Fixed point results in b-metric spaces using families of control functions and their application to dynamic programming

Zoran Kadelburga,1, Hemant Kumar Nashineb,2,3, Saroj Kumar Padhan, G.V.V. Jagannadha Rao Faculty of Mathematics, University of Belgrade, Studentski trg. 16, 11000 Beograd, Serbia kadelbur@matf.bg.ac.rs Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA drhknashine@gmail.com Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Sambalpur, Odisha-768018, India skpadhan_math@vssut.ac.in; gvvjagan1@gmail.com


Introduction
Many researchers extended Banach contraction principle by considering more general contractive mappings on various distance spaces -b-metric spaces [5,8] are one of the examples.Alber and Guerre-Delabrière [3] were the first to introduce weak contractive conditions in the setup of Hilbert spaces.On the other hand, Khan, Swaleh and Sessa [9] introduced the concept of an altering distance function, which is a control function that alters the distance between two points in a metric space.After the appearance of Rhoades theorem (see [12]), many results have been obtained involving contractivity conditions in which some families of functions play a key role.In particular, Agarwal et al. [1] unified most of these results for mappings acting in metric spaces using certain families of altering distance functions.
The aim of the present manuscript is to study what kind of altering distance functions might we include in an efficient contractivity condition in the case when we are given mappings acting in a b-metric space.Moreover, the concept of α-admissibility is used, it was introduced by Samet et al. [13] and extended by Sintunavarat [14] as α-admissibility of type S. Using these concepts, we study sufficient conditions on the functions that appear in very complex contractivity conditions with α-admissibility of type S in the framework of b-metric spaces.
We furnish an illustrative example to demonstrate the validity of the hypotheses of our results and necessity of some assumptions.Our results generalize and improve several fixed point results in metric spaces and b-metric spaces.As an application, the existence of solution for functional equations arising in dynamic programming is discussed, followed by suitable examples.

Preliminaries
In this section, we will introduce some essential notations, definitions and preliminary results that will be used in the article.Throughout this paper, we denote by N, R + and R the sets of positive integers, nonnegative real numbers and real numbers, respectively.
Recall that a function Λ : R + → R + is called an altering distance function [9] if the following properties hold: Example 1.Let (X, d) be a metric space, and let the mapping d b : X × X → R + be defined by d b (x, y) = [d(x, y)] p for all x, y ∈ X, where p > 1 is a fixed real number.Then (X, d b ) is a b-metric space with coefficient s = 2 p−1 .The triangular inequality (B 3 ) can easily be checked using the convexity of function R + x → x p .
The concepts of b-convergent sequence, b-Cauchy sequence, b-continuity and b-completeness in b-metric spaces are introduced in the same way as in metric spaces (see, e.g., [7]).In particular, a function f : Each b-convergent sequence in a b-metric space has a unique limit, and it is also a b-Cauchy sequence.However, a b-metric itself might not be continuous.Hence, the following lemma about b-convergent sequences is required in the proof of our main results.
Lemma 1. (See [2].)Let (X, d) be a b-metric space with coefficient s 1, and let {x n } and {y n } be b-convergent to points x, y ∈ X, respectively.Then The concept of α-admissibility was first introduced by Samet et al. [13] and extended as admissibility of type S by Sintunavarat [14] in the framework of metric spaces and b-metric spaces, respectively.Definition 2. (See [13,14].)Let X be a nonempty set, let α : X × X → R + and f : X → X be two mappings, and s 1 be a given real number.Then we say that this is denoted as f ∈ WP(X, α); this is denoted as f ∈ WP s (X, α).
2. α-admissibility of type S ⇒ weak α-admissibility of type S, that is, None of these inclusions can be reversed.Moreover, P(X, α) = P s (X, α), that is, the classes of α-admissible mappings and α-admissible mappings of type S are, in general, independent.

