The Stability and Well-Posedness of Fixed Points for Relation-Theoretic Multi-Valued Maps

: The purpose of this study is to present ﬁxed-point results for Suzuki-type multi-valued maps using relation theory. We examine a range of implications that arise from our primary discovery. Furthermore, we present two substantial cases that illustrate the importance of our main theorem. In addition, we examine the stability of ﬁxed-point sets for multi-valued maps and the concept of well-posedness. We present an application to a speciﬁc functional equation which arises in dynamic programming


Introduction
Mathematical analysis has witnessed a significant surge in interest regarding the examination of fixed-point outcomes for diverse maps in recent times.The Banach contraction principle (BCP) is a fundamental theorem in classical mathematics.Drawing upon this initial framework, other scholars have expanded and broadened the concept of the BCP to incorporate a wide range of circumstances and maps (see [1][2][3][4][5][6][7][8]).
Suzuki's [9] generalization of the BCP introduced a new class of contractive maps that satisfy contraction conditions only for specific elements of the underlying space.Subsequently, Alam and Imdad [10] expanded the boundaries of the BCP by considering a complete metric space (CMS) equipped with a binary relation.They introduced the concept of relation-theoretic contraction, which applies to elements related under the binary relation rather than the entire space.Other researchers, such as Song-il Ri [11], further extended the BCP for a new class of contractive maps.
Motivated by the works of Alam and Imdad [10], Kikkawa and Suzuki [15], and others, we present some new fixed-point results for multi-valued maps in relational metric spaces.These results extend and generalize the findings from previous studies by Alam and Imdad [10], Ciric [13], Kikkawa and Suzuki [15], Nadler [12], and others.Furthermore, the paper provides illustrative examples to support these findings and explores the stability of fixed-point sets for multi-valued maps within the framework of relational metric spaces.Lastly, by applying the presented results, the paper establishes the existence and uniqueness of solutions for a class of functional equations arising in dynamic programming.

Preliminaries
In this section, we recapitulate relevant notation, definitions, and results from the literature [12,13,18].Throughout this paper, we denote a metric space (MS) as (L, γ), where L is a set and γ is a metric on L. We use CB(L) to represent the collection of all nonempty closed and bounded subsets of L, and C(L) to denote the collection of all nonempty compact subsets of L. The Hausdorff metric Γ H induced by γ is and it is a strict fixed point of F if {ϑ} = F ϑ. We denote the sets of fixed points and strict fixed points of F as F(F ) and SF(F ), respectively.Theorem 1 ([12]).Consider a CMS (L, γ) and a multi-valued map where κ ∈ [0, 1), then F possesses a fixed point.
Definition 6 ([19]).Given a binary relation ℵ defined on a nonempty set L, the image of an element a ∈ L under the relation ℵ is denoted as Im(a, ℵ) and is defined as {ω ∈ L : (a, ω) ∈ ℵ or ω = a}.
Suppose lim η→∞ γ η = γ > 0 and γ η = γ + ξ η with ξ η > 0. Since for all t > 0, lim sup for (t η ) with t η ↓ γ + , we have lim sup Therefore, for any ε > 0 there exists κ ∈ N such that Assume that (ω η ) is not a Cauchy sequence in L.Then, for each positive integer κ, there exists an ε > 0 and sequences of positive integers {m κ }, {η κ } such that κ ≤ m κ < η κ and the following assertions hold: Without loss of generality, we may assume that η κ is the smallest integer greater than m κ satisfying the inequality (8) and Then, by triangle inequality and using inequality (9), we have Making κ → ∞ and using (6), we obtain From ( 7) and ( 8), we have Then, from condition (4) and by triangle inequality, we have Making κ → ∞ and using (6) and (10), we obtain is lower semi-continuous at the point ϑ, then we have The closedness of F ϑ implies ϑ ∈ F ϑ.
On the other hand, if hypothesis (d) holds, then the sequence {ω η } has a subsequence for κ ∈ N. By inference and contradiction, we assume that for each η ∈ N. As a result of the triangle inequality, we have This contradicts itself.The inequality ( 12) is valid for η ∈ N. Since the first scenario, by ( 4), we have We obtain by adding κ → ∞, Then, by lim sup we obtain Therefore, unless Γ = 0 or Γ(ϑ, F ϑ) = 0, is a contradiction.This suggests that ϑ ∈ F ϑ.In the other scenario, we can conclude that ϑ ∈ F ϑ.
Considering F := f as a single-valued map, we obtain the following result: Theorem 5. Let (L, γ) be a CMS and ℵ be a binary relation on L. If f : L → L is a map and the following conditions are satisfied: , then, f has a fixed point.
If we assume ℵ := L × L as a universal relation on L, then we obtain the following result: Theorem 6.Let (L, γ) be a CMS and F : L → CB(L) a multi-valued map such that for any ω ∈ L, ∈ F ω, where m(ω, ) is as in Theorem 2 and ϕ is as in Definition 2, then F has a fixed point in L.
Corollary 1.Let (L, γ) be a CMS endowed with a binary relation ℵ on L. If F : L → CB(L) is a multi-valued map and satisfying the following conditions: then F has a fixed point.
Corollary 2. Let (L, γ) be a CMS endowed with a binary relation ℵ on L. If F : L → CB(L) is a multi-valued map and satisfying the following conditions: then F has a fixed point.
However, for ω = p and ϕ = s, the mapping F does not satisfy contraction conditions (1), (2), and . Consequently, Theorems 1-3 cannot be applied to this particular example.
We consider the followings cases.
Case 1: ω, < 0.Then, Thus, in all the cases, Γ H (F ω, F ) ≤ ϕ(m(ω, )), and (4) is satisfied.Since under universal relation that is, ℵ = L × L, conditions (b) and (d) both are obviously true.Thus, all the assertions of Theorem 4 are fulfilled, leading to the conclusion that 0 ∈ F (0) ⊂ L is a fixed point for the map F .

