Some fixed point results on ultrametric spaces endowed with a graph

Özlem Acar; Aybala Sevde Özkapu; Mahpeyker Öztürk

doi:10.1515/dema-2023-0132

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Some fixed point results on ultrametric spaces endowed with a graph

Özlem Acar , Aybala Sevde Özkapu and Mahpeyker Öztürk

Published/Copyright: February 5, 2024

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From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The present article deals with new fixed point theorems by means of G -strongly contractive maps. The research findings are demonstrated in a spherically complete ultrametric space with a graph for single-valued mappings. The special cases of the results that extend the current ones are offered, along with some examples that illustrate our results. Besides, an application utilized in dynamic programming that endorses the acquired observations is also provided.

Keywords: dynamic programming; fixed point; F-contraction; functional equation; ultrametric space

MSC 2010: 54H25; 47H10

1 Introduction

Fixed point theory is an active and important branch of mathematics, and this theory is a powerful tool for investigating solutions to mathematical problems with various types of applications. Because of its simple application to several disciplines of mathematics, Banach [1] proposed the Banach fixed point theorem a century ago, which is a crucial source for the development of metric fixed point theory. Accordingly, there has been and continues to be a great deal of interest and demand for this hypothesis. The Banach fixed point theorem not only guarantees the existence and uniqueness of a fixed point of contraction self-mapping but also gives an effective approach to find the fixed point and can be expressed as “the contraction self-mapping T on a complete metric space ( X , d ) , i.e., for all x , y ∈ X , the inequality

(1) d ( T x , T y ) ≤ μ d ( x , y ) , where μ ∈ ( 0 , 1 )

is satisfied, then T owns a unique fixed point, and for every x 0 ∈ X , the sequence { T n x 0 } n ∈ N converges to this fixed point.”

When investigating metric fixed point theory, generalizing the contraction mapping is a particularly interesting area of research. Many generalizations in this sense are available in the literature, and the main ones are [2–4]. Aside from such mappings, fixed point theorems employing contraction mappings established with auxiliary functions have been among the theory’s building blocks, and new functions are continuously being defined nowadays. For more details, see [5–10].

The study of the fixed point occupies a prominent role in many aspects of a metric space equipped with a graph structure. Echenique [11] implemented graphs to demonstrate Tarski’s fixed point theorem, integrating fixed point theory with graph theory. Espinola and Kirk [12] then utilized the fixed point findings in graph theory. Two key findings for fixed point theory with a graph have recently been presented. The first result was given by Jachymski [13] for single-valued mappings, and subsequently, Beg et al. [14] extended Jachymski’s result for set-valued mappings. Following that, Sultana and Vetrivel established a fixed point theorem for Mizoguchi-Takahashi contraction in [15], while Sistani and Kazemipour [16] proved several theorems for ( α , φ ) -contractions in these directions. One can also see [17,18,38,39].

Specifically, we concentrate on one of the generalized metric spaces, the ultrametric space theory, an exciting field of research that has emerged in mathematics in the last 35 years. The notion of ultrametric space arose in many different works [19–24]. On the other hand, Gajić [25], considering the spherically completeness of a given ultrametric space, obtained a fixed point theorem with uniqueness for the mappings where the distance between T x and T y satisfies the following inequality:

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } for all x , y ∈ X , x ≠ y .

We must mention that Gajić [26] extended this result for multivalued maps.

Another important study in fixed point theory is determining coincidence point results for single-valued and set-valued mappings. The analog of such results in ultrametric spaces was obtained by Rao et al. [27] for coincidence point theorems from single-valued maps to set-valued contractive maps. In addition, Zhang and Song [28] put forward some results for weak contraction in ultrametric space.

This study seeks to provide fixed point theorems for single-valued mappings on a spherical complete ultrametric space equipped with a graph using the principles outlined above. An application in dynamic programming is provided to convey the validity of the results. The outcomes of this study generalize, extend, improve, and unify several known results in ultrametric space settings.

