{\mathrm{d}s}{\phi \left(s\right)},;z>0,\end{array}$(39) ) and (49(49) $\begin{array}{r}W\left(\omega \right(u_0\left)\right)=m^{\lambda }\left(U^{\star }\right),V\left(\omega \right(u\left)\right)\le W\left(\omega \right(u\left)\right),\end{array}$(49) ), we get (53) $\begin{array}{rl}& \mathrm{\Omega }\left(W\right(\omega \left(u\right)\left)\right)\\ & \le \mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{u}l\left(\omega \right(s\left)\right)\mathrm{d}s\\ & +{\int }_{u_0}^u\frac{\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s\\ & \le \mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s\\ & +{\int }_{u_0}^u\frac{\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s,\end{array}$(53) for $u<U^{\star }$. Denoting (54) $\begin{array}{r}{V}_1\left(u\right)={\int }_{u_0}^u\frac{\begin{array}{c}\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \\ \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)\end{array}}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s,\end{array}$(54) (53(53) $\begin{array}{rl}& \mathrm{\Omega }\left(W\right(\omega \left(u\right)\left)\right)\\ & \le \mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{u}l\left(\omega \right(s\left)\right)\mathrm{d}s\\ & +{\int }_{u_0}^u\frac{\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s\\ & \le \mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s\\ & +{\int }_{u_0}^u\frac{\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s,\end{array}$(53) ), (54(54) $\begin{array}{r}{V}_1\left(u\right)={\int }_{u_0}^u\frac{\begin{array}{c}\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \\ \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)\end{array}}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s,\end{array}$(54) ) gives (55) $\begin{array}{rl}W\left(\omega \right(u\left)\right)& \le {\mathrm{\Omega }}^{-1}\left(V_1\right(u\left)\right),\\ V_1\left(u_0\right)& =\mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s,\end{array}$(55) differentiate (54(54) $\begin{array}{r}{V}_1\left(u\right)={\int }_{u_0}^u\frac{\begin{array}{c}\lambda j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mu \\ \left(W\right(\omega \left(s\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(s\left)\right)\end{array}}{\phi \left(W\right(\omega \left(s\right)\left)\right)}\mathrm{d}s,\end{array}$(54) ) and applying (55(55) $\begin{array}{rl}W\left(\omega \right(u\left)\right)& \le {\mathrm{\Omega }}^{-1}\left(V_1\right(u\left)\right),\\ V_1\left(u_0\right)& =\mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s,\end{array}$(55) ), we observe that (56) $\begin{array}{rl}{V}_1^{\mathrm{\prime }}\left(u\right)& =\frac{\lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(W\right(\omega \left(u\right)\left)\right)W^{(\varrho +\lambda -1)}\left(\omega \right(u\left)\right)}{\phi \left(W\right(\omega \left(u\right)\left)\right)}\\ & \le \frac{\begin{array}{c}\lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right){\mathrm{\Omega }}^{-1}\\ \left(V_1^{(\varrho +\lambda -1)}\right(u\left)\right)\end{array}}{\phi \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right)},\end{array}$(56) implies (57) $\begin{array}{r}\frac{\phi \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right)V_1^{\mathrm{\prime }}\left(u\right)}{\mu \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right){\mathrm{\Omega }}^{-1}\left(V_1^{(\varrho +\lambda -1)}\right(u\left)\right)}\le \lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right),\end{array}$(57) inequality (57(57) $\begin{array}{r}\frac{\phi \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right)V_1^{\mathrm{\prime }}\left(u\right)}{\mu \left({\mathrm{\Omega }}^{-1}\right(V_1\left(u\right)\left)\right){\mathrm{\Omega }}^{-1}\left(V_1^{(\varrho +\lambda -1)}\right(u\left)\right)}\le \lambda j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right),\end{array}$(57) ) by integrating and using (40(40) $\begin{array}{rl}\mathrm{\Lambda }\left(z\right)& ={\int }_1^z\frac{\phi \left({\mathrm{\Omega }}^{-1}\right(s\left)\right)\mathrm{d}s}{\left(\mu \right({\mathrm{\Omega }}^{-1}\left(s\right)\left)\right)\left({\mathrm{\Omega }}^{-1}\right(s){)}^{(\varrho +\lambda -1)}},;z>0,\end{array}$(40) ), (55(55) $\begin{array}{rl}W\left(\omega \right(u\left)\right)& \le {\mathrm{\Omega }}^{-1}\left(V_1\right(u\left)\right),\\ V_1\left(u_0\right)& =\mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s,\end{array}$(55) ) yields the estimate (58) $\begin{array}{rl}\mathrm{\Lambda }\left(V_1\right(u\left)\right)& \le \mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(U^{\star }\right))+{\int }_{u_0}^{U^{\star }}l(\omega \left(s\right)\left)\mathrm{d}s\right)\\ & +\lambda {\int }_{u_0}^{u}j\left(\omega \right(s\left)\right){\omega }^{\mathrm{\prime }}\left(s\right)\mathrm{d}s,\end{array}$(58) (59) $\begin{array}{rl}{V}_1\left(t\right)& \le {\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(U^{\star }\right))\\ & +{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j(s)\mathrm{d}s),\end{array}$(59) we conclude from(55(55) $\begin{array}{rl}W\left(\omega \right(u\left)\right)& \le {\mathrm{\Omega }}^{-1}\left(V_1\right(u\left)\right),\\ V_1\left(u_0\right)& =\mathrm{\Omega }\left(m^{\lambda }\right(U^{\star }\left)\right)+{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s,\end{array}$(55) ) and (59(59) $\begin{array}{rl}{V}_1\left(t\right)& \le {\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(U^{\star }\right))\\ & +{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j(s)\mathrm{d}s),\end{array}$(59) ) that (60) $\begin{array}{rl}{W}^{\varrho }\left(\omega \right(u\left)\right)& \le ({\mathrm{\Omega }}^{-1}[{\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(U^{\star }\right))\\ & +{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j(s)\mathrm{d}s)]{)}^{\varrho }\\ & =Q\left(u\right),\end{array}$(60) substitute (60(60) $\begin{array}{rl}{W}^{\varrho }\left(\omega \right(u\left)\right)& \le ({\mathrm{\Omega }}^{-1}[{\mathrm{\Lambda }}^{-1}(\mathrm{\Lambda }\left(\mathrm{\Omega }\right(m^{\lambda }\left(U^{\star }\right))\\ & +{\int }_{u_0}^{U^{\star }}l\left(\omega \right(s\left)\right)\mathrm{d}s)+\lambda {\int }_{u_0}^{\omega \left(u\right)}j(s)\mathrm{d}s)]{)}^{\varrho }\\ & =Q\left(u\right),\end{array}$(60) ) in (47(47) $\begin{array}{rl}{V}^{\mathrm{\prime }}\left(u\right)& =j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(r\right(\omega \left(u\right)\left)\right)\\ & \times (r^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(r\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)\\ & \times (V^{\lambda }\left(\omega \right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}l\left(s\right)\phi \left(V\right(s\left)\right)\mathrm{d}s{)}^{\varrho }\\ & \le j\left(\omega \right(u\left)\right){\omega }^{\mathrm{\prime }}\left(u\right)\mu \left(V\right(\omega \left(u\right)\left)\right)W^{\varrho }\left(\omega \right(u\left)\right),\end{array}$(47) ), take integral from $u_0$ to u in the resulting inequality, use (41(41) $\begin{array}{rl}\xi \left(z\right)& ={\int }_1^z\frac{\mathrm{d}s}{\mu \left(s\right)},;z>0,\end{array}$(41) ) and let $u=U^{\star }$, we obtain (61) $\begin{array}{r}V\left(U^{\star }\right)\le {\xi }^{-1}(\xi \left(m\right(U^{\star }\left)\right)+{\int }_{u_0}^{\omega \left(U^{\star }\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s),\end{array}$(61) therefore from (46(46) $\begin{array}{r}r\left(u\right)\le V\left(u\right),V\left(u_0\right)=m\left(U^{\star }\right),\end{array}$(46) ) (62) $\begin{array}{r}r\left(U^{\star }\right)\le {\xi }^{-1}(\xi \left(m\right(U^{\star }\left)\right)+{\int }_{u_0}^{\omega \left(U^{\star }\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s),\end{array}$(62) the arbitrary nature of $U^{\star }$ in (62(62) $\begin{array}{r}r\left(U^{\star }\right)\le {\xi }^{-1}(\xi \left(m\right(U^{\star }\left)\right)+{\int }_{u_0}^{\omega \left(U^{\star }\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s),\end{array}$(62) ) produces the acquired inequality in (37(37) $\begin{array}{r}r\left(u\right)\le {\xi }^{-1}(\xi \left(m\right(u\left)\right)+{\int }_{u_0}^{\omega \left(u\right)}j\left(s\right)Q\left(s\right)\mathrm{d}s),\end{array}$(37) ).

