On new fixed point theorems for three types of (α, β)-(ψ, θ, ϕ)-multivalued contractive mappings in metric spaces

Abstract

Our goal in this paper is to present three classes of cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mappings, and then prove the existence of fixed point in a complete metric space. Our results generalize and extend several works. Two examples were added to support our work.

Keywords:

• cyclic (α, β)-admissible mapping
• cyclic (α, β)-(ψ, θ, ϕ)-admissible multivalued mapping
• cyclic (α, β)-(ψ, θ, ϕ)-admissible Meir-Keeler-Khan multivalued mapping
• fixed point
• metric spaces

Public Interest Statement

The study of fixed points for multivalued contraction maps using the Hausdorff metric was initiated by Nadler that extended the Banach contraction principle to set-valued mappings. After that several authors have studied fixed points for set-valued maps. The theory of set-valued maps has many applications in control theory, convex optimization, differential equations and economics. In this paper, we present three contractive conditions and proved the existence of fixed point theorems in complete metric space. Two examples were added to support our work.

1. Introduction and preliminaries

One of the most important tools in fixed point theory is Banach contraction principle. A lot of authors have extended or generalized this contraction and proved the existence of fixed and common fixed point theorems (see Aydi, Postolache, & Shatanawi, 2012; Aydi, Shatanawi, & Vetro, 2011; Khan, 1976; Samet, Vetro, & Vetro, 2012; Shatanawi & Postolache, 2013 and references therein). The theory of multivalued mapping has an important tool in various fields of mathematics, for example, control and optimal theory as well as real and complex analysis. Several authors have also extended and presented many forms of multivalued mapping conditions on self-mappings as a generalization of Banach contraction principle using the concept of Hausdorff metric and proved the existence of fixed and common fixed point theorems in metric space and other spaces. We recite some notations, needed definitions and elementary results, for the purpose of the sequel, and $\mathbb{N}$ and $\mathbb{R}$ the sets of positive integers and real numbers respectively. Let (Y, d) be a metric space, we denote CL(Y) the family of closed subsets of Y,  by CB(Y) the class of all nonempty closed bounded subsets of Y. and F(f) is the set of all fixed points of f. For $B,C\in CL(Y),$ let the $H:CL(Y)\times CL(Y)\to {\mathbb{R}}_{+}\cup \{\infty \}$ be defined by$$\begin{array}{c}\hfill H(B,C)=\left\{\begin{array}{cc}max\{{sup}_{b\in B}d(b,C),{sup}_{c\in C}d(c,B)\},\hfill & \text{if the maximum exists}\hfill \\ \infty ,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

such map H is called the generalized Hausdorff metric induced by the metric d.

Let $\mathrm{\Phi }$ be a set of functions, $\mathit{\varphi }:[0,+\infty )\to [0,+\infty )$ is lower semi-continuous with $\mathit{\varphi }(k)=0\iff k=0$ and let $\mathrm{\Psi }$ be a set of a family of nondecreasing functions, $\mathit{\psi }:[0,+\infty )\to [0,+\infty )$ such that${\sum }_{n=1}^{\infty }{\mathit{\psi }}^n(k)<\infty$ for each $k>0,$ where ${\mathit{\psi }}^n$ is the nth iterate of $\mathit{\psi }$. It is known that for each $\mathit{\psi }\in \mathrm{\Psi }$, we have $\mathit{\psi }(k)<k$ for each $k>0$ and $\mathit{\psi }(k)=0$ for $k=0$. More details about $\mathit{\psi }$ function can be found in e.g (Karapinar & Sametm, 2012; Samet et al., 2012)

Samet et al. (2012) presented the concept $\mathit{\alpha }$-admissible mappings as the following:

Definition 1.1

Let $f:Y\to Y$ and $\mathit{\alpha }:Y\times Y\to [0,\infty ).$f is called $\mathit{\alpha }$-admissible when if $y,z\in Y$ such that $\mathit{\alpha }(y,z)\ge 1$ then we have $\mathit{\alpha }(fy,fz)\ge 1$.

After that, many authors used the concept of $\mathit{\alpha }$-admissible contractive-type mappings to study the existence of fixed point in many spaces (see Hussain, Ahmad, & Azam, 2014; Hussain, Parvaneh, & Hoseini Ghoncheh, 2014; Jleli, Samet, Vetro, & Vetro, 2015 and references cited therein).

Alizadeh, Moradlou, and Salimi (2014) introduced the concept of cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping as the following:

Definition 1.2

Let $g:Y\to CL(Y)$ be and $\mathit{\alpha },\mathit{\beta }:Y\to {\mathbb{R}}^{+}$ be two functions. f is said to be a cyclic $(\mathit{\alpha },\mathit{\beta })-$admissible mapping if

(1)	$\mathit{\alpha }(y)\ge 1$ for some $y\in Y$$\Rightarrow$$\mathit{\beta }(fy)\ge 1,$
(2)	$\mathit{\beta }(y)\ge 1$ for some $y\in Y$$\Rightarrow$$\mathit{\alpha }(fy)\ge 1.$

Example 1.3

(Alizadeh et al., 2014)   Let $f:\mathbb{R}\to \mathbb{R}$ by defined by $fy=-(y+y^3).$ Assume that $\mathit{\alpha },\mathit{\beta }:\mathbb{R}\to {\mathbb{R}}^{+}$ are given by $\mathit{\alpha }(y)=e^y$ for all $y\in \mathbb{R}$ and $\mathit{\beta }(z)=e^{-z}$ for all $z\in \mathbb{R}.$ Then f is a cyclic $(\mathit{\alpha },\mathit{\beta })-$ admissible mapping. Indeed, if $\mathit{\alpha }(y)=e^y\ge 1,$ then $y\ge 0$ which implies $-f(y)\ge 0.$ Therefore, $\mathit{\beta }(fy)=e^{-fy}\ge 1.$ Also, if $\mathit{\beta }(z)=e^{-z}\ge 1,$ then $z\le 0$ which implies $f(z)\ge 0.$ So, $\mathit{\alpha }(fz)=e^{fz}\ge 1.$