2 Preliminaries

We present some fundamental definitions and outcomes that will be utilized throughout the study.

2.1 Some aspects of ultrametric spaces

This subsection explains fundamental concepts and terminologies related to ultrametric spaces.

Definition 2.1

[25] Let ( X , d ) be a metric space. If the metric d fulfils the ensuing inequality such that

d ( x , y ) ≤ max { d ( x , ω ) , d ( ω , y ) }

for all x , y , ω ∈ X , then d is called an ultrametric in X , and the pair ( X , d ) is an ultrametric space.

Example 2.2

[25] Suppose that d is a discrete metric on X ( X ≠ ∅ ) . Then, d is ultrametric on X .

Example 2.3

[25] Consider [ x ] is the entire part of x for x ∈ R . For any e ∈ R \ Q and for all x , y ∈ Q ,

d ( x , y ) = inf { 2 − n : n ∈ Z , [ 2 n ( x − e ) ] = [ 2 y ( y − e ) ] }

is ultrametric on Q .

Example 2.4

[29] Consider that the family of all sequences of non-negative integers is represented by X . For x = { x n } n ∈ N , y = { y n } n ∈ N ∈ X , set

u ( x , y ) = inf { n ∈ N : x n ≠ y n }

and

d ( x , y ) = 0 , if x = y 1 u ( x , y ) , if x ≠ y .

Then, the pair ( X , d ) is a complete ultrametric space.

Remark 1

[25] Every ultrametric space is a metric space; however, the converse need not be true. For instance, ( R , d ) is a usual space and is not an ultrametric space.

Definition 2.5

[30] Let ( X , d ) be an ultrametric space, x ∈ X and r > 0 . Then,

B ( x , r ) = { y ∈ X : d ( x , y ) ≤ r } ,

with B ( x , 0 ) = { x } , is called ball. It is denoted by B ( x , r ) .

Remark 2

[30] A well-known characteristic property of an ultrametric space is the following:

if x , y ∈ X , 0 ≤ r ≤ ε and B ( x , r ) ∩ B ( x , ε ) ≠ ∅ , then B ( x , r ) ⊂ B ( x , ε ) .

Definition 2.6

[30] Let ( X , d ) be an ultrametric space. If every shrinking collection of balls in X has a nonempty intersection, then an ultrametric space ( X , d ) is said to be spherically complete.

Definition 2.7

[31] Let X be a nonempty set. A pseudo metric on X is a function:

d : X × X → R + ∪ { 0 }

satisfying the following conditions for any x , y , ω ∈ X :

d ( x , y ) = 0 ⇒ x = y ,
d ( x , y ) = d ( y , x ) ,
d ( x , y ) ≤ d ( x , ω ) + d ( ω , y ) .

Definition 2.8

[32] A pseudo metric on X is an ultra pseudo metric if, in addition to conditions given in Definition 2.7, for any x , y , ω ∈ X ,

d ( x , y ) ≤ max { d ( x , ω ) , d ( ω , y ) } .

Proposition 2.9

[32] A pseudo metric ( X , d ) is an ultra pseudo metric if and only if, for any three points x , y , ω ∈ X , one of the following conditions is satisfied:

d ( ω , y ) ≤ d ( x , y ) = d ( x , ω ) or
d ( x , ω ) ≤ d ( y , x ) = d ( y , ω ) or
d ( x , y ) ≤ d ( ω , x ) = d ( ω , y ) .

Remark 3

Every spherically complete ultrametric space is a complete metric space. The converse is not true in general. As an example, we mention the completion of C p of the algebraic closure of the field of rational p -adic numbers. According to Krasner, this field has nice algebraic properties because it is algebraically closed and even isomorphic to complex numbers (cf. [33], pp. 134–145), but it has also been shown that C p is not spherically complete. This is mainly due to the fact that the complex p -adic numbers are a separable, complete ultrametric space with dense valuation (cf. [33], pp. 143–144).