3. Application

This segment is about to indicate a prompt use of Theorem 2.3 for analysing the boundedness of the following retarded integral equation of Volterra type is indicated (63) $\begin{array}{rl}& y^3\left(u\right)-{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)(y\left(s\right)+{\int }_{u_0}^{s}z\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu {)}^{\frac{1}{2}}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)({\int }_{u_0}^{s}n\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s=c\left(u\right),\end{array}$(63)

Example 3.1

Let $y\left(u\right)$, $t\left(u\right)$, $z\left(u\right)$, $n\left(u\right)$ and $c\left(u\right)$ be continuous functions on $[0,\mathrm{\infty })$, $\omega \left(u\right)$ be as defined in (H2). If $y\left(u\right)$ satisfies (63(63) $\begin{array}{rl}& y^3\left(u\right)-{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)(y\left(s\right)+{\int }_{u_0}^{s}z\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu {)}^{\frac{1}{2}}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)({\int }_{u_0}^{s}n\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s=c\left(u\right),\end{array}$(63) ), then (64) $\begin{array}{r}|y\left(u\right)|\le [|c\left(u\right)|+\frac{K\left(u\right)\exp \left(\frac{1}{6}{\int }_{u_0}^{\omega \left(u\right)}\mathrm{\Theta }\right(s\left)\mathrm{d}s\right)}{\begin{array}{c}1-\frac{1}{6}K\left(u\right){\int }_{u_0}^{\omega \left(u\right)}N\left(s\right)\\ \exp \left(\frac{1}{6}{\int }_{u_0}^s\mathrm{\Theta }\right(\nu \left)\mathrm{d}\nu \right)\mathrm{d}s\end{array}}{]}^{\frac{1}{3}},\end{array}$(64) provided with (65) $\begin{array}{rl}K\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|[\frac{1}{3}|c\left(s\right){|}^2+\frac{7}{3}|c\left(s\right)|\\ & +\frac{1}{3}|c\left(s\right)|\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|[\frac{4}{3}||c\left(s\right)|\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)\\ & +\frac{10}{3}+\frac{4}{3}\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|z\left(s\right)|[\frac{2}{3}|c\left(s\right)|\left({\int }_{u_0}^s|n\left(\nu \right)|c\left(\nu \right)|\mathrm{d}\nu \right)\\ & +\frac{8}{3}|c\left(s\right)|\left({\int }_{u_0}^s|n\left(\nu \right)\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|z\left(s\right)|\left[\frac{8}{3}\left({\int }_{u_0}^s|n\left(\nu \right)|\mathrm{d}\nu \right)\right]\mathrm{d}s,\end{array}$(65) (66) $\begin{array}{rl}\mathrm{\Theta }\left(u\right)& =|t\left(u\right)|[\frac{2}{3}|c\left(s\right)|+\frac{7}{3}+|c\left(s\right)|\\ & \times ({\int }_{u_0}^u\left(\frac{2}{3}|z\left(u\right)|+\frac{4}{3}|n\left(u\right)|\right)\mathrm{d}s)\\ & +{\int }_{u_0}^u\left(\frac{4}{3}|z\left(u\right)|+\frac{8}{3}|n\left(u\right)|\right)\mathrm{d}s],\end{array}$(66) (67) $\begin{array}{rl}N\left(u\right)& =|t\left(u\right)|\left[\frac{1}{3}+{\int }_{u_0}^u\left(\frac{1}{3}|z\left(u\right)|+\frac{2}{3}|n\left(u\right)|\right)\mathrm{d}s\right].\end{array}$(67)

Proof.