Concerning the class of cyclic $(\mathit{\alpha },\mathit{\beta })-$ admissible mapping, Alizadeh et al. (2014) presented the notation of $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\varphi })$-contractive mapping as following:

Definition 1.4

Let (Y, d) be a complete metric space and $f:Y\to Y$ be a cyclic $(\mathit{\alpha },\mathit{\beta })$- admissible mapping. f is called a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\varphi })$-contractive mapping if(1.1) $$\begin{array}{c}\hfill y,z\in Y,\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1\Rightarrow \mathit{\psi }(d(fy,fz))\le \mathit{\psi }(d(y,z))-\mathit{\varphi }(d(y,z)),\end{array}$$(1.1)

for $y,z\in Y,$ where $\mathit{\psi }:[0,+\infty )\to [0,+\infty )$ is continuous and increasing function and $\mathit{\varphi }:[0,+\infty )\to [0,+\infty )$ is a lower semi-continuous function with $\mathit{\varphi }(k)=0\Rightarrow k=0$.

Alizadeh et al. (2014) proved the existence of fixed point results for such mappings in complete metric spaces

Theorem 1.5

(Alizadeh et al., 2014)   Let (Y, d) be a complete metric space and $f:Y\to Y$ be a $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\varphi })$-admissible mapping. Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ such that $\mathit{\alpha }(y_0)\ge 1\text{and}\mathit{\beta }(y_0)\ge 1$
(2)	f is continuous, or
(3)	if $\{y_n\}$ is a sequence in Y such that $y_n\to y$ and $\mathit{\beta }(y_n)\ge 1$ for all $n\in \mathbb{N},$ then $\mathit{\beta }(y)\ge 1;$

then f has a fixed point. Moreover, if $\mathit{\alpha }(y)\ge 1$ and $\mathit{\beta }(z)\ge 1$ for all $y\in F(f),$ then f has a unique fixed point.

Ali, Kamran, and Karapinar (2014) presented the family $\mathrm{\Theta }$ of functions $\mathit{\theta }:[0,\infty )\to [0,\infty )$ satisfying the following conditions:

(1)	$\mathit{\theta }$ is continuous
(2)	$\mathit{\theta }$ is nondecreasing on ${\mathbb{R}}^{+}$
(3)	$\mathit{\theta }(0)=0$ and $\mathit{\theta }(k)>0$ for all $k\in (0,\infty )$
(4)	$\mathit{\theta }$ is subadditive

Lemma 1.6

(Ali et al., 2014)   Let (Y, d) is a metric space and $\mathit{\theta }\in \mathrm{\Theta }$. Then $(Y,\mathit{\theta }\circ d)$ is a metric space.

Lemma 1.7

(Ali et al., 2014)   Let (Y, d) be a metric space, let $\mathit{\theta }\in \mathrm{\Theta }$ and $C\in CL(Y)$. Suppose that there exists $y\in Y$ such that $\mathit{\theta }(d(y,C))>0.$ Then there exists $z\in C$ such that$$\mathit{\theta }(d(y,z))<\mathit{\tau }\mathit{\theta }(d(y,C)),$$

where $\mathit{\tau }>1.$

Meir and Keeler (1969) established a fixed point theorem on a metric space (Y, d). They studied the class of mappings satisfying the condition for each $\mathit{ϵ}>0$, there exists $\mathit{\delta }(\mathit{ϵ})>0$ such that$$\mathit{ϵ}\le d(y,z)<\mathit{ϵ}+\mathit{\delta }(\mathit{ϵ})\Rightarrow d(fy,fz)<\mathit{ϵ}$$

Khan (1976) proved the existence of fixed point in setting of metric space. Latif, Gordji, Karapinar, and Sintunavarat (2014) presented the concept $(\mathit{\alpha },\mathit{\psi })$-Meir-Keeler mapping. Lately, Redjel, Dehici, Karapinar, and Erhan (2015) presented the concept of $(\mathit{\alpha },\mathit{\psi })$-Meir-Keeler- Khan mappings as follows:

Definition 1.8

Let $f:Y\to Y$ be a mapping on a metric space (Y, d). f is said to be an $(\mathit{\alpha },\mathit{\psi })$-Meir-Keeler-Khan mapping if there exist $\mathit{\psi }\in \mathrm{\Psi }$ and $\mathit{\alpha }:Y\times Y\to [0,\infty )$ such that for every $\mathit{ϵ}>0$, there exists $\mathit{\delta }(\mathit{ϵ})$ such that if$$\mathit{ϵ}\le \frac{d(y,fy)d(y,fz)+d(z,fz)d(z,fy)}{d(y,fz)+d(z,fy)}<\mathit{ϵ}+\mathit{\delta }(\mathit{ϵ})$$

for any $y,z\in Y$ and $y\ne z\Rightarrow d(y,fz)+d(z,fy)\ne 0$, then$$\mathit{\alpha }(y,z)d(fy,fz)<\mathit{ϵ}.$$

It is easy to show that f is $(\mathit{\alpha },\mathit{\psi })$-Meir-Keeler-Khan mapping, then$$\begin{array}{c}\hfill \mathit{\alpha }(y,z)d(fy,fz)\le \mathit{\psi }(N(y,z)),\end{array}$$

where$$\begin{array}{c}\hfill N(y,z)=\left\{\frac{d(y,fy)d(y,fz)+d(z,fz)d(z,fy)}{d(y,fz)+d(z,fy)}\right\}\end{array}$$

for all $y,z\in Y.$

In this article, motivated and inspired by Alizadeh et al. (2014), Redjel et al. (2015), we present the class of cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$- admissible multivalued mapping and cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible Meir-Keeler-Khan multivalued mapping. After that, we establish the existence of fixed point for this class of mappings in metric space. Our work generalizes and extends some theorems in the literature. Two examples are given to support the obtained results.