2.2 Fundamentals of graph theory

Let X be a nonempty set and Δ denotes the diagonal of cartesian product X × X . A graph on X is an object G = ( V ( G ) , E ( G ) ) , where V ( G ) is a vertex set whose elements are called vertices, and E ( G ) is an edge set. We assume that G has no parallel edges and Δ ⊂ E ( G ) .

If x and y are vertices of G , then a path in G from x to y of length k ∈ N is a finite sequence { x n } n ∈ { 0 , 1 , 2 , … , k } of vertices such that

x 0 = x , x k = y and ( x i − 1 , x i ) ∈ E ( G ) for i ∈ { 1 , 2 , … , k } .

Note that a graph G is connected if there is a path between any two vertices, and it is weakly connected if G ˜ is connected, where G ˜ denotes the undirected graph obtained from G by ignoring the direction of edges.

Denote by G − 1 the graph obtained from G by reversing the direction of edges. Thus,

E ( G − 1 ) = { ( x , y ) } ∈ X × X : ( y , x ) ∈ E ( G ) .

As it is more convenient to treat G ˜ as a directed graph for which the set of its edges is symmetric, under this convention, we have

E ( G ˜ ) = E ( G ) ∪ E ( G − 1 ) .

The pair ( V ′ , E ′ ) is a subgraph of G if V ′ ⊆ V ( G ) and E ′ ⊆ E ( G ) , and for any edge ( x , y ) ∈ E ′ , x , y ∈ V ′ .

If G is symmetric and x is a vertex in G , then the subgraph G x consisting of all edges and vertices that are contained in some path beginning at x is called the component of G containing x . In this case, V ( G x ) = [ x ] G , where [ x ] G is the equivalence class of the following relation ℛ defined on V ( G ) by the rule:

y ℛ z if there is a path in G from y to z .

For detailed information about graph theory, see [34,40].

3 Main results

In this section, initially, we define a generalized G -strongly contractive map and then establish a fixed point theorem for this type of mapping. Besides, an illustrative example supports the validity of the main result.

Definition 3.1

Let ( X , d ) be an ultrametric space endowed with a graph G . A self-mapping T on X is a generalized G - strongly contractive mapping if

T preserves the edges of G , that is, ( x , y ) ∈ E ( G ) implies ( T x , T y ) ∈ E ( G ) for all x , y ∈ X ,
for all x , y ∈ X , x ≠ y with ( x , y ) ∈ E ( G ) ,
(1) d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } .

Example 3.2

d is ultrametric on N , and we defined d : N × N → [ 0 , + ∞ ) by

d ( x , y ) = 0 , x = y max 1 + 1 x , 1 + 1 y , x ≠ y .

Let T : N → N be a mapping such that

T x = x + k , x odd 1 , x even ,

where k ∈ Z + . Then, the conditions of Definition 3.1 hold. Indeed,

Case I . x and y are any odd integers ( x ≠ y , x < y ) ;

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max 1 + 1 x , 1 + 1 y = 1 + 1 x .

Case I I . x and y are any even integers ( x ≠ y , x < y ) ;

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max 1 + 1 x , 2 = 1 + 1 x .

Case I I I . x is any odd integer and y is any even integer ( x ≠ y , x < y ) ;

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max 1 + 1 x , 1 + 1 y = 1 + 1 x .

Let us recall the notion of the G-T-invariant before proceeding with our main theorem.

Definition 3.3

[35] Assume that ( X , d ) is an ultrametric space endowed with a graph G and a self-mapping T on X . We say that a B ( x , r ) is G-T-invariant if for any u ∈ B ( x , r ) such that ( u , x ) ∈ E ( G ) , then T u ∈ B ( x , r ) .

