Graphical structure of double controlled metric-like spaces with an application

The aim of the manuscript is to present the concept of a graphical double controlled metric-like space (for short, GDCML-space). The structure of an open ball of the proposed space is also discussed, and the newly presented ideas are explained with a new technique by depicting appropriately directed graphs. Moreover, we present some examples in a graph structure to prove that our results are sharp compared to those in the previous papers. Further, the existence of a solution to the boundary value problem originating from the transverse oscillations of a homogeneous bar (TOHB) is obtained theoretically.


Introduction
To resolve many problems in mathematics, we require to use fixed point theory. This theory is also considered a remarkable tool in applied sciences and several other disciplines. The Banach fixed point theorem [1] is known as the essential theorem in the metric fixed point theory. This theorem has been generalized and extended in several directions using different topologies and/or variant contractions. Among these generalizations and extensions, we cite [2][3][4][5][6][7][8][9][10].
On the other hand, graph theory is a branch of mathematics that deals with networks of elements connected by lines. A graph is considered a structure amounting to a set of objects in which some pairs of objects are related in some sense. The beginning of the graph theory field originated in number games. It has a significant contribution to mathematical research, with applications in social sciences, operations research, chemistry, and computer science. For more details, we refer to [11][12][13][14].
In 2005, Echenique [15] initiated the combination of graph theory and fixed point theory by presenting a proof of the fixed point result of Tarski via graphs (using ordinal numbers). A year later, Espinola and Kirk [16] gave some fixed point theorems via graph theory. Precisely, they established some fixed point results for nonexpansive mappings defined on R-trees and used the obtained theorems to give some results in graph theory. Note that metric graphs correspond to spaces followed by considering a connected graph and metrizing the nontrivial edges of the graph as bounded intervals of R. Going in the same direction, a fruitful contribution was made by Jachymski [17] in 2008 and Beg et al. [18] in 2010. Namely, variant generalizations of the Banach contraction principle to mappings on a metric space equipped with a graph have been provided. For further investigations of fixed point results using a graph, see [19][20][21][22][23].
Among the generalizations of a metric space, there is the concept of a double controlled metric-like space introduced by Mlaiki [24]. In this setting, double controlled control functions are considered on the right-hand-side of the modified triangular inequality. Also, self-distance may not be equal to zero. In this paper, we construct a suitable double controlled metric-like metric space equipped with a graph and some suitable graphical contraction mappings. We prove some related fixed point theorems. We also present some concrete examples and an application by ensuring the existence of a solution to a boundary value problem originating from TOHB.

Basic facts and primary definitions
This part is intended to review primary definitions of a controlled metric-type space, a double controlled metric-type space, and a double controlled metric-like space and some facts about the graph theory, which are very important for most of the statements of our paper.
Based on the results of Jachymski [17], let a set J = ∅ and be the diagonal of the Cartesian product J × J. Also, assume that = (ν( ), ϑ( )) is a directed graph without parallel edges, where ν( ) is the vertex set of so that it coincides with the set J, and the set of edges will be denoted by ϑ( ) so that it contains all the loops of , i.e., ⊆ ϑ( ).
The undirected graph is obtained from by ignoring the direction of edges denoted by the letter . Actually, it will be more convenient for us to treat as a directed graph for which the set of its edges is symmetric. Under this convention, where -1 is the graph obtained by reversing the direction of ϑ( ).
Assume that k and l are vertices of the directed graph , a path in described as a sequence {k i } m i=0 containing (m + 1) vertices so that k • = k, k m = l with (k i-1 , k i ) ∈ ϑ( ) for i = 1, 2, . . . , m. Further, if there is a path between any two vertices, then a graph is called connected, and is weakly connected if is connected.
The symbols below are due to Shukla et al. [27]: A relation on J is defined by: (k l) if there is a directed path from k to l in , and we write t ∈ (k l) if t contained in some directed path from k to l in . Moreover, they denote [k] n = {l ∈ J : there is a direct path from k to l in with the length n}.
Recall that a sequence {k i } ⊂ J is called -termwise connected ( -TWC) if (k i k i+1 ) , for all i ∈ N. Henceforward, we shall consider all graphs are directed unless otherwise stated.
Chuensupantharat et al. [28] applied a directed graph in metric fixed point theory by introducing a graphical b-metric space to generalize a b-metric space. They were able to study the topological properties of this space and get examples and results about the fixed points that contribute significantly to the development of graph theory. For more examples and explanations, see [29].