2. Main result

Now, we present the class of a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$- admissible multivalued mapping and prove some fixed point theorems on complete metric space.

Definition 2.1

Let (Y, d) be a metric space and $f:Y\to CL(Y)$ be a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping. We say that f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mapping of type A if there exists $\mathit{\alpha },\mathit{\beta }::Y\times Y\to [0,+\infty )$, $\mathit{\psi }\in \mathrm{\Psi }$, $\mathit{\theta }\in \mathrm{\Theta }$ and $\mathit{\varphi }\in \mathrm{\Phi }$ such that:(2.1) $$\begin{array}{c}\hfill y,z\in Y,\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1\Rightarrow \mathit{\theta }(H(fy,fz))\le \mathit{\psi }(\mathit{\theta }(M(y,z)))-\mathit{\varphi }(M(y,z)),\end{array}$$(2.1)

where$$\begin{array}{c}\hfill M(y,z)=max\left\{d(y,z),d(y,fy),d(z,fz),\frac{1}{2}[d(y,fz)+d(z,fy)]\right\}\end{array}$$

for all $y,z\in Y$.

Definition 2.2

Let (Y, d) be a metric space. The mapping $f:Y\to CL(Y)$ is called a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mapping of type B if there exists $\mathit{\alpha },\mathit{\beta }:Y\times Y\to [0,+\infty )$, $\mathit{\psi }\in \mathrm{\Psi }$, $\mathit{\theta }\in \mathrm{\Theta }$ and $\mathit{\varphi }\in \mathrm{\Phi }$ such that:(2.2) $$\begin{array}{c}\hfill y,z\in Y,\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1\Rightarrow \mathit{\theta }(H(fy,fz))\le \mathit{\psi }(\mathit{\theta }(P(y,z)))-\mathit{\varphi }(P(y,z)),\end{array}$$(2.2)

where$$\begin{array}{c}\hfill P(y,z)=max\left\{d(y,z),\frac{[1+d(y,fy)]d(z,fz)}{d(y,z)+1}\right\}\end{array}$$

for all $y,z\in Y.$

Theorem 2.3

Let (Y, d) be a complete metric space and $f:Y\to CL$ be cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mapping of type $\mathit{A}$. Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ and $y_1\in f{y}_0$ such that $$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$
(2)	if $\{y_n\}$ is a sequence in Y with $y_n\to y$ as $n\to \infty$ and $\mathit{\beta }(y_n)\ge 1$ for all $n\in \mathbb{N},$ then $\mathit{\beta }(y)\ge 1,$

then f has a fixed point.

Proof

By starting from $y_0$ and $y_1\in f{y}_0$ in conditions (1), we have$$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(y_1)=\mathit{\beta }(f{y}_0)\ge 1,\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(y_1)=\mathit{\alpha }(f{y}_0)\ge 1.\hfill \end{array}$$

Therefore, $\mathit{\alpha }(y_0)\ge 1$ and $\mathit{\beta }(y_1)\ge 1$, equivalently, $\mathit{\alpha }(y_0)\mathit{\beta }(y_1)\ge 1.$ If $y_0=y_1$, we derive that $y_1\in F(f)$ and so the proof is done. Now, we assume that $y_0\ne y_1$ and $y_1\notin f{y}_1$. From (2.1), we have(2.3) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))& \le \mathit{\theta }(H(f{y}_0,f{y}_1))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(M(y_0,y_1)))-\mathit{\varphi }(M(y_0,y_1)),\hfill \end{array}$$(2.3)

where(2.4) $$\begin{array}{cc}\hfill M(y_0,y_1)& =max\left\{d(y_0,y_1),d(y_0,f{y}_0),d(y_1,f{y}_1),\frac{1}{2}[d(y_0,f{y}_1)+d(y_1,f{y}_0)]\right\}\hfill \\ \hfill & =max\left\{d(y_0,y_1),d(y_0,y_1),d(y_1,f{y}_1),\frac{1}{2}d(y_0,f{y}_1)\right\}\hfill \\ \hfill & =max\left\{d(y_0,y_1),d(y_1,f{y}_1),\frac{1}{2}[d(y_0,y_1)+d(y_1,f{y}_1)]\right\}\hfill \\ \hfill & =max\left\{d(y_0,y_1),d(y_1,f{y}_1)\right\},\hfill \end{array}$$(2.4)

from (2.3) and (2.4) and by using the properties of $\mathit{\varphi }$, we get(2.5) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))\le \mathit{\psi }(\mathit{\theta }(max\{d(y_0,y_1),d(y_1,f{y}_1)\}))-\mathit{\varphi }(max\{d(y_0,y_1),d(y_1,f{y}_1)\}),\end{array}$$(2.5)

assume that $max\{d(y_0,y_1),d(y_1,f{y}_1)\}=d(y_1,f{y}_1),$ then we obtain$$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))& \le \mathit{\psi }(\mathit{\theta }(d(y_1,f{y}_1)))-\mathit{\varphi }(d(y_1,f{y}_1))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_1,f{y}_1))),\hfill \end{array}$$

which is a contradiction. Thus $max\{d(y_0,y_1),d(y_1,f{y}_1)\}=d(y_0,y_1).$ From (2.5), we obtain(2.6) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))& \le \mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))-\mathit{\varphi }(d(y_0,y_1))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))).\hfill \end{array}$$(2.6)

For $\mathit{\tau }>1$ by Lemma 1.7, there exists $y_2\in f{y}_1$ such that(2.7) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,y_2))<\mathit{\tau }\mathit{\theta }(d(y_1,f{y}_1)).\end{array}$$(2.7)