Theorem 3.4

Let ( X , d ) be an ultrametric space endowed with a graph G and a G-strongly contractive self-mapping T on X fulfills the subsequent statements:

there exists an x 0 ∈ X such that d ( x 0 , T x 0 ) < 1 ,
if x ∈ X is such that d ( x , T x ) < 1 , then there exists a path in G ¯ between x and Tx with vertices in B ( x , d ( x , T x ) ) ,
if B ( x n , d ( x n , T x n ) ) is a sequence of nonincreasing closed balls in X , and for each n ≥ 1 , there exists a path in G ¯ between x n and x n + 1 with vertices in B ( x n , d ( x n , T x n ) ) , then there exists a subsequence { x n k } k = 1 ∞ of { x n } n = 1 ∞ and a ź ∈ ∩ k = 1 ∞ B ( x n k , r n k ) such that for each k ≥ 1 , there exists a path in G ¯ between x n k and ź with vertices in B ( x n k , d ( x n k , T x n k ) ) .

In that case, T owns a fixed point in each closed ball of the form B ( x , d ( x , T x ) ) , where

x ∈ X 0 = { ź ∈ X : d ( ź , T ź ) < 1 } .

Proof

Presume that x ∈ X 0 , r = d ( x , T x ) , and u ∈ B ( x , r ) such that ( u , x ) ∈ E ( G ) . Then,

d ( u , x ) ≤ d ( u , T u ) ≤ max { d ( u , x ) , d ( x , T x ) , d ( T x , T u ) } < max d ( u , x ) , d ( x , T x ) , d ( u , x ) , d ( x , T x ) , d ( u , T u ) , d ( x , T u ) , d ( u , T x ) < max { d ( u , x ) , d ( x , T x ) } = d ( x , T x ) .

In addition,

B ( x , d ( x , T x ) ) ∩ B ( u , d ( u , T u ) ) ≠ ∅ .

Thus,

B ( u , d ( u , T u ) ) ⊂ B ( x , d ( x , T x ) ) ,

so T u ∈ B ( x , d ( x , T x ) ) . This means that B ( x , d ( x , T x ) ) is G-T-invariant for all x ∈ X 0 . Presume that x 0 ∈ X 0 is a fixed element. Put x 1 = x 0 , r 1 = d ( x 1 , T x 1 ) ; if r 1 = 0 , then x 1 is a fixed point of T such that x 1 = T x 1 and the proof is completed. Otherwise, put

E 1 = x ∈ B ( x 1 , r 1 ) ∣ there is a path in G ¯ between x and x 1 with vertices in B ( x 1 , r 1 ) .

It is evident that x 1 and T x 1 are elements of E 1 . Set

μ 1 = inf { d ( x , T x ) : x ∈ E 1 } .

If r 1 = μ 1 , then x 1 is a fixed point of T ; otherwise, owing to the fact that d ( x 1 , T x 1 ) < 1 and ( i i ) , there exists a path ( x 1 = y 0 , … , y n = T x 1 ) in B ( x 1 , r 1 ) from x 1 to T x 1 . Since B ( x 1 , r 1 ) is G-T-invariant, it ensures that T 2 x 1 ∈ B ( x 1 , r 1 ) , and we attain

μ 1 ≤ d ( T x 1 , T 2 x 1 ) < max { d ( T x 1 , T y 1 ) , d ( T y 1 , T y 2 ) , … , d ( T y n − 1 , T 2 x 1 ) } < max { d ( x 1 , y 1 ) , d ( y 1 , y 2 ) , … , d ( y n − 1 , T x 1 ) } ≤ d ( x 1 , T x 1 ) = r 1 ,

which causes a contradiction. Hence, finally, let μ 1 < r 1 . Assume that { ε n } n ∈ N is a sequence of positive numbers such that lim n → ∞ ε n = 0 . Choose an element x 2 ∈ B ( x 1 , r 1 ) such that there exists a path in G ¯ between x 1 and x 2 and

r 2 = d ( x 2 , T x 2 ) < min { r 1 , μ 1 + ε 1 } .