We note that Equation (63(63) $\begin{array}{rl}& y^3\left(u\right)-{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)(y\left(s\right)+{\int }_{u_0}^{s}z\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu {)}^{\frac{1}{2}}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}t\left(s\right)y\left(s\right)({\int }_{u_0}^{s}n\left(\nu \right)y\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s=c\left(u\right),\end{array}$(63) ) is transformed into (68) $\begin{array}{rl}|y\left(u\right){|}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)||y\left(s\right)|\\ & \times (|y\left(s\right)|+{\int }_{u_0}^s|z\left(\nu \right)||y\left(\nu \right)|\mathrm{d}\nu {)}^{1/2}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)||y\left(s\right)|({\int }_{u_0}^s|n\left(\nu \right)||y\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s,\end{array}$(68) Suppose that $|y\left(u\right)|=r\left(u\right)$, hence (68(68) $\begin{array}{rl}|y\left(u\right){|}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)||y\left(s\right)|\\ & \times (|y\left(s\right)|+{\int }_{u_0}^s|z\left(\nu \right)||y\left(\nu \right)|\mathrm{d}\nu {)}^{1/2}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)||y\left(s\right)|({\int }_{u_0}^s|n\left(\nu \right)||y\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s,\end{array}$(68) ) takes the form (69) $\begin{array}{rl}r(u{)}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)\\ & \times (r\left(s\right)+{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu {)}^{1/2}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s.\end{array}$(69) Using Lemma 2.1 on the right hand side of the inequality (69(69) $\begin{array}{rl}r(u{)}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)\\ & \times (r\left(s\right)+{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu {)}^{1/2}\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s.\end{array}$(69) ), we get (70) $\begin{array}{rl}r(u{)}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|g\left(s\right)|r\left(s\right)\\ & \times (\frac{1}{2}r\left(s\right)+\frac{1}{2}{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu +\frac{1}{2})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s,\end{array}$(70) put (71) $\begin{array}{rl}{r}_1\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)\\ & \times (\frac{1}{2}r\left(s\right)+\frac{1}{2}{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu +\frac{1}{2})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s,\end{array}$(71) from (70(70) $\begin{array}{rl}r(u{)}^3& \le |c\left(u\right)|+{\int }_{u_0}^{\omega \left(u\right)}|g\left(s\right)|r\left(s\right)\\ & \times (\frac{1}{2}r\left(s\right)+\frac{1}{2}{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu +\frac{1}{2})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s,\end{array}$(70) ), (71(71) $\begin{array}{rl}{r}_1\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)\\ & \times (\frac{1}{2}r\left(s\right)+\frac{1}{2}{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu +\frac{1}{2})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s,\end{array}$(71) ) and Lemma 2.1, we conclude (72) $\begin{array}{r}r\left(u\right)\le \left[|c\right(u)|+r_1(u){|}^{\frac{1}{3}}\le \frac{1}{3}[|c\left(u\right)|+r_1\left(u\right)]+\frac{2}{3},\end{array}$(72) substitute (72(72) $\begin{array}{r}r\left(u\right)\le \left[|c\right(u)|+r_1(u){|}^{\frac{1}{3}}\le \frac{1}{3}[|c\left(u\right)|+r_1\left(u\right)]+\frac{2}{3},\end{array}$(72) ) in (71(71) $\begin{array}{rl}{r}_1\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)\\ & \times (\frac{1}{2}r\left(s\right)+\frac{1}{2}{\int }_{u_0}^s|z\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu +\frac{1}{2})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|r\left(s\right)({\int }_{u_0}^s|n\left(\nu \right)|r\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s,\end{array}$(71) ), we gather (73) $\begin{array}{rl}{r}_1\left(u\right)& \le {\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left[\frac{1}{6}\left(|c\right(s)|+r_1(s\left)\right)+\frac{1}{3}\right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left[\frac{1}{6}{\int }_{u_0}^s|z\left(\nu \right)|\left(|c\right(\nu )|+r_1(\nu \left)\right)\mathrm{d}\nu \right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left(\frac{1}{3}{\int }_{u_0}^s|z\left(\nu \right)|\mathrm{d}\nu \right)+\frac{1}{9}\left(|c\right(s)|+r_1(s\left)\right)+\frac{1}{3})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{6}{\int }_{u_0}^s|z\left(\nu \right)|\left(|c\right(\nu )|\\ & +r_1\left(\nu \right))\mathrm{d}\nu +\frac{1}{3}{\int }_{u_0}^s|z(\nu )|\mathrm{d}\nu +\frac{1}{6}{r}_1(s))\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)||c\left(s\right)|\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|\\ & \times (\left(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\right)\left[\frac{2}{3}{\int }_{u_0}^s|n\left(\nu \right)|\mathrm{d}\nu \right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\left(\frac{1}{3}+\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\right)\\ & \times [\frac{1}{3}{\int }_{u_0}^s|n\left(\nu \right)|\left(|c\right(\nu )|+r_1(\nu \left)\right)\mathrm{d}\nu )\mathrm{d}s\end{array}$(73) $\begin{array}{rl}& +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{2}{3}[\frac{1}{3}{\int }_{u_0}^s|n\left(\nu \right)|\left(|c\right(\nu )|\\ & +r_1\left(\nu \right))\mathrm{d}\nu +\frac{2}{3}{\int }_{u_0}^s|n(\nu )\mathrm{d}\nu ])\mathrm{d}s\\ & \le {\int }_{u_0}^{\omega \left(u\right)}\frac{1}{18}|t\left(s\right)||c\left(s\right)\\ & \times |(|c\left(s\right)|+r_1\left(s\right)+2+{\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{18}|t\left(s\right)||c\left(s\right)|\\ & \times ({\int }_{u_0}^s|z\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu +2{\int }_{u_0}^s|z\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{18}|t\left(s\right)|r_1\left(s\right)\\ & \times (|c\left(s\right)|+r_1\left(s\right)+2+{\int }_{u_0}^s|z\left(\nu \right)||c(\nu |\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{18}|t\left(s\right)|r_1\left(s\right)\\ & \times ({\int }_{u_0}^s|z\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu +2{\int }_{u_0}^s|z\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{9}|t\left(s\right)|(|c\left(s\right)|+r_1\left(s\right)+2\\ & +{\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu +{\int }_{u_0}^s|z\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s\end{array}$ $\begin{array}{rl}& +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{9}|t\left(s\right)|(2{\int }_{u_0}^s|z\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|(|c\left(s\right)|+r_1\left(s\right)+2)\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{9}|t\left(s\right)||c\left(s\right)|({\int }_{u_0}^s|n\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \\ & +{\int }_{u_0}^s|n\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|(2|c\left(s\right)|{\int }_{u_0}^s|n\left(\nu \right)\mathrm{d}\nu \\ & +r_1\left(s\right){\int }_{u_0}^s|n\left(\nu \right)|c\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|(r_1\left(s\right){\int }_{u_0}^s|n\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu \\ & +2{\int }_{u_0}^s|n\left(\nu \right)|c\left(\nu \right)|\mathrm{d}\nu )\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|(2r_1\left(s\right){\int }_{u_0}^s|n\left(\nu \right)\mathrm{d}\nu \\ & +2{\int }_{u_0}^s|n\left(\nu \right)r_1\left(\nu \right)\mathrm{d}\nu +4{\int }_{u_0}^s|n\left(\nu \right)\mathrm{d}\nu )\mathrm{d}s\\ & \le K\left(u\right)+\frac{1}{6}{\int }_{u_0}^{\omega \left(u\right)}\mathrm{\Theta }\left(s\right)r_1\left(s\right)\mathrm{d}s\\ & +\frac{1}{6}{\int }_{u_0}^{\omega \left(u\right)}N\left(s\right)r_1^2\left(s\right)\mathrm{d}s.