From (2.6) and (2.7), we get(2.8) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,y_2))<\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))).\end{array}$$(2.8)

By applying $\mathit{\psi }$ in (2.8), we have(2.9) $$\begin{array}{c}\hfill 0<\mathit{\psi }(\mathit{\theta }(d(y_1,y_2)))<\mathit{\psi }(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))).\end{array}$$(2.9)

Set ${\mathit{\tau }}_1=\frac{\mathit{\psi }(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))))}{\mathit{\psi }(\mathit{\theta }(d(y_1,y_2)))}$. Then ${\mathit{\tau }}_1\ge 1$. Since f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping, from condition (1) and $y_2\in f{y}_1$, we have$$\begin{array}{cc}\hfill \mathit{\alpha }(y_1)\ge 1& \Rightarrow \mathit{\beta }(y_2)=\mathit{\beta }(f{y}_1)\ge 1,\hfill \\ \hfill \mathit{\beta }(y_1)\ge 1& \Rightarrow \mathit{\alpha }(y_2)=\mathit{\alpha }(f{y}_1)\ge 1.\hfill \end{array}$$

So, $\mathit{\alpha }(y_1)\ge 1$ and $\mathit{\beta }(y_2)\ge 1.$ Equivalently, $\mathit{\alpha }(y_1)\mathit{\beta }(y_2)\ge 1.$ If $y_2\in f{y}_2,$ then $y_2\in F(f).$ So, we assume that $y_2\notin f{y}_2$. From (2.1), we deduce(2.10) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_2,f{y}_2))& \le \mathit{\theta }(H(f{y}_1,f{y}_2))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(M(y_1,y_2)))-\mathit{\varphi }(M(y_1,y_2)),\hfill \end{array}$$(2.10)

where$$\begin{array}{cc}\hfill M(y_1,y_2)& =max\left\{d(y_1,y_2),d(y_1,f{y}_1),d(y_2,f{y}_2),\frac{1}{2}[d(y_1,f{y}_2)+d(y_2,f{y}_1)]\right\}\hfill \\ \hfill & =max\left\{d(y_1,y_2),d(y_1,y_2),d(y_2,f{y}_2),\frac{1}{2}d(y_1,f{y}_2)\right\}\hfill \\ \hfill & =max\left\{d(y_1,y_2),d(y_2,f{y}_2),\frac{1}{2}[d(y_1,y_2)+d(y_2,f{y}_2)]\right\}\hfill \\ \hfill & =max\{d(y_1,y_2),d(y_2,f{y}_2)\}.\hfill \end{array}$$

If $M(y_1,y_2)=d(y_2,f{y}_2)$ and by using properties of $\mathit{\varphi }$, we have:$$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_2,f{y}_2))& \le \mathit{\psi }(\mathit{\theta }(d(y_2,f{y}_2)))-\mathit{\varphi }(d(y_2,f{y}_2))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_2,f{y}_2))),\hfill \end{array}$$

which is a contradiction. Thus, if $M(y_1,y_2)=d(y_1,y_2)$, we get(2.11) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_2,f{y}_2))& \le \mathit{\theta }(H(f{y}_1,f{y}_2))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(d(y_2,y_2)))-\mathit{\varphi }(d(y_1,y_2))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_1,y_2))).\hfill \end{array}$$(2.11)

For ${\mathit{\tau }}_1>1$ by Lemma 1.7, then there exists $y_3\in f{y}_2$ such that(2.12) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_2,y_3))<{\mathit{\tau }}_1\mathit{\theta }(d(y_2,f{y}_2)).\end{array}$$(2.12)

From (2.11)and (2.12), we obtain(2.13) $$\begin{array}{c}\hfill 0<d(y_2,y_3))<{\mathit{\tau }}_1\mathit{\psi }(\mathit{\theta }(d(y_2,f{y}_2)))=\mathit{\psi }(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))).\end{array}$$(2.13)

By applying $\mathit{\psi }$ in (2.13), we have(2.14) $$\begin{array}{c}\hfill 0<\mathit{\psi }(\mathit{\theta }(d(y_2,y_3)))<{\mathit{\psi }}^2(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))).\end{array}$$(2.14)

By continuing this procedure, we construct the sequence$\{y_n\}$ in Y such that $y_{n+1}\ne y_n\in f{y}_n$, again, since f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping, we have$$\begin{array}{c}\hfill \mathit{\alpha }(y_n)\ge 1\text{and}\mathit{\beta }(y_n)\ge 1\end{array}$$

for all $n\in \mathbb{N}$. This implies that$$\begin{array}{c}\hfill \mathit{\alpha }(y_n)\mathit{\beta }(y_{n+1})\ge 1,\end{array}$$

and$$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_n,y_{n+1})){\mathit{\psi }}^n(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))),\end{array}$$

for all $\mathbb{N}\cup \{0\}$.

Let $m,n\in \mathbb{N}$ such that $m>n$. By the triangle inequality, we get$$\begin{array}{c}\hfill \mathit{\theta }(d(y_m,y_n))\le \sum _{l=n}^{m-1}\mathit{\theta }(d(y_l,y_{l+1}))\le \sum _{l=n}^{m-1}{\mathit{\psi }}^{l-1}(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))).\end{array}$$

From the $\mathit{\psi }$ properties, this implies that ${lim}_{n,m\to \infty }\mathit{\theta }(d(y_m,y_n))=0$ and by the continuity of $\mathit{\theta }$, we have $li{m}_{n,m\to \infty }d(y_m,y_n)=0$. Thus $\{y_n\}$ is Cauchy sequence in (Y, d) such that $y_n\to y$ as $n\to \infty$ for all $n\in \mathbb{N}$. For all $n\in \mathbb{N}$, we suppose that condition (2) hold. Hence $\mathit{\alpha }(y_n)\mathit{\beta }(z)\ge 1.$ From (2.1), we have(2.15) $$\begin{array}{c}\hfill \mathit{\theta }(H(f{y}_n,fz))\le \mathit{\psi }(\mathit{\theta }(M(y_n,z)))-\mathit{\varphi }(M(y_n,z))\end{array}$$(2.15)

for all $n\in \mathbb{N}$. where$$max\left\{d(y_n,z),d(f{y}_n,y_n),d(z,fz),\frac{1}{2}[d(y_n,fz)+d(z,f{y}_n)]\right\}.$$

