Existence, uniqueness, Ulam–Hyers–Rassias stability, well-posedness and data dependence property related to a ﬁxed point problem in γ -complete metric spaces with application to integral equations

. In this paper, we study a ﬁxed point problem for certain rational contractions on γ -complete metric spaces. Uniqueness of the ﬁxed point is obtained under additional conditions. The Ulam–Hyers–Rassias stability of the problem is investigated. Well-posedness of the problem and the data dependence property are also explored. There is a corollary of the main result. Finally, our ﬁxed point theorem is applied to solve a problem of integral equation. There is no continuity assumption on the mapping.


Introduction and mathematical background
In this paper, we consider a rational contraction on metric spaces and investigate the fixed point problem associated with it. We assume that the metric space is γ-complete, which is a concept introduced by Kutbi and Sintunavarat in the paper [11]. The uniqueness of the fixed point is obtained under additional conditions. Rational contractions were first introduced by Dass et al. [6] and have been considered in fixed point theory in recent works like [4,8]. Our investigation of the different aspects of the fixed point problem is performed in a metric space without completeness property. In most of the works on similar problems, the results are obtained by employing metric completeness. Instead, we assume the weaker concept of γ-completeness. There is a flexibility in such assumption since the choice of γ can be different subject to certain restrictions. This is one of the main motivations behind our considerations of the problems discussed in this paper. We impose an admissibility condition on the concerned mapping. The assumption of the contractive inequality is restricted to certain pairs of points. These assumptions are in tune with certain recent trends appearing in metric fixed point theory. Further, there are scopes of extending our present results, which are discussed at the end of the paper.
We investigate Ulam-Hyers-Rassias stability of the fixed point problem. It is a general type of stability, which is considered in several areas of mathematics. Introduced by Ulam [25] through a mathematical question posed in 1940 and later elaborated by Hyers [9] and Rassias [18], such stabilities have a very large literature today [10,16,19].
Well-posedness and data dependence property associated with this problem are also investigated.
Finally, we have an application of our results to a problem of a nonlinear integral equation.
Definition 1. (See [1].) An element s ∈ Z is called a fixed point of a mapping F : Z → Z if s = F s.
Several sufficient conditions have been discussed for the existence of fixed points of F : Z → Z, where Z has a metric d defined on it. The study is a part of the subject domain known as metric fixed point theory. The subject is widely recognized to have been originated in the work of Banach in 1922 [1], which is known as the Banach's contraction mapping principle and is instrumental to the proofs of many important results. In subsequent times, many metric fixed point results were proved and applied to different problem arising in mathematics. Today fixed point methods are recognized as strong mathematical methods. References [12,13] describe this development to a considerable extent.
Admissibility conditions have recently been used for obtaining fixed point results. Various admissibility criteria were introduced in the study of fixed points of mappings. We refer the reader to [21][22][23] for some details on admissibility conditions. Definition 3. (See [22].) Let Z be a nonempty set, F : Z → Z and γ : Definition 4. Let Z be a nonempty set and F : Z → Z. A function γ : Z × Z → [0, ∞) is said to have F -directed property if for every a, b ∈ Z, there exists u ∈ Z with γ(u, F u) 1 such that γ(a, u) 1, γ(b, u) 1. [23].) A metric space (Z, d) is said to have regular property with respect to a mapping γ : Z × Z → [0, ∞) if for any sequence {a n } in Z with limit a ∈ Z, γ(a n , a n+1 ) 1 implies γ(a n , a) 1 for all n. Example 1. Let Z = (−2, 2) be equipped with usual metric. Let F : Z → Z and γ : Z × Z → [0, ∞) be respectively defined as follows:

Definition 5. (See
Here (i) F is a γ-admissible mapping; (ii) γ has triangular property; (iii) Z has regular property with respect to γ.
Recently, Kutbi and Sintunavarat coined the concept of γ-complete metric space in the paper [11]. Definition 6. Let (Z, d) be a metric space and γ : Z × Z → [0, ∞). A Cauchy sequence {a n } in Z is called a γ-Cauchy sequence if γ(a n , a n+1 ) 1 for all n.
is a complete metric space, then Z is also a γ-complete metric space for any γ : Z × Z → [0, ∞), but the converse is not true.
is not a complete metric space. Indeed, if {a n } is a Cauchy sequence in Z such that γ(a n , a n+1 ) 1 for all n ∈ N , then a n ∈ [2019, 2020] for all n ∈ N . As [2019, 2020] is a closed subset of R, it follows that there exists a ∈ [2019, 2020] such that a n → a as n → ∞.
are two metric spaces. The mapping F is said to be γ-continuous at c ∈ Z if for any sequence {t n } in Z, ρ(c, t n ) → 0 as n → ∞ and γ(t n , t n+1 ) 1 for all n imply that d(F c, F t n ) → 0 as n → ∞.
Remark 2. The continuity of a mapping implies its γ-continuity for any γ : Z × Z → [0, ∞). In general, the converse is not true.
Problem P. Let F : Z → Z be a mapping, where (Z, d) is a metric space. Consider the problem of finding a point s ∈ Z satisfying s = F s.
Our paper is characterized by the following features.
1. We consider rational contractions in our theorems. 2. We prove our main result with a generalized notion of completeness assumption of the underlying space and without the continuity assumption on the mapping.
3. We investigate Ulam-Hyers-Rassias stability of the fixed point problem. 4. We investigate well-posedness of the problem. 5. We investigate data dependence of fixed point set and solution of the integral equation. 6. We apply our theorem to a problem of an integral equation.

Main results
In this section, we establish some fixed point results and illustrate them with examples. We discuss the uniqueness of the fixed point under some additional assumptions. We deduce a corollary of the main result.
Theorem 1. Let (Z, d) be a metric space and γ : Z × Z → [0, ∞) be a function such that (Z, d) is γ-complete. Let F : Z → Z be a γ-admissible mapping and there exists k ∈ (0, 1) such that for x, y ∈ Z with γ(x, y) 1, If there exists z 0 ∈ Z such that γ(z 0 , F z 0 ) 1 and property (A1) holds, then F has a fixed point in Z.
By repeated application of (6) we have With the help of (7), we have which implies that {z n } is a γ-Cauchy sequence in Z. As Z is γ-complete, there exists s ∈ Z such that lim n→∞ z n = s.
By (3), (8) and property (A1) we have γ(z n , s) 1 for all n 0. Using (2), we have Taking limit as n → ∞ in (9) and using (8), we have We present the following illustrative example in support of Theorems 1.
Example 3. Using the metric space Z, mappings γ and F as in Example 1, we see that Z = (−2, 2) is regular with respect to γ (see Example 1), that is, property (A1) holds, and F is a γ-admissible mapping. Let k = 1/4. Let x, y ∈ Z with γ(x, y) 1. Then x ∈ [0, 1] and y ∈ [0, 1/8]. Therefore, it is required to verify the inequality in Theorem 1 for x ∈ [0, 1] and y ∈ [0, 1/8]. Now, d(x, y) = |x − y| and Hence, all the conditions of Theorem 1 are satisfied, and 0 is a fixed point of F . Note 1. Theorem 1 is still valid if one considers the γ-continuity of F instead of taking property (A1). Then the portion of the proof just after (8) of Theorem 1 is replaced by the following portion: Using the γ-continuity assumption of F , we have Hence, s = F s, that is, s is a fixed point of F . We present the following illustrative example in view of Note 1.
be respectively defined as follows: is not a complete metric space. Let us consider the sequence {t n }, where t n = 1/(2n). Here t n → 0 as n → ∞, and γ(t n , t n+1 ) 1 for all n. But γ(t n , 0) = 0, and hence, Z is not regular with respect to γ. Also, F is γ-admissible. Here the function F is not continuous but γ-continuous. Choose k = 1/4. Let x, y ∈ Z with γ(x, y) 1. Then x, y ∈ (0, 1). In view of the above and Note 1, it only remains to be verified that the inequality in Theorem 1 is valid for all x, y ∈ (0, 1).
Here 0 is a fixed point of F . Theorem 2. In addition to the hypothesis of Theorem 1, suppose that properties (A2) and (A3) hold. Then F has a unique fixed point.
Proof. By Theorem 1 the set of fixed points of F is nonempty. If possible, let x and x * be two fixed points of F . Then x = F x and x * = F x * . Our aim is to show that x = x * . By property (A3) there exists u ∈ Z with γ(u, F u) 1 such that γ(x, u) 1, γ(x * , u) 1.
Put u 0 = u and let u 1 = F u 0 . Then γ(x, u 0 ) 1 and γ(u 0 , u 1 ) 1. Similarly, as in the proof of Theorem 1, we define a sequence {u n } such that u n+1 = F u n ∀n 1.
Arguing similarly as in proof of Theorem 1, we prove that {u n } is a γ-Cauchy sequence in Z, and there exists p ∈ Z such that lim n→∞ u n = p.
By (1) and (12) we have, for all n 0, Taking limit as n → ∞ in (13) and using (11), we have which is a contradiction unless d(x, p) = 0, that is, d(x, p) = 0, that is, Similarly, we can show that From (14) and (15) we have x = x * . Therefore, fixed point of F is unique.
We present some special cases illustrating the applicability of Theorem 1.
Corollary 1. Let (Z, d) be a complete metric space. Then F : Z → Z has a unique fixed point if for some k ∈ (0, 1) and for all x, y ∈ Z, one of the following inequalities holds: (iv) d(F x, F y) k max{d(x, y), (d(x, F x)+d(y, F y))/2, (d(x, F y)+d(y, F x))/2}.