Main results
Before discussing our main result, we will introduce the following four families of functions.They are defined in a similar way as in the paper [1], but adapted for the use in b-metric spaces.In what follows, s 1 will be a given real number.
We will present fixed point results for mappings belonging to the class WP s (X, α).Throughout this paper, Fix(f ) denotes the set of all fixed points of a self-mapping f on a nonempty set X, that is, Fix(f ) = {x ∈ X: f x = x}.Also, for all elements x and y in a b-metric space (X, d) with coefficient s 1 and the given functions be a given mapping, and let Λ 1 , Λ 2 : R + → R + be two altering distance functions.We say that a mapping f : X → X is an (α, Λ 1 , Λ 2 ) s -contraction mapping if the following condition holds: x, y ∈ X with α(x, y) s for some functions R + → R + be two altering distance functions, and let α : X × X → R + and f : X → X be given mappings.Suppose that the following conditions hold: ) α has a transitive property of type S, that is, for x, y, z ∈ X, α(x, y) s and α(y, z) s =⇒ α(x, z) s; Then Fix(f ) = ∅.
Nonlinear Anal.Model.Control, 22(5):719-737 Proof.By the given condition (S 2 ) there exists x 0 ∈ X such that α(x 0 , f x 0 ) s. Define the sequence {x n } by x n+1 = f x n for all n ∈ N ∪ {0}.If there is n ∈ N ∪ {0} so that x n = x n+1 , then we have x n ∈ Fix(f ), and hence, the conclusion holds.So we assume that x n = x n+1 for all n ∈ N ∪ {0}.It follows that . Now, we need to prove that It follows by induction from f ∈ WP s (X, α) and α(x 0 , f x 0 ) s that for all n ∈ N ∪ {0}.Note that for each n ∈ N ∪ {0}, we have ) for all n ∈ N 0 .Taking into account the properties of considered functions in ) for all n ∈ N ∪ {0}.From (4) we have for all n ∈ N ∪ {0}.Since Λ 1 is a nondecreasing mapping, {d(x n , x n+1 )} is a decreasing sequence in R. Since {d(x n , x n+1 )} is bounded from below, there exists r 0 such that Letting n → ∞ in (5), we get This is only possible if Λ 2 (r) = 0 and thus r = 0. Hence, (2) is proved.
Next, we prove that {x n } is a b-Cauchy sequence in X. Assume to the contrary that there exists > 0 for which we can find subsequences {x m(k) } and for all k ∈ N. By (B 3 ) and ( 6) we get Taking the upper limit in (7) as k → ∞ and using (2), we get Also from (B 3 ) we obtain and Taking the upper limit as k → ∞ in ( 9) and (10), from ( 2) and (8) we get i.e., Similarly, we can show that Finally, we obtain ) .
By taking the upper limit as k → ∞ in the above inequality, we have Using (B 3 ) again, we have Taking the upper limit as k → ∞ in (14), from ( 2) and (8) we get From ( 13) and (15) we get https://www.mii.vu.lt/NAUsing the transitivity property of type S of α, we get α( where Taking the upper limit as k → ∞ in the above inequality and using ( 2), ( 8), ( 11) and ( 12), we get Nonlinear Anal.Model.Control, 22(5):719-737 Therefore, Similarly, we can show that Taking the upper limit as k → ∞ in (16), we have This implies that Λ 2 ( ) = 0 and then = 0, which is a contradiction.Therefore, {x n } is a b-Cauchy sequence.By b-completeness of the b-metric space (X, d) there exists x ∈ X such that x n → x as n → ∞.By b-continuity of f we get x n+1 = f x n → f x as n → ∞, and since the limit of a sequence is unique, we deduce that f x = x.This shows that Fix(f ) = ∅.Now we present a result that does not use continuity of the given mapping.Theorem 2. Let (X, d) be a b-complete b-metric space with coefficient s 1, let Λ 1 , Λ 2 : R + → R + be two altering distance functions, and let α : X × X → R + and f : X → X be given mappings.Suppose that the following conditions hold: for all n ∈ N and x n → x ∈ X as n → ∞, then α(x n , x) s for all n ∈ N.
https://www.mii.vu.lt/NAProof.As in the proof of Theorem 1, we obtain a b-Cauchy sequence {x n } in the b-complete b-metric space (X, d) satisfying α(x n , x n+1 ) s for all n ∈ N. Hence, there exists x ∈ X such that lim By the α s -regularity of X we have α(x n , x) s for all n ∈ N. where Taking the upper limit as n → ∞ in (18) and using Lemma 1, we get which implies that Λ 2 (d(x, f x)) = 0.It follows that d(x, f x) = 0, equivalently, f x = x and thus Fix(f ) = ∅.This completes the proof.
The following example will demonstrate the use of our results.