Assume that $d(z,fz)\ne 0.$ Let $\mathit{ϵ}=\frac{d(z,fz)}{2}$. Since $y_n\to z$ as $n\to \infty$, we can find $t_1\in \mathbb{N}$ such that(2.16) $$\begin{array}{c}\hfill d(z,y_n)<\frac{d(z,fz)}{2},\end{array}$$(2.16)

for all $n\ge t_1.$ Also, we get(2.17) $$\begin{array}{c}\hfill d(y_n,fz)\le d(y_n,z)+d(z,fz)<\frac{d(z,fz)}{2}+d(z,fz)=\frac{3d(z,fz)}{2}\end{array}$$(2.17)

for all $n\ge t_2.$ Furthermore, we obtain(2.18) $$\begin{array}{c}\hfill d(y_n,f{y}_n)\le d(y_n,y_{n+1})<\frac{d(z,fz)}{2}\end{array}$$(2.18)

for all $n\ge t_3.$ Using (2.16)–(2.18), we have(2.19) $$\begin{array}{c}\hfill M(y_n,z)=max\left\{d(y_n,z),d(f{y}_n,y_n),d(z,fz),\frac{1}{2}[d(y_n,fz)+d(z,f{y}_n)]\right\}=d(z,fz)\end{array}$$(2.19)

for all $n\ge t=:\{t_1,t_2,t_3\}$. For $n\ge t$. From (2.15) and the triangle inequality and the properties of $\mathit{\varphi }$, we obtain$$\begin{array}{cc}\hfill \mathit{\theta }(d(z,fz))& \le \mathit{\theta }(d(z,y_{n+1}))+\mathit{\theta }(H(f{y}_n,fz))\hfill \\ \hfill & \le \mathit{\theta }(d(z,y_{n+1}))+\mathit{\psi }(\mathit{\theta }(M(y_n,z)))-\mathit{\varphi }(M(y_n,z))\hfill \\ \hfill & \le \mathit{\theta }(d(z,y_{n+1}))+\mathit{\psi }(\mathit{\theta }(d(z,fz))-\mathit{\varphi }(d(z,fz))\hfill \\ \hfill & <\mathit{\theta }(d(z,y_{n+1}))+\mathit{\psi }(\mathit{\theta }(d(z,fz)),\hfill \end{array}$$

taking $n\to \infty$ in the above inequality, we get$$\mathit{\theta }(d(z,fz))\le \mathit{\psi }(\mathit{\theta }(d(z,fz))<\mathit{\theta }(d(z,fz)),$$

which is a contradiction. Thus $d(z,fz)=0$, that is, $z\in fz.$$\square$

Corollary 2.4

Let (Y, d) be a complete metric space and $f:Y\to CL(Y).$ Assume that there exists four functions $\mathit{\alpha },\mathit{\beta }:Y\to [0,\infty ),\mathit{\psi }\in \mathrm{\Psi }$ and $\mathit{\varphi }\in \mathrm{\Phi }$ such that$$\begin{array}{c}\hfill \mathit{\alpha }(y)\mathit{\beta }(z)\mathit{\theta }(H(fy,fz))\le \mathit{\psi }(\mathit{\theta }(M(y,z)))-\mathit{\varphi }(M(y,z)),\end{array}$$

Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ and $y_1\in f{y}_0$ such that $$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$
(2)	if $\{y_n\}$ is a sequence in Y with $y_n\to y\in Y$ as $n\to \infty$ and $\mathit{\beta }(y_n)\ge 1$ for all $n\in \mathbb{N},$ then $\mathit{\beta }(y)\ge 1,$ then f has a fixed point.

Proof

Let $\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1$ for every $y,z\in Y$. Then by 2.4, we have:$$\begin{array}{cc}\hfill \mathit{\theta }(H(fy,fz))& \le \mathit{\alpha }(y)\mathit{\beta }(z)\mathit{\theta }(H(fy,fz))\hfill \\ \hfill & \le \mathit{\psi }(M(y,z))-\mathit{\varphi }(M(y,z)),\hfill \end{array}$$

this guides that f cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mapping. So, by following the proof Theorem 2.3 we obtain the desired outcome. $\square$

If we put $\mathit{\psi }(k)=\mathit{\theta }(k)=k$ and $\mathit{\varphi }(k)=(1-h)k$ in Theorem 2.3, we have the following corollary.

Corollary 2.5

Let (Y, d) be a complete metric space and $f:Y\to CL(Y).$ Assume that there exists four functions $\mathit{\alpha },\mathit{\beta }:Y\to [0,\infty )$,$\mathit{\psi }\in \mathrm{\Psi }$, $\mathit{\theta }\in \mathrm{\Theta }$ and $\mathit{\varphi }\in \mathrm{\Phi }$ such that$$\begin{array}{c}\hfill y,z\in Y,\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1\Rightarrow H(fy,fz)\le hM(y,z),\end{array}$$

where $h\in [0,1)$ Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ and $y_1\in f{y}_0$ such that $$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$
(2)	if $\{y_n\}$ is a sequence in Y with $y_n\to y\in Y$ as $n\to \infty$ and $\mathit{\beta }(y_n)\ge 1$ for all $n\in \mathbb{N},$ then $\mathit{\beta }(y)\ge 1,$

then f has a fixed point.