\end{array}$where $K\left(u\right)$, $\mathrm{\Theta }\left(u\right)$, $N\left(u\right)$ are as mentioned in (65(65) $\begin{array}{rl}K\left(u\right)& ={\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|[\frac{1}{3}|c\left(s\right){|}^2+\frac{7}{3}|c\left(s\right)|\\ & +\frac{1}{3}|c\left(s\right)|\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)|[\frac{4}{3}||c\left(s\right)|\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)\\ & +\frac{10}{3}+\frac{4}{3}\left({\int }_{u_0}^s|z\left(\nu \right)||c\left(\nu \right)|\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|z\left(s\right)|[\frac{2}{3}|c\left(s\right)|\left({\int }_{u_0}^s|n\left(\nu \right)|c\left(\nu \right)|\mathrm{d}\nu \right)\\ & +\frac{8}{3}|c\left(s\right)|\left({\int }_{u_0}^s|n\left(\nu \right)\mathrm{d}\nu \right)]\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|z\left(s\right)|\left[\frac{8}{3}\left({\int }_{u_0}^s|n\left(\nu \right)|\mathrm{d}\nu \right)\right]\mathrm{d}s,\end{array}$(65) ), (66(66) $\begin{array}{rl}\mathrm{\Theta }\left(u\right)& =|t\left(u\right)|[\frac{2}{3}|c\left(s\right)|+\frac{7}{3}+|c\left(s\right)|\\ & \times ({\int }_{u_0}^u\left(\frac{2}{3}|z\left(u\right)|+\frac{4}{3}|n\left(u\right)|\right)\mathrm{d}s)\\ & +{\int }_{u_0}^u\left(\frac{4}{3}|z\left(u\right)|+\frac{8}{3}|n\left(u\right)|\right)\mathrm{d}s],\end{array}$(66) ) and (67(67) $\begin{array}{rl}N\left(u\right)& =|t\left(u\right)|\left[\frac{1}{3}+{\int }_{u_0}^u\left(\frac{1}{3}|z\left(u\right)|+\frac{2}{3}|n\left(u\right)|\right)\mathrm{d}s\right].\end{array}$(67) ), respectively. The required bound (64(64) $\begin{array}{r}|y\left(u\right)|\le [|c\left(u\right)|+\frac{K\left(u\right)\exp \left(\frac{1}{6}{\int }_{u_0}^{\omega \left(u\right)}\mathrm{\Theta }\right(s\left)\mathrm{d}s\right)}{\begin{array}{c}1-\frac{1}{6}K\left(u\right){\int }_{u_0}^{\omega \left(u\right)}N\left(s\right)\\ \exp \left(\frac{1}{6}{\int }_{u_0}^s\mathrm{\Theta }\right(\nu \left)\mathrm{d}\nu \right)\mathrm{d}s\end{array}}{]}^{\frac{1}{3}},\end{array}$(64) ) can be attained by the suitable application of Theorem 2.3 in (73(73) $\begin{array}{rl}{r}_1\left(u\right)& \le {\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left[\frac{1}{6}\left(|c\right(s)|+r_1(s\left)\right)+\frac{1}{3}\right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left[\frac{1}{6}{\int }_{u_0}^s|z\left(\nu \right)|\left(|c\right(\nu )|+r_1(\nu \left)\right)\mathrm{d}\nu \right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\\ & \times \left(\frac{1}{3}{\int }_{u_0}^s|z\left(\nu \right)|\mathrm{d}\nu \right)+\frac{1}{9}\left(|c\right(s)|+r_1(s\left)\right)+\frac{1}{3})\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\frac{1}{6}{\int }_{u_0}^s|z\left(\nu \right)|\left(|c\right(\nu )|\\ & +r_1\left(\nu \right))\mathrm{d}\nu +\frac{1}{3}{\int }_{u_0}^s|z(\nu )|\mathrm{d}\nu +\frac{1}{6}{r}_1(s))\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}\frac{1}{6}|t\left(s\right)||c\left(s\right)|\mathrm{d}s+{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|\\ & \times (\left(\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\right)\left[\frac{2}{3}{\int }_{u_0}^s|n\left(\nu \right)|\mathrm{d}\nu \right])\mathrm{d}s\\ & +{\int }_{u_0}^{\omega \left(u\right)}|t\left(s\right)|(\left(\frac{1}{3}+\frac{1}{3}\left[|c\right(s)|+r_1(s\left)\right]\right)\\ & \times [\frac{1}{3}{\int }_{u_0}^s|n\left(\nu \right)|\left(|c\right(\nu )|+r_1(\nu \left)\right)\mathrm{d}\nu )\mathrm{d}s\end{array}$(73) ) with some alterations, therefore we omit the details.

4. Conclusion

The intent of this paper is to formulate some new nonlinear Gronwall–Bellman–Pachpatte type of inequalities of one independent variable. Unlike some former results in the literature, the integral inequalities taken into consideration right here include the retarded term, which brings about challenges in the estimation of unknown functions on the upper bounds. It can be accompanied that the received inequalities generalize and expand some current consequences and permit us to contemplate the stability, boundedness and asymptotic behaviour of the solutions of a class of more general retarded nonlinear differential, integral and integro-differential equations. Using our method, the integral inequality can be further studied with more dimensions.

Acknowledgements

Editor comments and those of the reviewers were highly insightful and enabled us to greatly improve the quality of our manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.