Some fixed point theorems for convex contractive mappings in complete metric spaces with applications

Abstract

In this research work, convexity condition is introduced to some classes of contraction mappings such as Chatterjea and Hardy and Rogers contractive mappings. The fixed points of these maps are proved in complete metric spaces. Example is equally provided to support the main result. The unique solution of nonlinear Fredholm integral equation is obtained via the Hardy and Rogers convex contraction mappings of type 2. The results obtained in this paper, extend and generalize some related works in the literature.

Keywords:

• convex contraction mapping
• Chatterjea two-sided convex contraction mapping
• Hardy and Rogers convex contraction mappings of type 2
• metric spaces
• fixed point
• nonlinear fredholm integral equation

PUBLIC INTEREST STATEMENT

Banach contraction principle is a fundamental tool for solving the existence solution for integral equations in mathematics and other branches of sciences. Many authors have generalized the contraction mappings; such are Chatterjea and Hardy and Rogers contractive mappings. The introduction of convexity condition to Chatterjea and Hardy and Rogers contractive mappings and the proof of their fixed points in metric spaces are achieved in this paper. The existence of the solution of Fredholm integral equation is obtained via Hardy and Rogers convex contractive operators. This result has many potentials because the fixed point of these maps can be proved in different abstract spaces and can equally be applied to solve integral and differential equations.

1. Introduction and preliminaries

In (Banach, 1922), the Banach contraction principle is an essential tool for solving problems in mathematics and applied mathematical fields. In the course of establishing the fixed point of Banach contraction mapping in a complete metric space $X$, it was observed that the mapping is continuous in the entire domain $X$. However, Kannan (Kannan, 1968) introduced contractive mappings which does not force the mapping to be continuous in the entire domain $X$ for the existence of the fixed point. Chatterjea (Chatterjea, 1972) equally introduced contractive mapping which is independent of Kannan contractive mappings and does not also force the mapping to be continuous in the entire domain $X$ for the existence of the fixed point. Many authors generalized Banach contraction mappings and proved the existence of the fixed point without the continuity of the mapping in the whole domain $X$ (see, Choudhury, 2009, Eke, 2016a, Eke, Oghonyon, & Davvaz, 2018, Eke, 2016b, Eke, Imaga, & Odetunmibi, 2017, Parvaneh & Hosseinzadeh, 2012, Chandok & Dinu, 2013, Muresan & Muresan, 2015).

In (Hardy & Rogers, 1973), the existence of the fixed point of Hardy and Rogers contractive mappings does not require the mapping to be continuous in the entire domain $X$. Rather, a mapping satisfying Hardy and Rogers contraction turns out to be continuous at the fixed point. To illustrate this, let $z=fz$ be a fixed point of $f$ and $x_n\to z$. Then

$\begin{array}{rl}& d(f{x}_n,z)=d(f{x}_n,fz)\\ & \le ad(x_n,z)+bd(x_n,f{x}_n)+cd(z,fz)+ed(x_n,fz)+fd(z,f{x}_n)\\ & \le ad(x_n,z)+ed(x_n,fz)+fd(z,f{x}_n)\end{array}$

$(1-f)d(f{x}_n,z)\le ad(x_n,z)+ed(x_n,fz)$

$d(f{x}_n,z)\le \frac{a+e}{1-f}d(x_n,z)$

Take the limit of above inequality as $n\to \mathrm{\infty }$ yields

$li{m}_{n\to \mathrm{\infty }}d(f{x}_n,z)=0$

This implies $f{x}_n\to z=fz$ and $f$ is continuous at the fixed point.

In 1981, Istratescu (Istratescu, 1981) inaugurated the class of convex contraction mappings by introducing convexity condition with the iterates of the mapping to Banach contraction mappings.

Definition 1.1 (Istratescu, 1981): A continuous mapping $f:X\to X$ is said to be a convex contraction of order 2 if there exist $a,b\in (0,1)$ such that for all $x,y\in X$,

$d(f^2(x),f^2(y))\le ad(f(x),f(y))+bd(x,y)$

and where $a+b<1$ .

The existence of the fixed point of convex contraction mapping requires the continuity of the mapping in the whole domain.

The following example showed that convex contraction mapping of order 2 is not a contraction mapping.

Example 1.2 (Istratescu, 1981) Let $X=[0,1]$ with the usual metric and define $f:X\to X$ by the relation .