This implies that (1) holds and thus
Also, we can easily see that (X, d) is α s -regular, and there is Therefore, all the conditions of Theorem 2 are satisfied.Then we can conclude that Fix(f ) = ∅ (indeed, 0 ∈ Fix(f )).
Observe that, for x, y > 1, condition (1) might not hold; hence, using of the function α, is necessary.
Finally, we use Remark 1 to establish the following results for the class P s (X, α).Corollary 1.Let (X, d) be a b-complete b-metric space with coefficient s 1, let Λ 1 , Λ 2 : R + → R + be altering distance functions, and let α : X × X → R + and f : X → X be given mappings.Suppose that the following conditions hold: Corollary 2. Let (X, d) be a b-complete b-metric space with coefficient s 1, let Λ 1 , Λ 2 : R + → R + be two altering distance functions, and let α : X × X → R + and f : X → X be given mappings.Suppose that the following conditions hold: Now by hypothesis (D 3 ), for all x ∈ W and all y ∈ D, g(x, y) + G x, y, u τ (x, y) g(x, y) + G x, y, u τ (x, y) g(x, y) + G x, y, u τ (x, y) − G x, y, 0 + G(x, y, 0) As a result, for all x ∈ W , we have that This implies that f u is a bounded function on W , that is, the operator f is well defined.Define a function α : where s = 2 p−1 and η ∈ (0, s).It is easy to see that α has a transitive property.It follows from (D 1 ) that f ∈ P s (X , α) and so f ∈ WP s (X , α).From (D 2 ) and ( 21) we get that α(u 0 , f u 0 ) s for some u 0 ∈ X .To prove condition (S 4 ) in Theorem 2, let {u n } be an increasing sequence in X ; then by (21), α(u n , u n+1 ) s for all n ∈ N. If u n → u ∈ X as n → ∞, then we get in a standard way that u n (x) u(x) for any x ∈ W .Therefore, by (21), α(u n , u) s for all n ∈ N. Thus, condition (S 4 ) holds.
Next, we show that f ∈ ∆ s (X, α, Λ 1 , Λ 2 ).Let u, v ∈ X be such that α(u, v) s, that is, u(x) v(x) for all x ∈ W ; then from (D 4 ) we have Notice that the last inequality does not depend on x ∈ W , and therefore we obtain Nonlinear Anal.Model.Control, 22(5):719-737 Thus, all the conditions of Theorem 2 are fulfilled, and there exists a fixed point of f , i.e., a bounded solution u * ∈ X such that f u * = u * .In other words, for all x ∈ W , This completes the proof.
We state the following consequence of Theorem 3.

Conclusion
Condition (1) considered in this paper is a generalized weakly contraction condition that includes several types of conditions based on various forms of control functions, and the obtained fixed point results include several results known thus far.In particular, it is shown that the term (d(x, f y) + d(f x, y))/(2s) usually appearing in contraction conditions can be replaced by a more general term φ 1 (d(x, f y), d(f x, y)) Also, our results extend Alber and Guerre-Delabrière [3], Rhoades [12] and Agarwal et al. [1] fixed point results from metric to the setup of b-metric spaces.Furthermore, as it has been observed in some studies, fixed point results in b-metric spaces endowed with partial order, graph, binary relation or cyclic mappings can be derived from results under some suitable (weak) α-admissible conditions of type S. We have applied our results to get existence of solution for functional equations arising in dynamic programming.
f u(x) − f v(x) p = sup y∈D g(x, y) + G x, y, u τ (x, y) − sup y∈D g(x, y) + G x, y, v τ (x, y) p sup y∈D G x, y, u τ (x, y) − G x, y, v τ (x, y) variable and satisfies G(x, y, t) − G(x, y, r) 2 2/p 16 |t − r| for some p > 1 and all x ∈ W , y ∈ D and t, r ∈ R. Then the functional equation (19) has a solution u * ∈ B(W ).