In a similar way, if r 2 = μ 2 , then x 2 is a fixed point of T ; otherwise, x 3 ∈ B ( x 2 , r 2 ) exists such that there exists a path in G ¯ between x 2 and x 3 and

r 3 = d ( x 3 , T x 3 ) < min { r 2 , μ 2 + ε 2 } .

Define x n ∈ X . Let

E n = x ∈ B ( x n , r n ) ∣ there is a path in G ¯ between x and x n with vertices in B ( x n , r n )

and

μ n = inf { d ( x , T x ) : x ∈ E n } .

If r n = 0 or r n = μ n , using the same way for n = 1 , the proof is completed. Otherwise, choose element x n + 1 ∈ B ( x n , r n ) such that there exists a path between x n and x n + 1 and

r n + 1 = d ( x n + 1 , T x n + 1 ) < min { r n , μ n + ε n } .

If this processing ends after a finite number of steps, then the proof is completed. Otherwise, we obtain a nonincreasing sequence of nontrivial closed { B ( x n , d ( x n , T x n ) ) } . Because { r n } n ∈ N is nonincreasing, r = lim n → ∞ r n exists and { μ n } is nondecreasing and bounded above, thus μ = lim n → ∞ μ n . Therefore, by (iii), there exists a subsequence { x n k } k = 1 ∞ of { x n } n = 1 ∞ and ź ∈ ∩ k = 1 ∞ B ( x n k , r n k ) such that for each k ∈ N , a path exists in G ¯ between x n k and ź with vertices in B ( x n k , d ( x n k , T x n k ) ) . T ź ∈ B ( x n k , r n k ) for all k ≥ 1 because B ( x n k , r n k ) is G-T-invariant for all k ≥ 1 . Thus,

d ( ź , T ź ) < max { d ( ź , x n k ) , d ( x n k , T ź ) } ≤ r n k

for all k ≥ 1 . Therefore,

μ n k ≤ d ( ź , T ź ) ≤ r ≤ r n k + 1 ≤ μ n k + ε n k

for all k ≥ 1 . Letting k → ∞ , we achieve d ( ź , T ź ) = r = μ . Moreover, if x ∈ B ( ź , d ( ź , T ź ) ) , then for each k ∈ N ,

d ( x , ź ) ≤ d ( ź , T ź ) ≤ r n k

for all k ≥ 1 . Thus,

d ( x , x n k ) < max { d ( x , ź ) , d ( ź , x n k ) } ≤ r n k

for all k ≥ 1 . Consequently, x ∈ B ( x n k , r n k ) for all k ≥ 1 . Let x ∈ B ( ź , d ( ź , T ź ) ) , and there exists a path between x and ź . Therefore, a path exists in B ( x n k , r n k ) between x n k and x for all k ≥ 1 . In consequence, μ n k ≤ d ( x , T x ) for all k ≥ 1 . Thereby, for each k ∈ N , μ n k ≤ r n k . Thus,

inf { d ( x , T x ) : x ∈ B ( ź , d ( ź , T ź ) ) } = d ( ź , T ź ) = r .

For all k ≥ 1 , we have

d ( ź , T ź ) ≤ r n k < 1 ,

and thus, it follows that by (ii), there exists a path in B ( ź , d ( ź , T ź ) ) from ź to T ź . We assert that r = 0 . Assume on the contrary that r > 0 and assume ( ź = y 0 , y 1 , … , y N = T ź ) is a path in B ( ź , d ( ź , T ź ) ) between ź and T ź . T 2 ź ∈ B ( ź , d ( ź , T ź ) ) because B ( ź , d ( ź , T ź ) ) is G-T-invariant, and consequently, we obtain

d ( T ź , T 2 ź ) ≤ max { d ( T ź , T y 1 ) , … , d ( T y N − 1 , T 2 ź ) } < max { d ( ź , y 1 ) , … , d ( y N − 1 , T ź ) } ≤ d ( ź , T ź ) ,

which causes a contradiction. Therefore, r = 0 and ź = T ź . As a result, the proof is completed.□

Example 3.5

Let X = { 0 , 1 , 2 , 3 , … } and

d ( x , y ) = 0 , x = y x + y , x ≠ y .