$f(x)={\left(x^2+\frac{1}{2}\right)}^2.$

It is obvious that this is continuous and not a contraction.

If $x,y\in X$ are arbitrary chosen, then

$|f^2(x)-f^2(y)|=\frac{(x^2+y^2)}{4}|(f(x)-f(y))|+\frac{(x+y)}{8}|(x-y)|$

Thus $f$ is a convex contraction of order 2.

Definition 1.3 (Istratescu, 1981): A continuous mapping $f:X\to X$ defined on a complete metric space $X$ is called two-sided convex contraction mappings if there exist positive numbers $a_1,a_2,b_1,b_2\in (0,1)$ such that the following inequality holds:

$d(f^2(x),f^2(y))\le a_{1}d(x,f(x))+a_{2}d(f(x),f^2(x))+b_{1}d(y,f(y))+b_{2}d(f(y),f^2(y))$

for all $x,y\in X$ and $a_1+a_2+b_1+b_2<1$.

Istratescu (Istratescu, 1981) in this pioneer work proved the fixed point of several convex contraction mappings in metric spaces. Among these results is the following theorem.

Theorem 1.4: Let $f$ be a continuous self -mapping of a complete metric space $(X,d)$ satisfying the condition;

$d(f^{2}x,f^{2}y)\le a_{0}d(x,y)+a_{1}d(fx,fy)+b_{0}d(x,fx)+b_{1}d(fx,f^{2}x)+c_{0}d(y,fy)+c_{1}d(fy,f^{2}y)$

where $0\le a_0+a_1+b_0+b_1+c_0+c_1<1$, for distinct $x,y\in X$. Then $f$ has a unique fixed point.

Alghamdi et al. (Alghamdi, Alnafei, Radenovic, & Shahzad, 2011) proved the analog of some fixed point theorems for Kannan two-sided convex contraction mappings and Reich convex contraction mappings of type 2 on non-normal cone metric spaces. Bisht et al. (Bisht & Hussian, 2017) proved that the assumption of the continuity condition of Theorem 1.4 can be replaced by a weaker condition of orbital continuity. Ciric (Ciric, 1971) proved the existence and uniqueness of the fixed point of generalized contraction mappings in a metric space by assuming the orbitally continuous of the mapping. Roy and Saha (Roy & Saha, 2018) established some fixed point theorems for Ciric contractive mappings of type1 and type 2 over a c*-algebra-valued metric space by emphasizing the necessity of the mapping to be orbitally continuous. In (Ciric, 2003), Ciric established several fixed point theory for generalized and quasi-contraction mappings in abstract spaces. In Kirk and Shahzad (Kirk & Shahzad, 2014), it was proved that every nonexpansive mappings in a bounded hyperconvex metric space has a fixed point. Several fixed point theory for contractive mappings in metric spaces and its generalizations are found in Agarwal et al. (Agarwal, Karapinar, O’Regan, & -de Hierro, 2015). Bisht and Rakocevic (Bisht & Rakocevic, 2018) proved fixed points of convex contraction and generalized convex contraction by assuming the K-continuity of the operator under various settings. Georgescu (Georgescu, 2017) established that every iterated function system consisting of generalized convex contractions in a complete metric space is a Picard operator. Ampadu (Ampadu, 2017a) proved that the fixed point of tri-cyclic convex contraction mapping of order 2 in a complete b-metric space has a unique fixed point. The fixed point theorem for convex type contraction of order 2 in various settings is proved by Ampadu (Ampadu, 2017b) in 2017. In this work, Chatterjea two-sided convex contraction mappings and Hardy and Rogers convex contraction mappings of type 2 are introduced, respectively. It is established in this work that the fixed points of these newly introduced convex contraction mappings of type 2 are unique by employing the orbital continuity of the mappings at the fixed point. Observe that the set $O(x;f)=\{f^{n}x;n\ge 0\}$ is known as orbit of the self-mapping $f$ at the point $x\in X$.

Definition 1.5 (Ciric, 1971): A self-mapping $f$ of a metric space $(X,d)$ is called orbitally continuous at a point $z\in X$ if for any sequence $\{x_n\}\subset O(x,f)$ (for some $x\in X$) $x_n\to z$ implies $f{x}_n\to fz$ as $n\to \mathrm{\infty }$.