Graphical double controlled metric-like spaces
Most of the results in metric fixed point theory depend on the fact that if the contractive condition holds for comparable elements p and q, and for q and r, it necessarily holds for p and r. Transitivity can be avoided while working in the structure of the graph.
Because of the importance of graphs in the metric fixed point, we commence the graphical version of double-controlled metric-like spaces as the following: Then (J, d σ ξ ) is called a graphical double-controlled metric-like space (GDCML-space).

Remark 3.2
(i) If we take σ = ξ , then we have a new space called a graphical controlled metric-like space.
(ii) It is clear that a GDCML-space is more general than a DCML-space.
It should be noted that (J, d σ ξ ) is not a graphical metric space and hence not a metric space because Remark 3.5 A GDCML-space can be obtained from its ordered version.
Let d σ ξ be an ordered double-controlled metric-like and Hence, we conclude that every ordered double-controlled metric-like space (ODCML-space) is a GDCML-space.
The following example confirms Remark 3.5.
Example 3.6 Consider J = {2, 4, 6, 8, 10} equipped with the partial order described as Now, assume the graph d σ ξ is endowed with the partial order as illustrated in Fig. 3. Then (J, d σ ξ ) is a GDCML-space with the same functions σ and ξ .
It should be noted that a GDCML-space is not necessarily a DCML-space. In order to confirm this statement, we will explain the following example.
To summarize the above results together with the remarkable work that has been completed in lead manuscripts [24,27,30,31], we present the next flow diagram (Fig. 5) in order to gain a better understanding of the corresponding concepts.  Encompassed by a GDCML, the set = {O d σ ξ (k, δ) | k ∈ J, δ > 0} forms a neighborhood system for the topology τ on J. Furthermore Proof Assume that m ∈ O d σ ξ (k, δ ρ ). If m = k, then we select = δ ρ . Now, suppose that m = k, then we get d σ ξ (m, k) = 0. Selecting 0 < = 1 ρ (δd σ ξ (m, k)) and consider l ∈ O d σ ξ (m, ), then based on the given assertions, we have (k m) σ ξ and (m l) σ ξ and hence (k l) σ ξ . According to the property of the GDCML-space, we get . This completes the proof.
exists and is finite.

Recent fixed point theorems
Assume that = (ν( ), ϑ( )) is a weighted graph containing all the loops. We say that a sequence {k i } ∈ J with the initial value k 0 ∈ J is an -Picard sequence ( -PS) for an operator : J → J if k i = k i-1 = i k 0 , ∀i ∈ N. Furthermore, = (ν( ), ϑ( )) is said to justify the property (P) [27] if a -termwise connected -PS {k i } converging in J guarantees that there are a limit k ∈ J of {k i } and k 0 ∈ N so that (k i , k) ∈ ϑ( ) or (k, k i ) ∈ ϑ( ), for all i > i 0 . Now, we demonstrate our first main results by defining a graphical σ ξ -contraction as follows: Definition 4.1 Let (J, d σ ξ ) be a GDCML-space endowed with a graph containing all the loops. A mapping : J → J is called a graphical σ ξ -contraction on a GDCML-space (J, d σ ξ ) if the stipulations below hold: ( σ ξ S1) J preserves edges of , i.e., ∀k, l ∈ J, If (k, l) ∈ ϑ( ), then ( k, l) ∈ ϑ( );