Then, d be an ultra pseudo metric on X with Proposition 2.9 because the defined d is a pseudo metric on X . Consider the graph given by V ( G ) = X and E ( G ) = X × X ⧹ { ( 0 , 1 ) , ( 1 , 0 ) } . Let T : X → X mapping be defined as follows:

T x = { x } , x = 0 , x = 1 { 0 , 1 , 2 , … x − 1 } , x ≥ 2 .

The mapping T satisfies all the conditions of Theorem 3.4, and even when the graph structure on X is removed, T has no fixed point. Indeed,

Case I . Let y = 0 and x > 1 ,

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max { x , 2 x − 1 } = 2 x − 1 .

Case I I . Let y = 1 and x > 1 ,

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max { x + 1 , 2 x − 1 } .

Case I I I . Let x , y > 1 ,

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max { x + y − 1 , 2 x − 1 } = 2 x − 1 .

The fixed points of T are 0 and 1. On the other hand, if we assume that there is no graph on X , we see that the contraction condition is not satisfied. Indeed,

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } = max { 1 , 0 } = 1

is obtained for x = 0 and y = 1 .

Corollary 3.6

Theorem 3.4 holds if the condition 3.1 of Definition 3.1 is replaced by

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } ,

∀ x , y ∈ X , x ≠ y with ( x , y ) ∈ E ( G ) .

Corollary 3.7

Theorem 3.4 holds if the condition 3.1 of Definition 3.1 is replaced by

d ( T x , T y ) < d ( x , y )

∀ x , y ∈ X , x ≠ y with ( x , y ) ∈ E ( G ) .

Corollary 3.8

Let ( X , d ) be a spherically complete ultrametric space, and T is a self-mapping on X if T is a mapping such that

d ( T x , T y ) < max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }

∀ x , y ∈ X , x ≠ y . Then, T admits a unique fixed point in X .

3.1 An application to dynamic programming

In this section, we implement our findings to solve the subsequent functional equations applied in dynamic programming

(1) q ( x ) = max y ∈ D { f ( x , y ) + G ( x , y , q ( η ( x , y ) ) ) } , x ∈ W ,

where f : W × D → R and G : W × D × R → R are bounded, η : W × D → W and X and Y are Banach spaces such that W ⊂ X and D ⊂ Y .

Equations of the type ( 1 ) find their application in optimization theory, computer programming, and dynamic programming. As an application, dynamic programming has been researched by several researchers. Klim and Wardowski [36] extended their concept of F - contraction mapping to nonlinear F -contractions, proving a fixed point theorem through dynamic processes. Asif et al. [37], working on the common fixed point problem in F -metric spaces (F-MS), proved a unique common solution to the functional equations widely used in computer programming and optimization theory. Dynamic programming is organized into two sections: state space and decision space. Suppose that W and D are state space and decision space, respectively.

Let B ( W ) define the set of all bounded real-valued functions on W . The pair ( B ( W ) , ∥ ⋅ ∥ ) , where

∥ h ∥ = max x ∈ W ∣ h ( x ) ∣ , h ∈ B ( W ) ,

is a Banach space and d is the metric defined as follows:

d ( h , k ) = max x ∈ W ∣ h ( x ) − k ( x ) ∣ .

By Proposition 2.9, d ( h , k ) is an ultra pseudo metric.

We aim to indicate the existence of a solution of equation (1), and we consider the operator T : B ( W ) → B ( W ) as follows:

(2) ( T h ) ( x ) = max y ∈ D { f ( x , y ) + G ( x , y , h ( η ( x , y ) ) ) }

for all h ∈ B ( W ) and x ∈ W . It is evident that the operator T is well-defined since f and G are bounded.