2. Main results

The definition of Chatterjea two-sided convex contraction mappings and Hardy and Rogers convex contraction mappings of type 2 with the existence and uniqueness of their fixed points in a metric space is discussed as follows.

Definition 2.1: A continuous mapping $f:X\to X$ defined on a complete metric space $X$ is called Chatterjea two-sided convex contraction mappings if there exist positive numbers $a_1,a_2,b_1,b_2\in (0,1)$ such that the following inequality holds:

$d(f^2(x),f^2(y))\le a_{1}d(x,f(y))+a_{2}d(f(y),f^2(y))+b_{1}d(y,f(x))+b_{2}d(f(x),f^2(x))$

for all $x,y\in X$ and $a_1+a_2+b_1+b_2<1$.

Definition 2.2: A continuous mapping $f:X\to X$ defined on a complete metric space $X$ is called Hardy and Rogers convex contraction mappings of type 2 if there exist positive numbers $a_1,a_2,b_1,b_2,c_1,c_2,e_1,e_2,f_1,f_2\in (0,1)$ such that the following inequality holds:

$d(f^2(x),f^2(y))\le a_{1}d(x,y)+a_{2}d(f(x),f(y))+b_{1}d(x,f(x))+b_{2}d(f(x),f^2(x))+c_{1}d(y,f(y))+c_{2}d(f(y),f^2(y))+e_{1}d(x,f(y))+e_{2}d(f(y),f^2(y))+f_{1}d(y,f(x))+f_{2}d(f(x),f^2(x))$

for all $x,y\in X$ and $a_1+a_2+b_1+b_2+c_1+c_2+e_1+e_2+f_1+f_2<1$.

Example 2.3: Let $f:X\to X$, where $X=[0,1]$ with usual metric $d(x,y)=|x-y|$. Define $fx=\frac{x^2}{4}+\frac{1}{8}$ for all $x\in X$. Then for all $x,y\in X$ with $x\ne y$ we obtain

$|fx-fy|=\frac{1}{2}|x^2-y^2|\le |x-y|$

and

$\begin{array}{rl}& |f^{2}x-f^{2}y|=\frac{1}{256}(|4x^4-4y^4+4x^2-4y^2|)\\ & =\frac{1}{64}(|x^4-y^4+x^2-y^2|)\\ & \le \frac{1}{64}(|x^4-y^4|+|x^2-y^2|)\\ & \le \frac{1}{2}|fx-fy|+\frac{1}{4}|x-y|\\ & \le \frac{1}{7}d(x,y)+\frac{1}{14}d(fx,fy)+\frac{1}{8}d(x,fx)+\frac{1}{15}d(fx,f^{2}x)\\ & +\frac{1}{7}d(y,fy)+\frac{1}{14}d(fy,f^{2}y)+\frac{1}{8}d(x,fy)+\frac{1}{20}d(fy,f^{2}y)\\ & +\frac{1}{15}d(y,fx)+\frac{1}{20}d(fx,f^{2}x).\end{array}$

Thus $f$ is Hardy and Rogers convex contraction of type 2.

Remarks 2.4: This example shows that Hardy and Rogers convex contraction mapping of type 2 generalizes all the convex contraction mappings of Istratescu (Istratescu, 1981) and Chatterjea convex contraction mapping defined in this work.

Theorem 2.5: Let (X, d) be a complete metric space and $f$ be a self- mapping satisfying the conditions;

$d(f^2(x),f^2(y))\le a_{1}d(x,f(y))+a_{2}d(f(y),f^2(y))+b_{1}d(y,f(x))+b_{2}d(f(x){,}^{2}f(x))$

for all $x,y\in X$ and $a_1+a_2+b_1+b_2<1$. Suppose $f$ is orbitally continuous . Then f has a unique fixed point in X. For any $x_0\in X$, the Picard iteration $\{x_n{\}}_{n=0}^{\mathrm{\infty }}$ given by $x_{n+1}=f{x}_n$, $n\ge 0$, converges to the fixed point of $X$.

Proof: Let $x_0\in X$ be arbitrarily choosen. Develop the orbit of $x_0$ under $f$. Let $x=x_0$, $y=f(x_0)$ and set $k=max\{d(x_0,f(x_0)),d(f(x_0),f^2(x_0))\}$.