Theorem 2.6

Let (Y, d) be a complete metric space and $f:Y\to CL(Y)$ be a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible multivalued mapping of type B. Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ and $y_1\in f{y}_0$ such that $$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$
(2)	if $\{y_n\}$ is a sequence in Y with $y_n\to y\in Y$ as $n\to \infty$ and $\mathit{\beta }(y_n)\ge 1$ for all $n\in \mathbb{N},$ then $\mathit{\beta }(y)\ge 1,$ then f has a fixed point.

Proof

By similar method in Theorem 2.3, we start from $y_0$ and $y_1\in f{y}_0$ in condition (1), we have$$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(y_1)=\mathit{\beta }(f{y}_0)\ge 1,\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(y_1)=\mathit{\alpha }(f{y}_0)\ge 1.\hfill \end{array}$$

Therefore, $\mathit{\alpha }(y_0)\ge 1$ and $\mathit{\beta }(y_1)\ge 1$, equivalently, $\mathit{\alpha }(y_0)\mathit{\beta }(y_1)\ge 1.$ If $y_0=y_1$, we derive that $y_1\in F(f)$ and so the proof is done. Now, we assume that $y_0\ne y_1$ and $y_1\in f{y}_1$ and hence $d(y_1,f{y}_1)>0.$ From (2.1), we have(2.20) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))& \le \mathit{\theta }(H(f{y}_0,f{y}_1))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(P(y_0,y_1)))-\mathit{\varphi }(P(y_0,y_1)),\hfill \end{array}$$(2.20)

where(2.21) $$\begin{array}{cc}\hfill P(y_0,y_1)& =max\left\{d(y_0,y_1),\frac{[1+d(y_0,f{y}_0)]d(y_1,f{y}_1)}{d(y_0,y_1)+1}\right\}\hfill \\ \hfill & =max\left\{d(y_0,y_1),\frac{[1+d(y_0,y_1)]d(y_1,f{y}_1)}{d(y_0,y_1)+1}\right\}\hfill \\ \hfill & =max\left\{d(y_0,y_1),(y_1,f{y}_1)\right\}.\hfill \end{array}$$(2.21)

To continue the proof after the above step, we will follow the same process as in Theorem 2.3. $\square$

Example 2.7

Let $y=[0,\infty )$ and $d:Y\times Y\to [0,\infty )$ be defined by $d(y,z)=|y-z|$ for all $y,z\in Y.$ Define $f:Y\to Y$ and $\mathit{\alpha },\mathit{\beta }:Y\to [0,\infty )$ by$$\begin{array}{cc}\hfill fy& =\left\{\begin{array}{cc}\frac{y}{2}\hfill & \text{if}y\in [0,1],\hfill \\ 2y\hfill & \text{if}y\in (1,\infty )\hfill \end{array}\right.\hfill \\ \hfill \mathit{\alpha }(y)& =\left\{\begin{array}{cc}\frac{y+3}{2}\hfill & \text{if}y\in [0,1],\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.\hfill \\ \hfill \mathit{\beta }(y)& =\left\{\begin{array}{cc}\frac{y+5}{3}\hfill & \text{if}y\in [0,1],\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

It’s easy to see that (Y, d) is a complete metric space.

Now, we want to show that the Theorem 2.3 can guarantee the existence of fixed point of f. Firstly, we will show that f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping

For $y,z\in Y,$ we have$$\mathit{\alpha }(y)\ge 1\Rightarrow y\in [0,1]\Rightarrow \mathit{\beta }(fy)=\mathit{\beta }\left(\frac{y}{2}\right)=\frac{y+10}{6}\ge 1$$

and$$\mathit{\beta }(y)\ge 1\Rightarrow y\in [0,1]\Rightarrow \mathit{\alpha }(fy)=\mathit{\alpha }\left(\frac{y}{2}\right)=\frac{y+6}{4}\ge 1.$$

Secondly, we will prove that f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-multivalued contractive mapping. Define functions $\mathit{\psi },\mathit{\varphi }:[0,\infty )\to [0,\infty )$ by$$\mathit{\psi }(k)=\frac{5k}{3},\mathit{\theta }(k)=\frac{3k}{7}\text{and}\mathit{\varphi }(k)=k,\text{for}\text{all}k\in [0,\infty ).$$

If $\{y_n\}$ is a sequence in Y such that $\mathit{\beta }(y_n)\ge 1$ and $y_n\to y$ as $n\to \infty .$ So, $y_n\in [0,1]$. Hence, i.e. $\mathit{\beta }(y)\ge 1.$

Let $\mathit{\alpha }(y)\mathit{\beta }(y)\ge 1.$ Then $y,z\in [0,1]$ and $\mathit{\theta }(k)=k$ therefore, we have$$\begin{array}{cc}\hfill \mathit{\theta }(H(fy,fz))& =\frac{1}{2}|y-z|\le \frac{5}{7}|y-z|\hfill \\ \hfill & =\frac{5}{7}d(y,z)\hfill \\ \hfill & \le \frac{5}{7}M(y,z)\hfill \\ \hfill & =\frac{5}{3}(\frac{3}{7}M(y,z))-\mathit{\varphi }(M(y,z))\hfill \\ \hfill & =\mathit{\psi }(\mathit{\theta }(M(y,z)))-\mathit{\varphi }(M(y,z))\hfill \end{array}$$

So, all conditions of Theorem 2.3 hold, which imply that f has fixed point.