We now establish the subsequent theorem.

Theorem 3.9

Let T : B ( W ) → B ( W ) be an operator defined by equation (2), and suppose that the following statements are met:

f and G are bounded,
∀ h , k ∈ B ( W ) , ∀ x ∈ W , ∀ y ∈ D ,
∣ G ( x , y , h ( x ) ) − G ( x , y , k ( x ) ) ∣ < M ( h ( x ) , k ( x ) ) ,
where
M ( h ( x ) , k ( x ) ) = max d ( h ( x ) , k ( x ) ) , d ( h ( x ) , T h ( x ) ) , d ( k ( x ) , T k ( x ) ) , d ( h ( x ) , T k ( x ) ) , d ( k ( x ) , T h ( x ) ) ,

then the functional equation (2) has a bounded solution.

Proof

Let λ ∈ R + be an arbitrary, x ∈ W , and h ∈ B ( W ) . Without loss of generality, we suppose that T h ≠ h . Then, y 1 , y 2 ∈ D exist such that

( T h ) ( x ) < f ( x , y 1 ) + G ( x , y 1 , h ( η ( x , y 1 ) ) ) + λ , ( T k ) ( x ) < f ( x , y 2 ) + G ( x , y 2 , k ( η ( x , y 2 ) ) ) + λ , ( T h ) ( x ) ≥ f ( x , y 2 ) + G ( x , y 2 , h ( η ( x , y 2 ) ) ) , ( T k ) ( x ) ≥ f ( x , y 1 ) + G ( x , y 1 , k ( η ( x , y 1 ) ) ) .

Then, we achieve

( T h ) ( x ) − ( T k ) ( x ) < G ( x , y 1 , h ( η ( x , y 1 ) ) ) − G ( x , y 1 , k ( η ( x , y 1 ) ) ) + λ ≤ ∣ G ( x , y 1 , h ( η ( x , y 1 ) ) ) − G ( x , y 1 , k ( η ( x , y 1 ) ) ) ∣ + λ < M ( h ( x ) , k ( x ) ) + λ

and

( T k ) ( x ) − ( T h ) ( x ) < G ( x , y 2 , k ( η ( x , y 2 ) ) ) − G ( x , y 2 , h ( η ( x , y 2 ) ) ) + λ ≤ ∣ G ( x , y 2 , k ( η ( x , y 2 ) ) ) − G ( x , y 2 , h ( η ( x , y 2 ) ) ) ∣ + λ < M ( h ( x ) , k ( x ) ) + λ

for all λ > 0 . Therefore,

∣ ( T h ) ( x ) − ( T k ) ( x ) ∣ < M ( h ( x ) , k ( x ) ) + λ

for all λ > 0 . Hence,

d ( ( T h ) ( x ) , ( T k ) ( x ) ) < M ( h ( x ) , k ( x ) ) .

Consequently, Corollary 3.8 implies the existence of a bounded solution of equation (2).□

4 Conclusion

Ultimately, in the context of ultrametric space, we bring on and carry forward the results by exploiting the space’s spherical completeness with a graph for single-valued G -strongly contractive mappings. We emphasize the significance of employing a distinct type of completeness. We specifically aim to extend the application of relevant fixed-point solutions to dynamic programming, which will assist in advancing the literature in ultrametric spaces.

Acknowledgements

The authors appreciate the anonymous reviewers’ recommendations for improving the study.

Funding information: The authors state that there is no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. The first and second authors have been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) (1002-Project 122F316).
Conflict of interest: The authors state that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-06-08

Revised: 2023-10-13

Accepted: 2023-11-12

Published Online: 2024-02-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2023-0132

Keywords for this article

dynamic programming; fixed point; F-contraction; functional equation; ultrametric space

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