Ulam-Hyers stability
In [19], one can find the following definition as well as some related notions concerning the Ulam-Hyers stability, which is relevant to the present considerations. Let (Z, d) be a metric space and T : Z → Z be a mapping. We say that the fixed point problem x = T x is Ulam-Hyers stable if there is > 0 such that for y ∈ Z with d(y, T y) , there exists x 0 ∈ Z satisfying x = T x and d(y, x 0 ) .
where c > 0 is a constant, then Definition 9 reduces to Definition in [10].
Let us consider the fixed point Problem P (x = F x) and the following inequation: In the next theorem, we take the following additional condition to assure the Ulam-Hyers stablity via γ-admissible mapping.
Theorem 3. In addition to the hypothesis of Theorem 2, suppose that (A4) holds. Then the fixed point Problem P is Ulam-Hyers stable.
Proof. By Theorem 2 there exists unique x * ∈ Z such that x * = F x * . So, x * is a solution of Problem P. Let u * ∈ Z be a solution of (16). Then d(u * , F u * ) . By property (A4) we have γ(u * , x * ) 1. With the help of (1), we have The function φ is monotone increasing, continuous, and φ(0) = 0. By (17) we have Therefore, the fixed point Problem P is Ulam-Hyers stable.

Well-posedness
The notion of well-posedness of a fixed point problem has evoked much interest to several mathematicians (see, for example, [16,17]). Let (Z, d) be a metric space and T : Z → Z be a mapping. The fixed point problem of T is said to be well-posed if T has a unique fixed point x ∈ Z and for any sequence {x n } in Z, d(x n , T x n ) → 0 as n → ∞ implies d(x n , x) → 0 as n → ∞.
Definition 10. (See [10].) Problem P is called well-posed if (i) F has a unique fixed point In the next theorem, we take the following condition to assure the well-posedness via γadmissible mapping.
(A5) If x * is any solution of Problem P and {x n } is any sequence in Z for which d(x n , F x n ) → 0 as n → ∞, then γ(x n , x * ) 1 for all n.
Theorem 4. In addition to the hypothesis of Theorem 2, suppose that (A5) holds. Then the fixed point Problem P is well-posed.
Proof. By Theorem 2 there exists unique x * ∈ Z such that x * = F x * . So, x * is a solution of Problem P. Let {x n } be a sequence in Z for which d(x n , F x n ) → 0 as n → ∞. As (A5) holds, we have γ(x n , x * ) 1 for all n. By (1) we have which implies that Thus, lim n→∞ d(x n , x * ) = 0, and hence, the fixed point Problem P is well-posed.