Stability of Fixed-Point Sets and Well-Posedness
The stability of fixed points is concerned with understanding whether small deviations from a fixed point will lead the system's solutions to stay close to the fixed point or diverge away from it.This topic has been explored in various works; see [4][5][6]14,16,[22][23][24][25][26][27].Here, we delve into the stability of fixed-point sets for multi-valued maps.Our exploration begins with the following lemma.Lemma 1 ([12]).In an MS (L, γ), for every ω ∈ L ∃ ∈ B ∈ C(L) such that γ(ω, ) = Γ(ω, B).
Theorem 7. Let ℵ be a binary relation on a CMS (L, γ) and F j : L → C(L) (j ∈ {1, 2}) are two multi-valued maps satisfying all the assumptions of Theorem 4 with Proof.The validity of Theorem 4 guarantees the existence of nonempty fixed-point sets F(F j ) = ∅ for j ∈ {1, 2}, satisfying condition (a).Moving on, let us assume . Continuing this iteration and following the proof strategy of Theorem 4, we generate an ℵ-preserving sequence {ϑ η } that fulfills Now, as we follow the proof of Theorem 4, it becomes evident that the sequence {ϑ η } is an ℵ-preserving Cauchy sequence.Thus, it inevitably converges to a point w ∈ L. Furthermore, it can be established that w is a fixed point of F 2 since Now, using the definition of L, we obtain With the triangle inequality and Equation ( 14), we obtain Taking the limit as η → ∞ and utilizing Equation ( 15), we derive Consequently, given ϑ 1 ∈ F(F 1 ), we find w ∈ F(F 2 ) satisfying γ(ϑ 1 , w) ≤ Ψ(L).Similarly, it can be proven that for any This concludes the proof of condition (b).
Lemma 2. Assume that (L, γ) is a CMS, ℵ is a binary relation on L, and F η : L → CB(L) (η ∈ N) is a sequence of multi-valued maps.If (F η ) converges uniformly to F : L → CB(L) for each η ∈ N and F η satisfies all the conditions of Theorem 4, then F also satisfies (4) and has a fixed point in L.
Proof.Let ω ∈ L and ∈ F ω be such that (ω, ) ∈ ℵ.Since each F η satisfies (4), we have for all ω ∈ L, ∈ F ω with (ω, ) ∈ ℵ, where By letting η → ∞ while maintaining uniform convergence, and following a similar argument as in the proof of Theorem 4, we conclude that for all ω ∈ L, ∈ F ω with (ω, ) ∈ ℵ, where m(ω, ) is as defined in Theorem 4. This implies that F satisfies (4).Since L is complete and F satisfies (4), F has a fixed point in L. From Theorem 7, we have Therefore, sets of fixed points of F η are stable.Now, we show that the fixed-point problem (fpp) is well-posed.We begin with the following definitions.
Definition 7. Assume that (L, γ) is an MS, ℵ is a binary relation, and F : L → CB(L) is a multi-valued map.We say fpp is well-posed for F with respect to Γ if Definition 8. Assume that (L, γ) is an MS, ℵ is a binary relation, and F : L → CB(L) is a multi-valued map.We say fpp is well-posed for F with respect to Γ H if Notice that when F(F ) = SF(F ) and fpp is well-posed for F with respect to Γ, then it is well-posed with respect to Γ H . Theorem 9. Let all the conditions of Corollary 1 be true along with assertions (i) SF(F ) = φ and (ii) all fixed points of F are comparative.Then, Proof.(a) Let u ∈ SF(F ) and ϑ ∈ F(F ) such that u = ϑ.This leads to 0 = 1 2 Γ(u, F u) < γ(u, ϑ).As all fixed points of F are comparative, so we have (u, ϑ) ∈ ℵ.Using (4) we find For η κ ≥ η 0 , we have 1 2 Γ(ω η κ , F ω η κ ) < ε < γ(ω η κ , ϑ).Utilizing (4), we obtain Taking κ → ∞ and using the properties of ϕ, we derive which is a contradiction.Therefore, lim η→∞ γ(ω η , ϑ) = 0, and the fpp is well-posed for F concerning Γ H .

An Application to Dynamic Programming
In the context of this section, we consider Banach spaces Ξ and Λ, with Π ⊂ Ξ and E ⊂ Λ, while R denotes the field of real numbers.We work with maps τ : Π × E → Π, f : Π × E → R, F : Π × E × R → R, and utilize the set B(Π) to represent all bounded real-valued functions on Π.
Our focus in this section is on investigating the existence and uniqueness of a solution for the functional equation where f and F are bounded functions, ω and symbolize the state and decision vectors, respectively, τ denotes the process transformation, and p(ω) signifies the optimal return function given an initial state ω.
To facilitate our analysis, we introduce a map F : B(Π) → B(Π), defined as: Our aim is to establish the existence and uniqueness of a solution for the functional Equation ( 16) using the framework provided by Theorem 4.
Then, the functional Equation ( 16) has a bounded solution in B(Π).
Therefore, all the conditions of Theorem 5 are fulfilled for the map F .As a result, the map F possesses a fixed point denoted as h(ω), signifying that h(ω) is a bounded solution for the functional Equation ( 16).