(1) $\begin{array}{rl}& d(f^2(x_0),f^3(x_0))\le a_{1}d(x_0,f^2(x_0))+a_{2}d(f^2(x_0),f^3(x_0))+b_{1}d(f(x_0),f(x_0))\\ & +b_{2}d(f(x_0),f^2(x_0))\\ & \le a_{1}d(x_0,f^2(x_0))+a_{2}d(f^2(x_0),f^3(x_0))+b_{2}d(f(x_0),f^2(x_0))\\ & \le a_{1}d(x_0,f(x_0))+a_{1}d(f(x_0),f^2(x_0))+a_{2}d(f^2(x_0),f^3(x_0))\\ & +b_{2}d(f(x_0),f^2(x_0))\\ & \le a_{1}d(x_0,f(x_0))+(a_1+b_2)d(f(x_0),f^2(x_0))+a_{2}d(f^2(x_0),f^3(x_0))\\ & \le (2a_1+b_2)max\{d(x_0,f(x_0)),d(f(x_0),f^2(x_0))\}\\ & +a_{2}d(f^2(x_0),f^3(x_0))\\ & \le \frac{(2a_1+b_2)k}{1-a_2}\end{array}$(1)

(1) $\begin{array}{rl}& d(f^3(x_0),f^4(x_0))\le a_{1}d(f(x_0),f^3(x_0))+a_{2}d(f^3(x_0),f^4(x_0))+b_{1}d(f^2(x_0),f^2(x_0))\\ & +b_{2}d(f^2(x_0),f^3(x_0))\\ & \le a_{1}d(f(x_0),f^3(x_0))+a_{2}d(f^3(x_0),f^4(x_0))+b_{2}d(f^2(x_0),f^3(x_0))\\ & \le a_{1}d(f(x_0),f^2(x_0))+a_{1}d(f^2(x_0),f^3(x_0))+a_{2}d(f^3(x_0),f^4(x_0))\\ & +b_{2}d(f^2(x_0),f^3(x_0))\\ & \le a_{1}d(f(x_0),f^2(x_0))+(a_1+b_2)d(f^2(x_0),f^3(x_0))+a_{2}d(f^3(x_0),f^4(x_0))\\ & \le (2a_1+b_2)max\{d(f(x_0),f^2(x_0)),d(f^2(x_0),f^3(x_0))\}+a_{2}d(f^3(x_0),f^4(x_0))\\ & \le \frac{(2a_1+b_2)k}{1-a_2}\end{array}$(1)

Consequently, for all n,

$d(f^n(x_0),f^{n+1}(x_0))\le {\left(\frac{(2a_1+b_2)}{1-a_2}\right)}^{n-2}k$

$d(f^n(x_0),f^{n+1}(x_0))\le {\gamma }^{n-2}k$ where $\gamma =\frac{(2a_1+b_2)}{1-a_2}$

For $m>n$ and using the triangle inequality we obtain,

$\begin{array}{rl}& d(f^n(x_0),f^m(x_0))\le d(f^n(x_0),f^{n+1}(x_0))+d(f^{n+1}(x_0),f^{n+2}(x_0))\\ & +d(f^{n+2}(x_0),f^{n+3}(x_0))+\cdots +d(f^{m-1}(x_0),f^m(x_0))\\ & \le {\gamma }^{n-2}k+{\gamma }^{n-1}k+{\gamma }^{n}k+\cdots +{\gamma }^{m+n-3}k\\ & \le ({\gamma }^{n-2}+{\gamma }^{n-1}+{\gamma }^n+\cdots +{\gamma }^{m+n-3})k\\ & \le {\gamma }^{n-2}(1+\gamma +{\gamma }^2+\cdots )k\\ & \le \frac{{\gamma }^{n-2}}{1-\gamma }k\end{array}$

As $n\to \mathrm{\infty }$ then $d(f^n(x_0),f^m(x_0))\to 0$. Therefore $\{d(f^n(x_0),f^{n+1}(x_0))\}$ is a Cauchy sequence. Since $X$ is complete, there exists a point $x\in X$ such that $x_n\to x$ as $n\to \mathrm{\infty }$. Also $f^n(x_0)\to x$ . Applying the orbital continuity of $f$, gives $li{m}_{n\to \mathrm{\infty }}{f}^{n+1}(x_0)=fx$ . By the uniqueness of limit, we obtain $x=fx$. Thus $x$ is the fixed point of $f$. The uniqueness of the fixed point follows with the property.

