1.1 Single-valued mappings

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

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

where a and b are constants which satisfy 0 < a, b < 1 and .

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

subject to a change of notation.

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

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 be equipped with the usual metric of . Define by

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

with and .

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

(1)

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

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

for all and .

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

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

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 is termed a Chatterjea two-sided convex contraction if there are constants and the subsequent inequality is satisfied:

for any and .

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

Example 1.9: Let with the metric . Define by:

Let us calculate :

where

Since the sum of the coefficients is:

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

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

for any and .

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 (metric space). For any given , a point is called an approximate fixed point of S if it fulfills the condition:

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

where 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 that adheres to the subsequent properties:

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

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

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

for all with d(Sg, Sh) > 0.

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

1. (K1)
2. (K2) such thatfor all , with .

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

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), , and then we have

Suppose that S is k-continuous for [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 ;

where the constants are non-negative and their sum satisfies . 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:

for all , where and . If S is orbitally continuous, then S admits a unique fixed point in G. Moreover, for any initial point , the Picard iterations sequence , defined by for , 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 has g₀ as it’s fixed point if .

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

H is the Hausdorff metric given by

,

where .

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

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

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

for every , and .

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

for all , , and

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

holds for each pair of points .

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 and a fixed constant q > 1. Then, for each , there exists an element such that

Proof: For , the result holds for b = a with .

Set where k < 1.

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

(*)

Putting the selected value of 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.

2.1 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 be a fixed real number. A function is said to be a b-metric if, for all , the subsequent conditions hold:

1. (b₁) , g = h
2. (b₂)
3. (b₃)

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 and S be a self-map on G satisfying the condition;

(2)

for all , , and .

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

Proof: Let be an arbitrary point. Define a sequence {g_n} by

for all .

Put g = g₀, h = S(g₀) in (2), we get

Now substitute g = S(g₀) and in (2), and get:

By continuing this process, we obtain:

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

As , in the view of , .

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

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 .

As   , 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 satisfying (2) in Theorem 2.2 with . Then both S and S² have a unique common fixed point (i.e. S(z) = S²(z) = z).

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

Now, we aim to show that S and S² have a unique common fixed point. Clearly, . Suppose and . Then S²(u) = u.

Assume, for a contradiction, that . Since S²(u) = u, therefore by (2), applied to the pair u and ,we get

Now, if , the above inequality implies that is smaller than itself, which is a contradiction. Thus, , and hence . Therefore, F(S) = F(S²), and since z is the only fixed point of S, it must also be the only fixed point of S². This shows that both S and S² 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 . S is -admissible if for ,

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 , S a generalized convex contraction on G characterized by the base mapping . Assume that S satisfies the -admissibility condition and that there exists a point such that . 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 be such that . Define {g_n} in G as before to get

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

Then .

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

Again put g = Sg₀ and in (1) to obtain.

Similarly,

And also,

By continuing this procedure, we get

where m = 2l or m = 2l  +  1. Consequently, as . 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 {g_n} is a Cauchy sequence, consider m and n as positive integers such that m < n. Let m = 2l and n = 2p, where and .

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

Similarly, let m = 2l + 1 and n = 2p + 1, where and .

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

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

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

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 and {g_n} represent a sequence in a b-metric space (G, d, s) such that

(3)

for all , where and

Then {g_n} is Cauchy.

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

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

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

Proof Let be arbitrary. Define a sequence {g_n}, as before, to get

Now,

for every , where a_i is non-negative, and the sum .

By Lemma 2.6, the sequence {g_n} forms a Cauchy sequence in the complete space G. Hence, there exists a point such that as . Furthermore, since , the orbital continuity of S ensures that . Consequently, we have , 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 denotes the class of all nonempty, closed, and bounded subsets of G. For any , the following are satisfied:

1. 
2. 

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

1. (a) ;
2. (b) ;
3. (c) ;
4. (d) .

The triplet 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 be an asymptotically regular convex contraction. Then there exists such that .

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

(4)

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

(5)

Now, is a closed and bounded subset of G, so there exists such that

As before,

(6)

Similarly,

(7)

Using (6), we have

In general,

For convenience, we set

so the above result can be written as

(8)

For , , we have

Using (8), we get

In the limiting case, when ,

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

By Lemma 2.8 (1),

In the limiting case, when ,

Now Sh is closed and so . 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 be a multivalued F-convex contraction satisfying:

(9)

for all , and . Then S has a fixed point.

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

Assume that . Then .

Let g = g₀, h = S(g₀) in (9) and set ,

(10)

Since F is strictly increasing and is greater than zero, (10) becomes

(11)

where . Substituting g = S(g₀), in (9),we have

Given that F is strictly increasing and

Similarly, we can show that

By following this process, we obtain

As F is strictly increasing and is greater than zero, we have

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

This demonstrates that the sequence {g_n} is Cauchy in G, implying the existence of an element such that

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

This shows that z is a fixed point of S.

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

for all and and and

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 a multivalued weak convex contraction. Then

1. (i)
2. (ii) For every initial point , there exists a sequence generated by the operator S that converges to a fixed point u of S, for which the following estimates hold:(12)(13)for a certain real number h < 1.

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

Case 1: If , then .

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

Case 2: Let . By Lemma 1.28, there exists such that .

We take such that

Hence

If , then , i.e., .

Assume that . Again by Lemma 1.28, there exists such that

In this way, we construct an orbit at g₀ for S satisfying

(14)

By (14), we obtain inductively

(15)

Hence

(16)

Similarly, by (15), we have

(17)

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

(18)

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

(19)

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

As Su is closed, so .

(ii) Let in (17). Then approaches 0 and so we get

This proves (12).

In the same way when in (16), we get

(20)

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.