Example 2.8

Let $y=\mathbb{R}$ and $d:Y\times Y\to [0,\infty )$ be defined by $d(y,z)=|y-z|$ for all $y,z\in Y.$ Define $f:Y\to Y$ such that $fy=4y$ and $\mathit{\alpha },\mathit{\beta }:Y\to [0,\infty )$ by$$\begin{array}{cc}\hfill \mathit{\alpha }(y)& =\left\{\begin{array}{cc}1,\hfill & \text{if}y\in [-1,0],\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.\hfill \\ \hfill \mathit{\beta }(y)& =\left\{\begin{array}{cc}1,\hfill & \text{if}y\in [0,1],\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

Firstly, we want to show that f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping. Let $\mathit{\alpha }(y)\ge 1$ for some $y\in Y$ Then $y\in [-1,0]$ and thus $fy\in [-1,0].$ Subsequently, $\mathit{\beta }(fy)\ge 1.$ In the same way, if $\mathit{\beta }(y)\ge 1$ then $\mathit{\alpha }(fy)\ge 1.$ Then f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping. If $\{y_n\}$ is a sequence in Y such that $\mathit{\beta }(y_n)\ge 1$ and $y_n\to y$ as $n\to \infty .$ So, $y_n\in [0,1]$. Hence, i.e. $\mathit{\beta }(y)\ge 1.$ Let $\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1.$ Then $y\in [-1,0]$ and $z\in [0,1].$ All conditions of Corollary 2.5 are satisfied. Here, 0 is the fixed point f. Notice also Nadler multivalued mapping principle (Nadler, 1969) is not applicable. Indeed, $d(y,z)=|y-z|$ for all $y,z\in Y$. Then we have $y\ne z$, $d(y,z)=4\mid y-z\mid >h\mid y-z\mid$ for all $h\in [0,1)$

3. Some results for Meir-Keeler-Khan multivalued mapping

In this section, we will introduce the class of cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-admissible Meir-Keeler-Khan multivalued mapping, we will prove the existence of fixed point for the class of mapping via cyclic ($\mathit{\alpha },\mathit{\beta }$)-admissible mapping. After that, all mappings $f:Y\to CL(Y)$ considered in the sequel of this article satisfying$$\forall y,z\in Y,y\ne z\Rightarrow d(y,fz)+d(z,fy)\ne 0$$

Definition 3.1

Let (Y, d) be a complete metric space and $f:Y\to CL(Y).$f is called $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-Meir-Keeler-Khan multivalued mapping if there exists $\mathit{\psi }\in \mathrm{\Psi }$, $\mathit{\theta }\in \mathrm{\Theta }$, $\mathit{\varphi }\in \mathrm{\Phi }$ and $\mathit{\alpha },\mathit{\beta }:[0,\infty )\to [0,\infty )$ such that(3.1) $$\begin{array}{c}\hfill y,z\in Y,\mathit{\alpha }(y)\mathit{\beta }(z)\ge 1\Rightarrow \mathit{\theta }(H(fy,fz))\le \mathit{\psi }(\mathit{\theta }(N(y,z))-\mathit{\varphi }(N(y,z)),\end{array}$$(3.1)

where$$\begin{array}{c}\hfill N(y,z)=\frac{d(y,fy)d(y,fz)+d(z,fz)d(z,fy)}{d(y,fz)+d(z,fy)}\end{array}$$

for all $y,z\in Y.$

Now, we will state our results in this section:

Theorem 3.2

Let $f:Y\to CL(Y)$ be a cyclic $(\mathit{\alpha },\mathit{\beta })$-$(\mathit{\psi },\mathit{\theta },\mathit{\varphi })$-Meir-Keeler-Khan multivalued mapping on metric space (Y, d). Assume that the following conditions hold:

(1)	there exists $y_0\in Y$ and $y_1\in f{y}_0$ such that $$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$
(2)	if f is continuous

then f has a fixed point.

Proof

We start from condition (1), we have $y_0\in Y$ and $y_1\in f{y}_0$ such that$$\begin{array}{cc}\hfill \mathit{\alpha }(y_0)\ge 1& \Rightarrow \mathit{\beta }(f{y}_0)=\mathit{\beta }(y_1)\ge 1\hfill \\ \hfill \mathit{\beta }(y_0)\ge 1& \Rightarrow \mathit{\alpha }(f{y}_0)=\mathit{\alpha }(y_1)\ge 1,\hfill \end{array}$$

therefore, $\mathit{\alpha }(y_0)\ge 1$ and $\mathit{\beta }(y_1)\ge 1$, equivalently, $\mathit{\alpha }(y_0)\mathit{\beta }(y_1)\ge 1.$ If $y_0=y_1$, we derive that $y_1\in F(f)$ and so the proof is done. Now, we assume that $y_0\ne y_1$ and $y_1\notin f{y}_1$ and hence $d(y_1,f{y}_1)\ne 0.$ From (3.1), we have(3.2) $$\begin{array}{cc}\hfill 0<d(y_1,f{y}_1)& \le \mathit{\theta }(H(f{y}_0,f{y}_1))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(N(y_0,y_1)))-\mathit{\varphi }(N(y_0,y_1)),\hfill \end{array}$$(3.2)

where(3.3) $$\begin{array}{cc}\hfill N(y_0,y_1)& =\frac{d(y_0,f{y}_0)d(y_0,f{y}_1)+d(y_1,f{y}_1)d(y_1,f{y}_0)}{d(y_0,f{y}_1)+d(y_1,f{y}_0)}\hfill \\ \hfill & =d(y_0,y_1),\hfill \end{array}$$(3.3)

from (3.4) and (3.5) and using the properties of $\mathit{\varphi }$, we get(3.4) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_1,f{y}_1))& \le \mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))-\mathit{\varphi }(d(y_0,y_1))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))),\hfill \end{array}$$(3.4)

For $\mathit{\sigma }>1$ by Lemma 1.7, there exists $y_2\in f{y}_1$ such that(3.5) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,y_2))<\mathit{\sigma }\mathit{\theta }(d(y_1,f{y}_1)).\end{array}$$(3.5)

From (3.4) and (3.5), we get(3.6) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,y_2))<\mathit{\psi }(\mathit{\sigma }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))),\end{array}$$(3.6)