Data dependence result
In this section, we investigate the data dependence of fixed points.
Definition 11. Let T 1 , T 2 : Z → Z be two mappings, where (Z, d) is a metric space such that d(T 1 x, T 2 x) η for all x ∈ Z, where η is some positive number. Then the problem of data dependence is to estimate the distance between the fixed points of these two mappings.
Several research papers on data dependence have been published in the recent literature, some of which we mention in references [3,5,20]. Proof. By Theorem 2 there exists unique s ∈ Z such that s = F s. Suppose t is a fixed point of T . Take x 0 = t. Then Applying the assumption of the theorem, we have γ(x 0 , x 1 ) 1. Let x 2 = F x 1 , then by admissibility property of F we have γ(x 1 , x 2 ) 1. Inductively, arguing similarly as in the proof of Theorem 1, we have a sequence {x n } in Z such that x n+1 = F x n and γ(x n , x n+1 ) 1 ∀n 0.
Arguing similarly as in proof of Theorem 1, we can prove that • (7) is satisfied; • {x n } is a γ-Cauchy sequence in the metric space (Z, d), and there exists u ∈ Z such that lim n→∞ x n = u; • u is a fixed point of F , that is, u = F u; as fixed point of F is unique, we have u = s and F s = s. Using triangular property, we have Taking limit as n → ∞ in the above inequality and using (18), we have