The existence and uniqueness of the fixed point of Hardy and Rogers convex contraction mappings is proved below.

Theorem 2.6: Let (X, d) be a complete metric space and $f$ be a self- mapping satisfying the conditions;

$\begin{array}{rl}& d(f^2(x),f^2(y))\le a_{1}d(x,y)+a_{2}d(f(x),f(y))+b_{1}d(x,f(x))+b_{2}d(f(x),f^2(x))\\ & +c_{1}d(y,f(y))+c_{2}d(f(y),f^2(y))+e_{1}d(x,f(y))+e_{2}d(f(y),f^2(y))\\ & +f_{1}d(y,f(x))+f_{2}d(f(x),f^2(x))\end{array}$

for all $x,y\in X$ and $a_1+a_2+b_1+b_2+c_1+c_2+e_1+e_2+f_1+f_2<1$. Suppose $f$ is orbitally continuous. Then f has a unique fixed point in X. For any $x_0\in X$, the Picard iteration $\{x_n{\}}_{n=0}^{\mathrm{\infty }}$ given by $x_{n+1}=f{x}_n$, $n\ge 0$, converges to the fixed point of $X$.

Proof: Let $x_0\in X$ be arbitrarily choosen. Develop the orbit of $x_0$ under $f$. Let $x=x_0$, $y=f(x_0)$ and set $k=max\{d(x_0,f(x_0)),d(f(x_0),f^2(x_0))\}$.

$\begin{array}{rl}& d(f^2(x_0),f^3(x_0))\le a_{1}d(x_0,f(x_0))+a_{2}d(f(x_0),f^2(x_0))+b_{1}d(x_0,f(x_0))\\ & +b_{2}d(f(x_0),f^2(x_0))+c_{1}d(f(x_0),f^2(x_0))+c_{2}d(f^2(x_0),f^3(x_0))\\ & +e_{1}d(x_0,f^2(x_0))+e_{2}d(f^2(x_0),f^3(x_0))+f_{1}d(f(x_0),f(x_0))\\ & +f_{2}d(f(x_0),f^2(x_0))\\ & =a_{1}d(x_0,f(x_0))+a_{2}d(f(x_0),f^2(x_0))+b_{1}d(x_0,f(x_0))\\ & +b_{2}d(f(x_0),f^2(x_0))+c_{1}d(f(x_0),f^2(x_0))+c_{2}d(f^2(x_0),f^3(x_0))\\ & +e_{1}d(x_0,f^2(x_0))+e_{2}d(f^2(x_0),f^3(x_0))+f_{2}d(f(x_0),f^2(x_0))\\ & \le a_{1}d(x_0,f(x_0))+a_{2}d(f(x_0),f^2(x_0))+b_{1}d(x_0,f(x_0))\\ & +b_{2}d(f(x_0),f^2(x_0))+c_{1}d(f(x_0),f^2(x_0))+c_{2}d(f^2(x_0),f^3(x_0))\\ & +e_{1}d(x_0,f(x_0))+e_{1}d(f(x_0),f^2(x_0))\\ & +e_{2}d(f^2(x_0),f^3(x_0))+f_{2}d(f(x_0),f^2(x_0))\\ & =(a_1+b_1+e_1)d(x_0,f(x_0))+(a_2+b_2+c_1+e_1+f_2)d(f(x_0),f^2(x_0))\\ & +(c_2+e_2)d(f^2(x_0),f^3(x_0))\\ & \le (a_1+b_1+e_1+a_2+b_2+c_1+e_1+f_2)\\ & max\{d(x_0,f(x_0)),d(f(x_0),f^2(x_0))\}+(c_2+e_2)d(f^2(x_0),f^3(x_0))\\ & \le \frac{\beta k}{1-c_2-e_2}\end{array}$

$where\beta =a_1+b_1+e_1+a_2+b_2+c_1+e_1+f_2<1.$

Similarly, we obtain

$d(f^3(x_0),f^4(x_0))\le \frac{\beta k}{1-c_2-e_2}$

Consequently, for all $n$,

$d(f^n(x_0),f^{n+1}(x_0))\le {\left(\frac{\beta }{1-c_2-e_2}\right)}^{n-2}k$

$d(f^n(x_0),f^{n+1}(x_0))\le {\gamma }^{n-2}k$ where $\gamma =\frac{\beta }{1-c_2-e_2}$

For $m>n$ and using the triangle inequality we obtain,

$\begin{array}{rl}& d(f^n(x_0),f^m(x_0))\le d(f^n(x_0),f^{n+1}(x_0))+d(f^{n+1}(x_0),f^{n+2}(x_0))\\ & +d(f^{n+2}(x_0),f^{n+3}(x_0))+\cdots +d(f^{m-1}(x_0),f^m(x_0))\\ & \le {\gamma }^{n-2}k+{\gamma }^{n-1}k+{\gamma }^{n}k+\cdots +{\gamma }^{m+n-3}k\\ & \le ({\gamma }^{n-2}+{\gamma }^{n-1}+{\gamma }^n+\cdots +{\gamma }^{m+n-3})k\\ & \le {\gamma }^{n-2}(1+\gamma +{\gamma }^2+\cdots )k\\ & \le \frac{{\gamma }^{n-2}}{1-\gamma }k\end{array}$

As $n\to \mathrm{\infty }$ then $d(f^n(x_0),f^m(x_0))\to 0$. Therefore $\{d(f^n(x_0),f^{n+1}(x_0))\}$ is a Cauchy sequence. Since $X$ is complete, there exists a point $x\in X$ such that $x_n\to x$ as $n\to \mathrm{\infty }$. Also $f^n(x_0)\to x$ . Applying the orbital continuity of $f$, gives $li{m}_{n\to \mathrm{\infty }}{f}^{n+1}(x_0)=fx$ . By the uniqueness of limit, we obtain $x=fx$. Thus $x$ is the fixed point of $f$.

To prove the uniqueness, let $u$ be a different fixed point of $f$ such that $x\ne u$. We need to prove that $x=u$. Suppose $x\ne u$ and using inequality (2) yields

$\begin{array}{rl}& d(f^2(x),f^2(u))\le a_{1}d(x,u)+a_{2}d(f(x),f(u))+b_{1}d(x,f(x))\\ & +b_{2}d(f(x),f^2(x))\\ & +c_{1}d(u,f(u))+c_{2}d(f(u),f^2(u))\\ & +e_{1}d(x,f(u))+e_{2}d(f(u),f^2(u))+f_{1}d(u,f(x))\\ & +f_{2}d(f(x),f^2(x))\\ & =a_{1}d(x,u)+a_{2}d(x,u)+b_{1}d(x,x)+b_{2}d(x,f(x))\\ & +c_{1}d(u,u)+c_{2}d(u,f(u))+e_{1}d(x,u)+e_{2}d(u,f(u))\\ & +f_{1}d(u,x)+f_{2}d(x,f(x))\\ & =a_{1}d(x,u)+a_{2}d(x,u)+b_{1}d(x,x)+b_{2}d(x,x)\\ & +c_{1}d(u,u)+c_{2}d(u,u)+e_{1}d(x,u)+e_{2}d(u,u)\\ & +f_{1}d(u,x)+f_{2}d(x,x)\\ & =(a_1+a_2+e_1+f_1)d(x,u)\end{array}.$

A contradiction since $a_1+a_2+e_1+f_1<1$. Hence $d(x,u)=0$. This implies that $x=u$. The uniqueness proved.

Remark 2.7: If $b_1=b_2=c_1=c_2=e_1=e_2=f_1=f_2=0$ in Theorem 2.5 then we obtain the result of Istratescu (Istratescu, 1981) without orbital continuity. If $e_1=e_2=f_1=f_2=0$ in Theorem 2.5 then we obtain fixed point theorem for two-sided convex contraction mappings of Bisht et al. (Bisht & Hussian, 2017). If $a_1=a_2=e_1=e_2=f_1=f_2=0$ in Theorem 2.5 then we obtain the result of Alghamdi et al. (Alghamdi et al., 2011) for two-sided convex contraction mappings without orbital continuity in the context of cone metric space.

Example 2.8: Let $X=[0,1]$ be a metric space with usual metric. If $f:[0,1]\to [0,1]$ is defined by

(1) $f(x)=\left\{\begin{array}{c}0forallx\in [0,\frac{1}{2})\\ \frac{x^2}{5}+\frac{1}{10}ifx=\frac{1}{2}\end{array}\right.$(1)

Then $f^{n}x=\frac{x^{2n}}{5}+\frac{1}{{10}^n}\to 0$ as $n\to \mathrm{\infty }$ . $f(\frac{1}{2})=\frac{3}{20}$, $f^n(\frac{1}{2})\to 0$ as $n\to \mathrm{\infty }$. $f(f^{n}x)\to f(0)=0$ as $n\to \mathrm{\infty }$.