Since f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping, from condition (1) and $y_2\in f{y}_2$, we have$$\begin{array}{cc}\hfill \mathit{\alpha }(y_1)\ge 1& \Rightarrow \mathit{\beta }(y_2)=\mathit{\beta }(f{y}_1)\ge 1,\hfill \\ \hfill \mathit{\beta }(y_1)\ge 1& \Rightarrow \mathit{\alpha }(y_2)=\mathit{\alpha }(f{y}_1)\ge 1.\hfill \end{array}$$

so, $\mathit{\alpha }(y_1)\ge 1$ and $\mathit{\beta }(y_2)\ge 1.$ Equivalently, $\mathit{\alpha }(y_1)\mathit{\beta }(y_2)\ge 1.$ If $y_2\in f{y}_2,$ then $y_2\in F(f).$ So, we assume that $y_2\notin f{y}_2;$ that is $d(y_2,f{y}_2)>0.$ From (3.1), we deduce(3.7) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_2,f{y}_2))& \le \mathit{\theta }(H(f{y}_1,f{y}_2))\hfill \\ \hfill & \le \mathit{\psi }(\mathit{\theta }(N(y_1,y_2)))-\mathit{\varphi }(N(y_1,y_2)),\hfill \end{array}$$(3.7)

where(3.8) $$\begin{array}{cc}\hfill N(y_1,y_2)& =\frac{d(y_1,f{y}_1)d(y_1,f{y}_2)+d(y_2,f{y}_2)d(y_2,f{y}_1)}{d(y_1,f{y}_2)+d(y_2,f{y}_1)}\hfill \\ \hfill & =d(y_1,y_2),\hfill \end{array}$$(3.8)

Using properties of $\mathit{\varphi }$, we have:(3.9) $$\begin{array}{cc}\hfill 0<\mathit{\theta }(d(y_2,f{y}_2))& \le \mathit{\theta }(H(f{y}_1,f{y}_2))\hfill \\ \hfill & <\mathit{\psi }(\mathit{\theta }(d(y_1,y_2))).\hfill \end{array}$$(3.9)

For ${\mathit{\sigma }}_1>1$ by Lemma 1.7, there exists $y_3\in f{y}_2$ such that(3.10) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_2,y_3))<{\mathit{\sigma }}_1\mathit{\theta }(d(y_2,f{y}_2)).\end{array}$$(3.10)

From (3.9)and (3.10), we obtain(3.11) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_1,y_2))<{\mathit{\psi }}^2(\mathit{\sigma }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1))),\end{array}$$(3.11)

By continuing this method, we construct the sequence$\{y_n\}\subset Y$ such that $y_{n+1}\ne y_n\in f{y}_n$, again, since f is a cyclic $(\mathit{\alpha },\mathit{\beta })$-admissible mapping , we have$$\begin{array}{c}\hfill \mathit{\alpha }(y_n)\ge 1\text{and}\mathit{\beta }(y_n)\ge 1,\end{array}$$

for all $n\in \mathbb{N}$. This implies that$$\begin{array}{c}\hfill \mathit{\alpha }(y_n)\mathit{\beta }(y_{n+1})\ge 1,\end{array}$$

and(3.12) $$\begin{array}{c}\hfill 0<\mathit{\theta }(d(y_n,y_{n+1}))<{\mathit{\psi }}^n(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))),\end{array}$$(3.12)

for all $\mathbb{N}\cup 0$.

Let $m,n\in \mathbb{N}$ such that $m>n$. By the triangle inequality, we get(3.13) $$\begin{array}{c}\hfill \mathit{\theta }(d(y_m,y_n))\le \sum _{l=n}^{m-1}\mathit{\theta }(d(y_l,y_{l+1}))\le \sum _{l=n}^{m-1}{\mathit{\psi }}^{l-1}(\mathit{\tau }\mathit{\psi }(\mathit{\theta }(d(y_0,y_1)))).\end{array}$$(3.13)

Since $\mathit{\psi }\in \mathrm{\Psi }$ and $\mathit{\theta }$ is continuous, we have$$\underset{n,m\to \infty }{lim}d(y_m,y_n)=0$$.

Thus $\{y_n\}$ is Cauchy sequence in (Y, d). By the completeness of (Y, d),  there exists $y^{\ast }\in Y$ such that $y_n\to y^{\ast }$ as $n\to \infty .$ Since f is continuous, we get$$d(y^{\ast },f{y}^{\ast })=\underset{n\to \infty }{lim}d(y_{n+1},f{y}^{\ast })\le \underset{n\to \infty }{lim}H(f{y}_n,f{y}^{\ast })=0$$

thus, we have $y^{\ast }\in f{y}^{\ast }$. $\square$

Additional information

Funding

The authors would like to acknowledge the grant: Universiti Kebangsaan Malaysia (UKM) [grant number DIP-2014-034] and Ministry of Education, Malaysia [grant number FRGS/1/2014/ST06/UKM/01/1] for financial support.

Notes on contributors

Habes Alsamir

Habes Alsamir received his MSc from Taibah University, Saudi Arabia in 2011 and is pursuing PhD from Kebangsaan University, Malaysia. His main research interest is: Fixed point Theory. He has published two papers in international journals.

Mohd Salmi Md Noorani

Mohd Salmi Md Noorani is a professor of mathematics in Kebangsaan University, Malaysia. He has supervised several PhD and master’s students in different areas of pure and applied mathematics and has published a lot of articles in international journals. Especially he has published many in metric space and its generalization.

Wasfi Shatanawi

Wasfi Shatanawi Wasfi Shatanawi, PhD, is a professor of Mathematics in the Department of Mathematics at Hashemite University. Shatanawi completed his PhD study from Carleton University/Canada in 2001. He published more than 100 papers. Shatanawi is one of the most influential scientific minds in the world; he has been listed in highly cited researchers for the years 2015 and 2016, according to Thomson Reuters.