Theorem 4.2 Let (J, d σ ξ ) be a -complete GDCML-space and : J → J be a graphical σ ξ -contraction. Assume that the hypotheses below hold:
(1) the graph verifies the property (p); (2) for some n ∈ N, there is k 0 ∈ J so that k 0 ∈ [k 0 ] n and where {k i } is -PS with initial value k 0 ; (3) for every k ∈ J, we have that lim i→∞ σ (k, k i ) and lim i→∞ ξ (k i , k) exist and are finite. Then there is k * ∈ J so that the -PS {k i } is -TWC and converges to both k * and k * .
Proof Assume that k 0 ∈ J so that for some n ∈ N, k 0 ∈ [k 0 ] n . Because {k i } is a -PS with initial value k 0 , there is a path {l j } n j=0 with k 0 = l 0 , k 0 = l n and (l j-1 , l j ) ∈ ϑ( ) for j = 1, 2, . . . , n. Based on ( σ ξ S1), for j = 1, 2, . . . , n we have ( l j-1 , l j ) ∈ ϑ( ). This implies that { l j } n j=0 is a path from l 0 = k 0 = k 1 to l n = 2 k 0 = k 2 having length n, and thus k 2 ∈ [k 1 ] n . By repeating the same approach, we find that { i l j } n j=0 is a path from i l 0 = i k 0 = k i to i l n = i k 0 = k i+1 of length n, and thus k i+1 ∈ [k i ] n , for all i ∈ N. This proves that {k i } is a -TWC sequence. Now, ( i l j-1 , i l j ) ∈ ϑ( ) for j = 1, 2, . . . , n and i ∈ N. By ( σ ξ S2), we get By continuing with the same scenario, we have Since {k i } is a -TWC sequence, by (4.3), we can write Since r is finite, letting a finite quantity, we have Again, since {k i } is a -TWC sequence for i, m ∈ N, i < m and using (4.4), we obtain Note that, we used the fact σ (k, l), ξ (k, l) ≥ 1. Assume that Then, we get From condition (4.2) and using the ratio test, we find that lim i→∞ i exists, and hence and the real sequence { i } is a -Cauchy. At the last, letting m, i → ∞ in (4.5), we have This proves that the sequence {k i } is a -Cauchy in (J, d σ ξ ). The completeness of (J, d σ ξ ) implies that there is a sequence {k i } converges in J and from stipulation (1), there is k * ∈ J, This assures that {k i } converges to k * . If (k i , k * ) ∈ ϑ( ), then by stipulation ( σ ξ S2), we get This implies that Also, if (k * , k i ) ∈ ϑ( ), then with the same arguments as above, we find that Therefore, {k i } converges to both k * and k * , and this finishes the proof.
In order to achieve the existence of the fixed point, we introduce the following results: Proof In Theorem 4.2, we were able to prove that the -PS {k i } with initial value k 0 converges to both k * and k * . As k * ∈ J and k * ∈ (J), therefore by our assumption, we have k * = k * . Hence, has a fixed point.
The below example supports Theorem 4.2.

The existence of a solution for transverse oscillations
In this section, we apply the obtained theoretical results to discuss the existence of a solution to the boundary value problem originating from transverse oscillations of a homogeneous bar (TOHB).
The TOHB is a problem of paramount importance. Suppose that we have a homogeneous bar that is fixed at one end and free at the other one so that the axis of the homogeneous bar corresponds to the segment (0, 1) of the x-axis and the deviation parallel to the z-axis at the point c. The differential equation that describes the TOHB is written as: where (c, u) is the Green function described by Assume that J = C(I, R) is the set of real continuous functions on I. Let Let be a graph defined by J = ν( ) and for all k, k * ∈ J with σ (k, k * ) = s + k + k * and σ (k, k * ) = t + k + k * , such that s, t > 2 and s = t. Using our assumptions, we take sup c∈I ( (ℵ, c)) 2 = η ∈ [0, 1), such that we get d σ ξ ( k, l) ≤ ηd σ ξ (k, l).
This verifies the stipulation ( σ ξ S2) of Theorem 4.2. Now, for all k, l ∈ so that (k, l) ∈ ϑ( ), we find that k, l ∈ and k(c) ≤ l(c), for all c ∈ I. By the assertion (H 2 ), we obtain inf c∈I (k) > 0, In other words, (k)(c) ∈ and ( (k)(c), (l)(c)) ∈ ϑ( ). Moreover, by condition (H 1 ), the existence of a lower solution of the integral equation (5.2) confirms that there is a solution, say κ ∈ , so that (κ) ∈ [κ] 1 . This implies that the hypothesis (2) of Theorem 4.2 is fulfilled. Also, the remaining conditions of Theorem 4.2 can be verified easily. So, the mapping has a fixed point, which is a solution of the ordinary differential equation (5.1).

Some open problems
• Is it possible to expand the triangle inequality to become quadrilateral or rectangular to obtain new spaces under the same topological properties? • Is it possible to extend the contraction condition given in our basic theorem, such as the equivalent results of Reich [32], Meir-Keelar [33], Kannan [34], Hardy-Rogers [35], Ciric [36], Edelstein [37], and De la Sen [38] in a GDCML-space? • Can we apply the theoretical results to discuss the existence solution of the following integro-differential equation: