Fixed points of multivalued convex contractions with application

TL;DR

This paper extends fixed point theory for convex contractions in b-metric spaces and applies it to solve a nonlinear integral equation.

Contribution

The paper introduces new fixed point results for multivalued convex contractions and applies them to a nonlinear Fredholm integral equation.

Findings

Fixed point outcomes are established for single-valued convex contraction mappings in b-metric spaces.

A new result for multivalued convex contractions is derived, analogous to Nadler’s fixed point theorem.

The results are applied to solve a nonlinear Fredholm integral equation using a Chatterjea convex contraction.

Abstract

In this work, we establish fixed point outcomes for single- valued convex contraction type mappings in the context of a b-metric space. Some of the new results are extended for a multivalued convex contraction and an F-convex contraction. Thereby, an analogue of the famous Nadler’s fixed point theorem for a multivalued convex contraction mapping is obtained. The relation among various contractions is presented in a diagram for an insight in this area of investigations. We apply a special case of Theorem 2.11, to solve a nonlinear Fredholm integral equation for a Chatterjea convex contraction.

Figures

Equations

Full text

1 Preliminaries and introduction

The exploration of fixed point theory represents a fundamental and highly impactful area in modern mathematics. The main theme in this subject is the celebrated principle of Banach contraction, which asserts that there is a unique fixed point for each contraction mapping on a complete metric space. The proof of this principle hinges on the iterates of the contraction and the principle itself has applications in a wide range of fields like partial differential equations, integral equations, image processing, optimization and artificial intelligence. In view of paramount importance of this principle, Browder and Petryshyn [7], Istratescu [11] and Berinde [2] have introduced and studied new classes of mappings enjoying higher powers of the mapping; in particular, Istratescu has coined the term “convex contraction”.

We now set out to develop results for convex contraction type single-valued as well as multi-valued mappings in the context of b-metric spaces.

1.1 Single-valued mappings

Definition 1.1: Consider a nonempty set G and a mapping $[eqn]$ . A point $[eqn]$ is referred to as a fixed point of S if it satisfies $[eqn]$ . The set consisting of all fixed points of the mapping S is represented by $[eqn]$ .

Definition 1.2: Let (G, d) be a complete metric space. A map $[eqn]$ is known as Istratescu convex contraction [11] if

$$[eqn]$$

where a and b are constants which satisfy 0 < a, b < 1 and $[eqn]$ .

Remark 1.3: ([17],Example 2.1) (i) If b = 0, the convex contraction condition reduces to the well-known Banach contraction condition:

$$[eqn]$$

subject to a change of notation.

(ii) If a = 0, the convex contraction condition reduces to the classial "asymptotic” contraction:

$$[eqn]$$

which confirms the presence of a fixed point, even when the number 2 is replaced with any arbitrary integer n.

Example 1.4: [17] Let $[eqn]$ be equipped with the usual metric of $[eqn]$ . Define $[eqn]$ by

$$[eqn]$$

Then S is not a Banach contraction, and $[eqn]$ . But S is a convex contraction, because for $[eqn]$ , we have

$$[eqn]$$

with $[eqn]$ and $[eqn]$ .

Definition 1.5: [16] A mapping $[eqn]$ is defined as a generalized convex contraction if there exist a function $[eqn]$ and constants $[eqn]$ satisfying $[eqn]$ , so that the subsequent condition is satisfied:

$$[eqn]$$

If a = 0 and $[eqn]$ in Definition 1.5, then it becomes asymptotic contraction condition of Remark 1.3.

Definition 1.6: [9] A continuous map $[eqn]$ on a complete metric space (G,d) is referred to as a two-sided convex contraction if there are constants $[eqn]$ and the following inequality is satisfied:

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ and $[eqn]$ .

Definition 1.7. [10] Let S be a mapping from a metric space G into itself. The set

$$[eqn]$$

is referred to as the orbit of S starting at g. We say that S is orbitally continuous at a point $[eqn]$ if for every sequence $[eqn]$ , with $[eqn]$ , the condition

$$[eqn]$$

It is important to note that every continuous self-map on a metric space is orbitally continuous, the converse does not necessarily hold [6].

Definition 1.8: [9] Let (G, d) denote a complete metric space. A continuous mapping $[eqn]$ is termed a Chatterjea two-sided convex contraction if there are constants $[eqn]$ and the subsequent inequality is satisfied:

$$[eqn]$$ $$[eqn]$$

for any $[eqn]$ and $[eqn]$ .

We now provide an example of a Chatterjea two-sided convex contraction.

Example 1.9: Let $[eqn]$ with the metric $[eqn]$ . Define $[eqn]$ by:

$$[eqn]$$

Let us calculate $[eqn]$ :

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

where

$$[eqn]$$

Since the sum of the coefficients is:

$$[eqn]$$

therefore, S is a Chatterjea two-sided convex contraction.

Definition 1.10: [9] A mapping $[eqn]$ (complete metric space) is called Hardy and Rogers convex contraction if there exist positive integers $[eqn]$ and the subsequent inequality is satisfied:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

for any $[eqn]$ and $[eqn]$ .

It is remarked that some of the above mentioned classes of convex contraction type mappings are independent of each other [9, 11]. In case, the constants are allowed to be zero in Definition 1.10, then it reduces to Istratescu convex contraction (Definition 1.2).

Definition 1.11: [16] Let $[eqn]$ (metric space). For any given $[eqn]$ , a point $[eqn]$ is called an approximate fixed point of S if it fulfills the condition:

$$[eqn]$$

Definition 1.12: [7] A map S defined on a metric space (G, d) is termed asymptotically regular at any point $[eqn]$ if

$$[eqn]$$

where $[eqn]$ denotes the n-th iterate of S at g.

Lemma 1.13: ([16],Lemma 2.1) Suppose that (G, d) is a metric space and S is an asymptotically regular mapping on G. Then S has an approximate fixed point.

Definition 1.14: [14] Consider the mapping $[eqn]$ that adheres to the subsequent properties:

(a) The function F is strictly increasing.(b) A sequence $[eqn]$ of positive real numbers satisfies $[eqn]$ if and only if $[eqn]$ .(c) If there exists a constant $[eqn]$ , then $[eqn]$ .

The class of functions F that satisfies conditions (a)-(c) is represented by $[eqn]$ .

Definition 1.15: An F-contraction is a self-map S defined over a metric space G if there is a function $[eqn]$ and a constant $[eqn]$ such that

$$[eqn]$$

for all $[eqn]$ with d(Sg, Sh) > 0.

Definition 1.16: [26] Let F be a mapping that meets requirements (a)–(c). A funtion $[eqn]$ is known as an F-Kannan mapping if the following hold:

(K1) $[eqn]$ (K2) $[eqn]$ such that

$$[eqn]$$

for all $[eqn]$ , with $[eqn]$ .

Example 1.17: ([26],Lemma 12) Consider a metric space (G, d) and F-Kannan mapping $[eqn]$ . Then,

$$[eqn]$$

So an F-Kannan mapping is asymptotically regular.

Here are some well known results for convex contraction type mappings.

Theorem 1.18: ([11], Theorem 1.2) Every convex contraction mapping defined on a complete metric space has a unique fixed point.

Theorem 1.19: ([5], Theorem 2.1) If S is a self-map on a complete metric space (G, d), $[eqn]$ , $[eqn]$ and $[eqn]$ then we have

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Suppose that S is k-continuous for $[eqn]$ [12]. Then S admits a unique fixed point.

Theorem 1.20: ([6],Theorem 2.1) Let S be a self-map of a complete metric space (G, d) such that for each $[eqn]$ ;

$$[eqn]$$

where the constants $[eqn]$ are non-negative and their sum satisfies $[eqn]$ . The map S has a unique fixed point if it is either orbitally continuous or k-continuous.

Theorem 1.21: ([9], Theorem 2.5) Let (G, d) be a complete metric space and S be a self-map on G satisfying the condition:

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ , where $[eqn]$ and $[eqn]$ . If S is orbitally continuous, then S admits a unique fixed point in G. Moreover, for any initial point $[eqn]$ , the Picard iterations sequence $[eqn]$ , defined by $[eqn]$ for $[eqn]$ , converges to a fixed point of S in G.

1.2 Multivalued mappings

Definition 1.22: Let (G, d) be a metric space and CB(G) represent the family of closed and bounded subsets of G. A mapping $[eqn]$ has g0 as it’s fixed point if $[eqn]$ .

Definition 1.23 [12] A map S on a metric space (G, d) is known as asymptotically regular at a point $[eqn]$ if

$$[eqn]$$

H is the Hausdorff metric given by

$[eqn]$ , $[eqn]$

where $[eqn]$ .

Definition 1.24: Consider the metric space (G, d). A mapping $[eqn]$ is called convex contraction if

$$[eqn]$$

for all $[eqn]$ , where a, b are the constants that fulfill 0 < a, b < 1 and $[eqn]$ .

Definition 1.25: Let (G, d) be a metric space. A mapping $[eqn]$ is called Chatterjea two-sided convex contraction if the following holds:

$$[eqn]$$ $$[eqn]$$

for every $[eqn]$ , $[eqn]$ and $[eqn]$ .

Definition 1.26: Let (G, d) be a complete metric space and $[eqn]$ be a mapping. The mapping S is said to be a weak convex contraction if it satisfies:

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ , $[eqn]$ , $[eqn]$ and $[eqn]$

Definition 1.27: [2] Let (G, d) be a metric space and $[eqn]$ be a multivalued map. If $[eqn]$ and $[eqn]$ , then S is a multivalued weak contraction if

$$[eqn]$$

holds for each pair of points $[eqn]$ .

For ease of reference, we provide proof of the result to follow.

Lemma 1.28: ([2], Lemma 1) Consider a metric space (G, d), two subsets $[eqn]$ and a fixed constant q > 1. Then, for each $[eqn]$ , there exists an element $[eqn]$ such that

$$[eqn]$$

Proof: For $[eqn]$ , the result holds for b = a with $[eqn]$ .

Set $[eqn]$ where k < 1.

According to the definitions of d(a, B) and H(A, B), for all $[eqn]$ , there exists an element $[eqn]$ such that

$$[eqn]$$

Putting the selected value of $[eqn]$ in the above inequality, we get the result with q = k^−1^.

Fixed point results for single-valued asymptotically regular and convex contraction type mappings have been considered by Berinde and Pacurar [3], Bisht and Hussain [4] and Khan and Oyetunbi [12] while Khan and Oyetunbi [13], Karakaya and Sekman [15] and Sgroi and Vetro [23] have studied these results for multivalued mappings.In this paper, we will extend Theorem 1.21 for a Chatterjea two-sided convex contraction in a b-metric space. On the one hand, we apply our new result to solve a non-linear Fredholm integral equation and on the other hand, we find it’s multivalued version. A fixed point result for generalized convex contraction on a b-metric space is proved in Theorem 2.5. We established a multivalued version of Theorem 1.18, the fundamental result of Istratescu for a convex contraction.

Recently, Nallaselli et al [19], Özkan et al [21] and Ricinschi et al [22] have studied convex contractions on metric spaces and b-metric spaces. It is remarked that our results and techniques are different from their ones and are more focused on the development of fixed point results for a multivalued convex contraction.

2 Fixed points results

2.1 Single-valued mappings

The concept of a b-metric space was presented by Czerwik [8] as follows:

Definition 2.1: Let G be a nonempty set and $[eqn]$ be a fixed real number. A function $[eqn]$ is said to be a b-metric if, for all $[eqn]$ , the subsequent conditions hold:

(b_1_) $[eqn]$ , $[eqn]$ g = h(b_2_) $[eqn]$ (b_3_) $[eqn]$

The triplet (G, d, s) is then referred to as a b-metric space.

Note that when s = 1, a b-metric space becomes a metric space. But, in general, the converse does not hold.

We extend Theorem 2.5 in [9] for b-metric spaces as follows.

Theorem 2.2: Let (G, d, s) be a complete b-metric space with coefficient $[eqn]$ and S be a self-map on G satisfying the condition;

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ , $[eqn]$ , $[eqn]$ and $[eqn]$ .

If S is orbitally continuous, then it admits a unique fixed point in G. For any initial point $[eqn]$ , the sequence $[eqn]$ , defined recursively by $[eqn]$ for $[eqn]$ , converges to the unique fixed point of S in G.

Proof: Let $[eqn]$ be an arbitrary point. Define a sequence {gn} by

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

for all $[eqn]$ .

Put g = g0, h = S(g0) in (2), we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Now substitute g = S(g0) and $[eqn]$ in (2), and get:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

By continuing this process, we obtain:

$$[eqn]$$ $$[eqn]$$

We now demonstrate that {gn} forms a Cauchy sequence in G. For m < n, we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

As $[eqn]$ , in the view of $[eqn]$ , $[eqn]$ .

Therefore $[eqn]$ is a Cauchy sequence. Since G is complete, there exists a point $[eqn]$ such that $[eqn]$ as $[eqn]$ . So applying orbital continuity of S, we get

$$[eqn]$$

This shows that g is a fixed point of S.

For it’s uniqueness, assume on the contrary that u is another fixed point of S such that $[eqn]$ .

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

As   $[eqn]$ , so d(g,u)=0 implies g = u which is a contradiction. Hence S has a unique fixed point in G.

An extension of Corollary 1 in [1] for b-metric spaces is presented below.

Corollary 2.3: Let S be a self-map on a complete b-metric space $[eqn]$ satisfying (2) in Theorem 2.2 with $[eqn]$ . Then both S and S^2^ have a unique common fixed point (i.e. S(z) = S^2^(z) = z).

Proof: By Theorem 2.2, there exists a unique fixed point z of S and so $[eqn]$ .

Now, we aim to show that S and S^2^ have a unique common fixed point. Clearly, $[eqn]$ . Suppose $[eqn]$ and $[eqn]$ . Then S^2^(u) = u.

Assume, for a contradiction, that $[eqn]$ . Since S^2^(u) = u, therefore by (2), applied to the pair u and $[eqn]$ ,we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Now, if $[eqn]$ , the above inequality implies that $[eqn]$ is smaller than itself, which is a contradiction. Thus, $[eqn]$ , and hence $[eqn]$ . Therefore, F(S) = F(S^2^), and since z is the only fixed point of S, it must also be the only fixed point of S^2^. This shows that both S and S^2^ possess a unique common fixed point.

Definition 2.4 [16] Consider a self-map S defined on a nonempty set G, along with a function $[eqn]$ . S is $[eqn]$ -admissible if for $[eqn]$ ,

$$[eqn]$$

We now extend ([16],Theorem 3.1) in the framework of b-metric spaces.

Theorem 2.5: Consider a b-metric space (G, d, s) with $[eqn]$ , S a generalized convex contraction on G characterized by the base mapping $[eqn]$ . Assume that S satisfies the $[eqn]$ -admissibility condition and that there exists a point $[eqn]$ such that $[eqn]$ . Under these assumptions, S possesses an approximate fixed point. Furthermore, if S is continuous and (G, d, s) is complete, then S has a fixed point.

Proof: Let $[eqn]$ be such that $[eqn]$ . Define {gn} in G as before to get

$$[eqn]$$

If $[eqn]$ for some n, the proof is complete. Otherwise, assume $[eqn]$ . Given that S is $[eqn]$ -admissible, we have the condition $[eqn]$ for all n. Let $[eqn]$ and $[eqn]$ .

Then $[eqn]$ .

Now put g = g0, h = S(g0) in (1) and get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Again put g = Sg0 and $[eqn]$ in (1) to obtain.

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Similarly,

$$[eqn]$$

And also,

$$[eqn]$$

By continuing this procedure, we get

$$[eqn]$$

where m = 2l or m = 2l  +  1. Consequently, $[eqn]$ as $[eqn]$ . This implies that S is asymptotically regular. By Lemma 1.13, S possesses an approximate fixed point. Now, suppose that S is continuous, and the space (G, d) is a complete b-metric space. To demonstrate that {gn} is a Cauchy sequence, consider m and n as positive integers such that m < n. Let m = 2l and n = 2p, where $[eqn]$ and $[eqn]$ .

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

For m = 2l + 1 and n = 2p, where $[eqn]$ and $[eqn]$ , with the condition that $[eqn]$ , we obtain:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Similarly, let m = 2l + 1 and n = 2p + 1, where $[eqn]$ and $[eqn]$ .

$$[eqn]$$

For $[eqn]$ , in all the cases, we obtain $[eqn]$ (in view of $[eqn]$ ). That is, {gn} is a Cauchy sequence in G. As G is complete, so there exists $[eqn]$ such that $[eqn]$ as $[eqn]$ . If S is continuous, then $[eqn]$ , i.e., $[eqn]$ . Hence Sz = z.

Now we need to prove that S has a unique fixed point in G. Assume, on the contrary that $[eqn]$ is a fixed point of S such that $[eqn]$ .

Taking z = g and z^*^ = h in (1), we get by hypothesis $[eqn]$

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

In view of (1–a–b) < 0, d(z,z^^) = 0.Hence z = z^^, which is a contradiction. Hence S has a unique fixed point in G.

Lemma 2.6: ([18], Lemma 4) Let $[eqn]$ and {gn} represent a sequence in a b-metric space (G, d, s) such that

$$[eqn]$$

for all $[eqn]$ , where $[eqn]$ and

$$[eqn]$$

Then {gn} is Cauchy.

The following result improves ([18],Theorem 1).

Theorem 2.7: Let $[eqn]$ be a convex contraction of order k defined on a complete b-metric space (G, d, s), such that

$$[eqn]$$

for all $[eqn]$ , where ai is non-negative and the sum satisfies $[eqn]$ . If S exhibits orbital continuity, then S admits a unique fixed point in G.

Proof Let $[eqn]$ be arbitrary. Define a sequence {gn}, as before, to get

$$[eqn]$$

Now,

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

for every $[eqn]$ , where ai is non-negative, and the sum $[eqn]$ .

By Lemma 2.6, the sequence {gn} forms a Cauchy sequence in the complete space G. Hence, there exists a point $[eqn]$ such that $[eqn]$ as $[eqn]$ . Furthermore, since $[eqn]$ , the orbital continuity of S ensures that $[eqn]$ . Consequently, we have $[eqn]$ , implying that z is a fixed point of S, and uniqueness of z follows as before.

2.2 Multi-valued mappings

Lemma 2.8: [25] Let (G, d, s) be a b-metric space and $[eqn]$ denotes the class of all nonempty, closed, and bounded subsets of G. For any $[eqn]$ , the following are satisfied:

$$[eqn]$$ $$[eqn]$$

Definition 2.9 (cf. [25],Definition 2.4) Let G be an arbitrary nonempty set and $[eqn]$ be a fixed real number. A strong b-metric on G is a function $[eqn]$ , satisfying the following axioms for $[eqn]$ :

(a) $[eqn]$ ;(b) $[eqn]$ ;(c) $[eqn]$ ;(d) $[eqn]$ .

The triplet $[eqn]$ denotes a strong b-metric space.

We establish a multivalued version of Theorem 1.18, a classical result of Istratescu [11].

Theorem 2.10: Let (G, d,s) be a complete strong b-metric space and $[eqn]$ be an asymptotically regular convex contraction. Then there exists $[eqn]$ such that $[eqn]$ .

Proof: Let $[eqn]$ . Then $[eqn]$ is a closed and bounded subset of G. Furthermore, let $[eqn]$ and $[eqn]$ be a closed and bounded subset of G. By Lemma 2.8 (2), there exists $[eqn]$ such that

$$[eqn]$$

Using the definition of convex contraction and asymptotic regularity of S, we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Now, $[eqn]$ is a closed and bounded subset of G, so there exists $[eqn]$ such that

$$[eqn]$$

As before,

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Similarly,

$$[eqn]$$ $$[eqn]$$

Using (6), we have

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

In general,

$$[eqn]$$

For convenience, we set

$$[eqn]$$

so the above result can be written as

$$[eqn]$$

For $[eqn]$ , $[eqn]$ , we have

$$[eqn]$$ $$[eqn]$$

Using (8), we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

In the limiting case, when $[eqn]$ ,

$$[eqn]$$

So {gn} is a Cauchy sequence in G. By completeness of G, there exists $[eqn]$ such that $[eqn]$ . Now we will prove that h is a fixed point of S.

$$[eqn]$$

By Lemma 2.8 (1),

$$[eqn]$$ $$[eqn]$$

In the limiting case, when $[eqn]$ ,

$$[eqn]$$ $$[eqn]$$

Now Sh is closed and so $[eqn]$ . Hence, h is a fixed point of S as desired.

Here is a multivalued version of Theorem 2.2 for a convex contraction (see also [20],[24],[27]).

Theorem 2.11: Let (G, d) be a complete metric space and let $[eqn]$ be a multivalued F-convex contraction satisfying:

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ , $[eqn]$ and $[eqn]$ . Then S has a fixed point.

Proof: Let $[eqn]$ be an arbitrary point of G and choose $[eqn]$ . If $[eqn]$ , then g1 is a fixed point of S and nothing to prove.

Assume that $[eqn]$ . Then $[eqn]$ .

$$[eqn]$$ $$[eqn]$$

Let g = g0, h = S(g0) in (9) and set $[eqn]$ ,

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Since F is strictly increasing and $[eqn]$ is greater than zero, (10) becomes

$$[eqn]$$ $$[eqn]$$

where $[eqn]$ . Substituting g = S(g0), $[eqn]$ in (9),we have

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Given that F is strictly increasing and $[eqn]$

$$[eqn]$$ $$[eqn]$$

Similarly, we can show that

$$[eqn]$$ $$[eqn]$$

By following this process, we obtain

$$[eqn]$$

As F is strictly increasing and $[eqn]$ is greater than zero, we have

$$[eqn]$$ $$[eqn]$$

For m > n, we need to show {gn} is a Cauchy sequence in G. Using the triangular inequality, we obtain

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

This demonstrates that the sequence {gn} is Cauchy in G, implying the existence of an element $[eqn]$ such that

$$[eqn]$$

Now we have to prove that z is a fixed point of S. As S is orbitally continuous, so we get

$$[eqn]$$

This shows that z is a fixed point of S.

Definition 2.12 Let (G, d) be a complete metric space. A mapping $[eqn]$ is a weak convex contraction if it satisfies

$$[eqn]$$ $$[eqn]$$

for all $[eqn]$ and $[eqn]$ and $[eqn]$ and $[eqn]$

Here is an extention of Theorem 3 of Berinde and Berinde [2] for a weak convex contraction.

Theorem 2.13 Let (G, d) be a complete metric space and $[eqn]$ a multivalued weak convex contraction. Then

(i) $[eqn]$ (ii) For every initial point $[eqn]$ , there exists a sequence $[eqn]$ generated by the operator S that converges to a fixed point u of S, for which the following estimates hold:

$$[eqn]$$ $$[eqn]$$

for a certain real number h < 1.

Proof: (i) Let q > 1. Let $[eqn]$ and $[eqn]$ . We consider two cases based on the Hausdorff distance between the iterates of S.

Case 1: If $[eqn]$ , then $[eqn]$ .

Now $[eqn]$ implies that $[eqn]$ (cf. Remark 1.3(i)). Therefore $[eqn]$ gives $[eqn]$ and the proof is complete.

Case 2: Let $[eqn]$ . By Lemma 1.28, there exists $[eqn]$ such that $[eqn]$ .

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

We take $[eqn]$ such that

$$[eqn]$$

Hence

$$[eqn]$$

If $[eqn]$ , then $[eqn]$ , i.e., $[eqn]$ .

Assume that $[eqn]$ . Again by Lemma 1.28, there exists $[eqn]$ such that

$$[eqn]$$

In this way, we construct an orbit $[eqn]$ at g0 for S satisfying

$$[eqn]$$

By (14), we obtain inductively

$$[eqn]$$

Hence

$$[eqn]$$

Similarly, by (15), we have

$$[eqn]$$

Considering $[eqn]$ , it can be concluded that the sequence $[eqn]$ constitutes a Cauchy sequence. Given that $[eqn]$ is a complete metric space, it follows that the sequence $[eqn]$ converges. Let

$$[eqn]$$

Then by (*) in the proof of Lemma 1.28, we get

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Letting $[eqn]$ in (19) and using the fact that $[eqn]$ imply by (18) that $[eqn]$ , as $[eqn]$ . So we get

$$[eqn]$$

As Su is closed, so $[eqn]$ .

(ii) Let $[eqn]$ in (17). Then $[eqn]$ approaches 0 and so we get

$$[eqn]$$

This proves (12).

$$[eqn]$$

In the same way when $[eqn]$ in (16), we get

$$[eqn]$$

At the end of this section, we present relation among various concepts, used in this work in the form of a diagram:

1- Contraction

2- Convex Contraction

3- Hardy and Rogers Convex Contraction

4- Weak Convex Contraction

5- Generalized Convex Contraction

6- F-Contraction

2.3 Diagram

Here the arrow stands for the inclusion.

$$[eqn]$$

3 Application

Let $[eqn]$ represent the vector space of all continuous real-valued functions on [a, b] endowed with the usual metric. Then (G, d) is a complete metric space.

The non-linear Fredholm integral equation is given as follows:

$$[eqn]$$

where $[eqn]$ , $[eqn]$ and $[eqn]$ are continuous and v(t) is a given function in G.

The solution of non-linear Fredhalm integral equation has been obtained for Hardy and Rogers convex contraction in [9]. We apply our Theorem 2.2 to solve non-linear Fredhalm integral equation for a Chatterjea convex contraction.

Theorem 3.1: Let $[eqn]$ be the equipped with the usual metric.

Assume that (i) $[eqn]$ is given by

$$[eqn]$$

(ii) For $[eqn]$ , $[eqn]$ with $[eqn]$ and $[eqn]$ , we have

$$[eqn]$$ $$[eqn]$$

If S is orbitally continuous, the integral operator defined by equation (22) possesses a unique solution $[eqn]$ . Moreover, for any initial value $[eqn]$ , it holds that $[eqn]$ for all $[eqn]$ . Consequently, we have $[eqn]$

Proof

Let $[eqn]$ be an arbitrary point. Define a sequence {gn} in G by $[eqn]$ for all $[eqn]$ . By (22), we have,

$$[eqn]$$

We must demonstrate that S is a Chatterjea two sided convex contraction on C[a, b]. Use of (22) and (23) yields

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$ $$[eqn]$$

where $[eqn]$ for all $[eqn]$ with $[eqn]$ .

Since G is a complete metric space, therefore the iterative process converges to a specific point $[eqn]$ (i.e. $[eqn]$ ). By orbital continuity of S, we can establish that z is a fixed point of S. Thus all the conditions of Theorem 2.2 are satisfied and so by it’s conclusion the non-linear Fredholm integral operator S defined by (22) has a unique solution.

Now we provide an example to demonstrate the application of Theorem 3.1

Example 3.2:

Let us consider the operator $[eqn]$ defined as:

$$[eqn]$$

where $[eqn]$ , and $[eqn]$ . The kernel $[eqn]$ is given by:

$$[eqn]$$

Let us choose the following constants:

$$[eqn]$$

which satisfy the requirement:

$$[eqn]$$

Let $[eqn]$ . We must demonstrate that:

$$[eqn]$$ $$[eqn]$$

Substituting the kernel $[eqn]$ , we get:

$$[eqn]$$

Substituting the expressions for $[eqn]$ and $[eqn]$ , we have:

$$[eqn]$$

By the given nature of g and h, the subsequent inequality holds:

$$[eqn]$$ $$[eqn]$$

So S satisfies (23). The operator S being continuous is orbitally continuous.

Thus, all the requirements of Theorem 3.1 are satisfied and so the operator S defined by (24) has a unique solution.

4 Conclusion

In this work, we have proved fixed point theorems for single-valued convex contraction mappings in b-metric spaces. Generalizing, some of these results for multivalued convex contraction mappings, an analogue of well-known theorems of Nadler and Istratescu are obtained. A result for an F-convex contraction is also established. A diagram is included here to provide an insight for the relationship among various convex contractions. A special case of Theorem 2.11 is applied to solve a non-linear Fredholm integral equation in the context of a Chatterjea two-sided convex contraction.

5 Open problems

Establish Theorems 2.2, 2.5, 2.10 and 2.13 for common fixed points and coincidences of convex contractions.