Also, for all $x,y\in X$ with $x\ne y$ we obtain

$|fx-fy|=\frac{1}{5}|x^2-y^2|\le |x-y|$

$\begin{array}{rl}& |f^{2}x-f^{2}y|=|\left(\frac{x^2}{5}+\frac{1}{10}\right){ }^2-{\left(\frac{y^2}{5}+\frac{1}{10}\right)}^2|\\ & =|\frac{x^4}{25}+\frac{x^2}{25}+\frac{1}{100}-\frac{y^4}{25}-\frac{y^2}{25}-\frac{1}{100}|\\ & =|\frac{x^4}{25}+\frac{x^2}{25}-\frac{y^4}{25}-\frac{y^2}{25}|\\ & =\frac{1}{25}|x^4-y^4+x^2-y^2|\\ & \le \frac{1}{5}|fx-fy|+\frac{1}{10}|x-y|\end{array}$

$\le \frac{1}{4}d(x,fy)+\frac{1}{7}d(fy,f^{2}y)+\frac{1}{4}d(y,fx)+\frac{1}{7}d(fx,f^{2}x)$

The conditions of Theorem 2.5 are satisfied. The unique fixed point of the operator is 0.

3. Applications

In this section, we apply our result to prove the existence theorem for nonlinear Fredholm integral equation.

Let $X=C[a,b]$ be a set of all real continuous functions on $[a,b]$ endowed with the usual metric. Then $(X,d)$ is a complete metric space.

Now, we consider the nonlinear Fredholm integral equation

(1) $x(t)=v(t)+\frac{1}{b-a}{\int }_a^{b}k(t,s,x(s)ds$(1)

where $t,s\in [a,b]$. Assume that $k:[a,b]\times [a,b]\times X\to R$ and $v:[a,b]\to R$ continuous, where $v(t)$ is a given function in $X$.

Theorem 3.1: Let $(X,d)$ be a metric space endowed with the usual metric and $f:X\to X$ be defined by

(2) $fx(t)=v(t)+\frac{1}{b-a}{\int }_a^{b}k(t,s,x(s)ds$(2)

If there are $a_1,a_2,b_1,b_2,c_1,c_2,e_1,e_2,f_1,f_2\in [0,1)$ such that for all $x,y\in X$ with $x\ne y$ and $s,t\in [a,b]$ satisfies

(3) $\begin{array}{rl}& |k(t,s,fx(s))-k(t,s,fy(s))|\le a_{1}d(x(s),y(s))+a_{2}d(fx(s),fy(s))\\ & +b_{1}d(x(s),fx(s))+b_{2}d(fx(s),f^{2}x(s))\\ & +c_{1}d(y(s),fy(s))+c_{2}d(fy(s),f^{2}y(s))\\ & +e_{1}d(x(s),fy(s))+e_{2}d(fy(s),f^{2}y(s))\\ & +f_{1}d(y(s),fx(s))+f_{2}d(fy(s),f^{2}y(s))\end{array}.$(3)

Suppose $f$ is orbitally continuous, then the integral operator defined by (2) has a unique solution $z\in X$ and for each $x_0\in X$, $f{x}_{\ne }{x}_n$ for all $n\in N\cup \{0\}$, we have $li{m}_{n\to \mathrm{\infty }}f{x}_n=z$.

Proof: Let $x_0\in X$ and define a sequence $\{x_n\}$ in $X$ by $x_{n+1}=f{x}_n=f^{n+1}{x}_0$ for all $n\ge 0$. By (2) we have

(4) $x_{n+1}=f{x}_n(t)=v(t)+\frac{1}{b-a}{\int }_a^{b}k(t,s,x_n(s)ds$(4)

We need to show that $f$ is Hardy and Rogers convex contraction mapping of type 2 on $C[a,b]$. Using (2) and (3) yields