Application
We have already mentioned in introduction that fixed point theorems in metric spaces are widely investigated and have applications in differential and integral equations (see [2,15,22]). In this section, we deal with a nonlinear integral equation. In the first part, we apply Theorems 1 and 2 to prove the existence and uniqueness of solution of Fredholm-type nonlinear integral equations. In the remaining part, we discuss three aspects of the same integral equation, namely, Ulam-Hyers stability, well-posedness and data dependence.
We consider the following Fredholm-type nonlinear integral equation: where Problem I. To find out a solution of the Fredholm-type integral equation under some appropriate conditions on g, h and K.
We take the following assumptions: Proof. Define a mapping F : Z → Z by Let x, y ∈ C ([a, b]) and x ≺ y. Then x(s) y(s), hence, by (I2) we have which implies F x ≺ F y.
Suppose that {x n } is a convergent sequence in C([a, b]) with limit x ∈ C ([a, b]) and x n ≺ x n+1 for all n. Then x n (s) x n+1 (s) for all n and for all s ∈ [a, b], which implies that x n (s) x(s) for all n and for all s ∈ [a, b], that is, x n ≺ x for all n. It follows that if {x n } is a convergent sequence in C([a, b]) with limit x ∈ C([a, b]) and γ(x n , x n+1 ) = 1, then γ(x n , x) = 1 for all n. Therefore, Z has γ-regular property, that is, property (A1) holds.
C ([a, b]) being complete, is a γ-complete metric space (see Remark 1). All the assumptions of Theorem 1 are satisfied. Therefore, F has a fixed point, that is, the integral equation (19) has a solution in Z.
Theorem 7. In addition to the hypothesis of Theorem 6, suppose that assumption (I6) holds. Then nonlinear integral equation (19) has a unique solution.
Proof. First, we show that Z has γ-triangular property. Let x, y, z ∈ Z and γ(x, y) 1 and γ(y, z) 1. By definition of γ we have x ≺ y and y ≺ z, that is, x(t) y(t) and Hence, Z has γ-triangular property. Therefore, property (A2) holds.
All the assumptions of Theorem 2 are satisfied. Thus, by Theorems 2 and 6 F has a unique fixed point, that is, the nonlinear integral equation (19) has a unique solution in C([a, b]).
Being motivated by Definition 9, we give definitions of Ulam-Hyers stability for the case of integral equation (19).
there exists a solution x * ∈ C([a, b]) of the integral equation (19) such that Let us consider the following the integral inequality: In the following next theorem, we add a new condition to assure the Ulam-Hyers stability of the integral equation (19): (I7) For any solution x * of (19) and any solution u * of (24), one has x * ≺ u * .
Proof. By Theorem 7 the integral equation (19) has a unique solution x * . Hence, it is a unique fixed point of the function F : Z → Z defined by (20). Let u * is a solution of the integral inequation (24), hence, u * is a solution of d(x, F x) , and by (I7), x * ≺ u * . By the definition of γ, γ(x * , u * ) 1, that is, property (A4) holds. By application of Theorem 3 the fixed point problem x = F x is Ulam-Hyers stable. Therefore, the solution of the integral equation (19) is Ulam-Hyers stable, and Being motivated by Definition 10, we give definitions of well-posedness for the case of integral equation (19). x n (t) − g(t) − λ b a K(t, s)h s, x n (s) ds → 0, n → ∞.
In the following theorem, we add a new condition to assure the well-posedness for integral equation (19).
(I8) If x * is a solution of the integral equation (19) and {x n } is any sequence in a K(t, s)h(s, x n (s)) ds| → 0 as n → ∞, then x n ≺ x * for all n.
Theorem 9. Let all the hypothesis of Theorem 7 hold. Then the integral equation (19) has a unique solution x * . Also suppose that (I8) holds. Then Problem I is well-posed.
Proof. By Theorem 7 the integral equation (19) has a unique solution x * . Hence, it is a unique fixed point of the function F : Z → Z defined by (20). Let {x n } be a sequence in C([a, b]) such that sup t∈[a,b] |x n (t) − g(t) − λ b a K(t, s)h(s, x n (s)) ds| → 0 as n → ∞.
Then by assumption (I8) we have x n ≺ x * for all n. From definition of γ we have γ(x n , x * ) = 1 for all n, that is, property (A5) holds. By application of Theorem 4 the fixed point Problem P, that is, the problem x = F x, is well-posed. Therefore, Problem I is well-posed.
Being motivated by Definition 11, we give definitions of data dependence for the case of integral equation.
Definition 14. Let x * ∈ C([a, b]) be the unique solution of the integral equation (19) and u * be the solution of the integral equation x(t) = p(t) + λ Theorem 10. Let all the hypothesis of Theorem 7 hold and x * be the unique solution of the integral equation (19). Also suppose that if x be any solution of the integral equation where p(t) − g(t) ν. Then Proof. By Theorem 7 the integral equation (19) has a unique solution x * . Let us define a map T : Z → Z by T (x)(t) = p(t) + λ Since x is a solution of (25), it is a fixed point of the mapping T defined by (26). By the assumption of the theorem we have x(t) F (x)(t), t ∈ [a, b], which implies that https://www.journals.vu.lt/nonlinear-analysis x ≺ F x. Then γ(x, F x) = 1. Also, for any x ∈ C([a, b]), we have

Conclusion
The result of the Theorem 1 is also valid if we replace the constant k by a Mizoguchi-Takahashi function [7,14]. Here we have not proceeded with it but this can be taken up in a future work. Also the corresponding problem with multivalued mappings and possible applications to integral inclusion problems is supposed to be of considerable interest. One reason for it is that, in general, the fixed point sets of multivalued mappings are mathematically complicated in their structures. This can also be taken up in future works.