$\begin{array}{rl}& |f^{2}x(t)-f^{2}y(t)|=\frac{1}{|b-a|}\left|{\int }_a^{b}k(t,s,fx(s))ds-{\int }_a^{b}k(t,s,fy(s))ds\right|\\ & \le \frac{1}{|b-a|}{\int }_a^b|k(t,s,fx(s))-k(t,s,fy(s))|ds\\ & \le \frac{1}{|b-a|}{\int }_a^b\{|a_1|x(s)-y(s)|+a_2|fx(s)-fy(s)|\\ & +b_1|x(s)-fx(s)|+b_2|fx(s)-f^{2}x(s)|\\ & +c_1|y(s)-fy(s)|+c_2|fy(s)-f^{2}y(s)|\end{array}$

$\begin{array}{rl}& +e_1|x(s)-fy(s)|+e_2|fy(s)-f^{2}y(s)|\\ & +f_1|y(s)-fx(s)|+f_2|fy(s)-f^{2}y(s)|\}ds\\ & \le \frac{a_1+a_2+b_1+b_2+c_1+c_2+e_1+e_2+f_1+f_2}{|b-a|}\\ & {\int }_a^{b}max\{|x(s)-y(s)|,|fx(s)-fy(s)|,|x(s)-fx(s)|,\\ & |fx(s)-f^{2}x(s)|,|y(s)-fy(s)|,|fy(s)-f^{2}y(s)|,\\ & |x(s)-fy(s)|,|fy(s)-f^{2}y(s|),|y(s)-fx(s)|,\\ & |fy(s)-f^{2}y(s)|\}ds\\ & \le \frac{K}{|b-a|}max\{|x(s)-y(s)|,|fx(s)-fy(s)|,\\ & |x(s)-fx(s)|,|fx(s)-f^{2}x(s)|,|y(s)-fy(s)|,\\ & |fy(s)-f^{2}y(s)|,|x(s)-fy(s)|,|fy(s)-f^{2}y(s|),\\ & |y(s)-fx(s)|,|fy(s)-f^{2}y(s)|\}{\int }_a^{b}ds\\ & =Kmax\{d(x,y),d(fx,fy),d(x,fx),d(fx,f^{2}x),\\ & d(y,fy),d(fy,f^{2}y),d(x,fy),d(fy,f^{2}y),d(y,fx),\\ & d(fx,f^{2}x)\}\end{array}.$

where $K=a_1+a_2+b_1+b_2+c_1+c_2+e_1+e_2+f_1+f_2<1$ for all $x,y\in X$ with $x\ne y$.

Since $X$ is a complete metric space then the iterative scheme converges to some point $z\in X$. That is, $li{m}_{n\to \mathrm{\infty }}{x}_n=z$. By the orbital continuity of $f$ we can prove that $f$ has a fixed point. Thus, the integral operator $f$ defined by (2) has a unique solution $z\in X$.

4. Conclusion

This study introduces convexity conditions to Chatterjea and Hardy and Rogers contraction mappings. This research proves that the fixed point for Chattterjea two-sided convex contraction mappings and Hardy and Rogers convex contraction mappings of type 2 in a complete metric space is unique. The solution of nonlinear Fredholm integral equation is obtained via Hardy and Rogers convex contraction mappings of type 2. This research work has many potentials as the fixed point for these newly introduced convex contraction mappings can be established in different abstract spaces.

Authors Contribution

Kanayo Eke developed formal analysis; Sheila Bishop supervised the work; Victoria Oliasma worked on writing-review and editing. All authors approved the final manuscript.

Acknowledgements

The authors are grateful to Covenant University for supporting this research work financially.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Kanayo Stella Eke

Kanayo Stella Eke research focuses on metrical and topological fixed point theory and applications. She introduced three generalized metric spaces; G-partial metric spaces, G-symmetric spaces and E-uniform spaces. She teaches abstract mathematics and functional analysis both at undergraduate and postgraduate levels. The author has over twenty five articles published in reputable international journals to her credit.

Victoria Olusola Olisama

Victoria Olusola Olisama research is in best proximity point theory. She introduced the best proximity point result of some Hardy and Rogers type proximal contractions in uniform spaces, E-Jav distance function in b-metric spaces. She has published about seven articles in reputable international journals.

sheila Amina Bishop

sheila Amina Bishop has teaching experience of about twelve years in undergraduate and postgraduate studies. Her research interest is on stochastic analysis and applications, differential equations and mathematical statistics. Most of these results have been published in several reputable journals.