Best proximity point results with their consequences and applications

In the commenced work, we establish some best proximity point results for multivalued generalized contractions on partially ordered complete metric spaces along with the tactic of altering distance function. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications of fractional calculus and also invoke our findings to the equation of motion modeling to differential equations.


Introduction and preliminaries
Estimating the solution of fixed point problems is well thought-out as one of the main problems in the metric fixed point theory. This forces the researchers to use the contractive conditions on underlying functions, to guarantee the existence of the fixed point. However, this issue becomes more interesting and challenging when mappings involved are non-self. This evolves the concept of best proximity point and related theorems. In fact a best proximity point theorem is principally dedicated to global minimization of the real-valued function y → σ (y, Sy), which measures the error involved for an approximate solution of the equation Sy = y (fixed point problem). In other words, a best proximity point theorem expounds sufficient conditions for the existence of an element y such that the error σ (y, Sy) is minimum. The more general version of best proximity point theorems having more than one non-self-mapping is known as common best proximity point theorems. In 2010, Basha [3] found a best proximity point with the help of the Banach contraction principle. Basha et al. [4] gave the existence of common best proximity points for pairs of non-self-mappings in metric spaces. Karapinar and Erhan [7] also studied best proximity for different types of contractions. Interestingly, these best proximity point theorems also serve as a natural generalization of fixed point theorems. If the mapping under consideration is a self-mapping, then a best proximity point becomes a fixed point. Note that one can convert optimization problems to the problem of finding the best proximity points. Hence, the existence of the best proximity points develops the theory of optimization. Through this theory, one can guarantee that a solution of the multi-objective global minimization problem proposed by a common best proximity point theorem, in turn, becomes a common approximate solution to the system of equations with the least probable error. The theory of fixed point for multivalued mappings plays a key role in the theory of integral inclusion which confirms the existence of solutions. Nadler [13] introduced the study of fixed point theory for multivalued mappings. To this end, the researcher can see notable works in [2,5,6,11,12,14,18]. In the setting of metric spaces, strict contractive conditions for self-mappings and multivalued mappings do not ensure the existence of the fixed point, one can refer to [1]. Recently, as a generalized contraction, Wardwoski [19] introduced the concept of F-contraction. Klim et al. [10] discussed F-contractions for dynamic process and proved fixed point theorems involving F-contractions. To address the above issues, our motivation to define a new concept of multivalued F-contraction on partially ordered complete metric spaces with the notion of altering distance function is to ensure the existence of best proximity point through best proximity point theorems.
Let W be a nonempty set and (W , σ , ≤) be a partially ordered metric space. Let L and M be nonempty subsets of the metric space (W , σ ). Let us assume that CB(W ) is the family of closed and bounded nonempty subsets of metric space W and B(W ) is the family of nonempty bounded subsets of W . The subsequent symbols used in our results are as follows: With the idea presented by Khan et al. [8], we use the following version of altering distance function.
Raj [17] firstly introduced the idea of P-property as follows.
In 2016, Pragadeeswarar et al. [16] established some results on best proximity point for multivalued mappings defined on a partially ordered metric space. Using the concept of F-contraction, in this paper we define a new concept of multivalued F-contractions on a partially ordered complete metric space with the notion of altering distance function and set the results for best proximity point garnished by examples with applications to differential equations and fractional calculus.

Main results
The multivalued F-contraction with altering distance function is defined as follows: Let L and M be nonempty closed subsets of a metric space (X, σ ) and T : L → CB(M) be a multivalued mapping, ϕ is an altering distance function such that Tu 0 is included in M 0 for all u 0 ∈ L 0 and In this condition if we take F(x) = log x, then we get which is a contraction using by Pragadeeswarar et al. [16].
Therefore, our newly defined contraction is a more generalized form of Pragadeeswarar et al. [16]. Theorem 2.1 Let (X, ≤, σ ) be a partially ordered complete metric space. Let L and M be nonempty closed subsets of the metric space (X, σ ) such that L 0 is nonempty and (L, M) satisfies the P-property. Let T : L → CB(M) be a multivalued F-contraction with altering distance function ϕ such that the following conditions are satisfied: Then there exists an element u in L such that D(u, Tu) = σ (L, M).
Proof As in given condition there exist two elements u 0 , u 1 in L 0 and v 0 ∈ Tu 0 such that If there exists n 0 such that u n 0 = u n 0 +1 , then σ (u n 0 +1 , v n 0 ) = D(u n 0 , Tu n 0 ) = σ (L, M). This means that u n 0 is the best proximity point of T and the proof is completed. Thus, we suppose that u n = u n+1 for all n. Since σ (u n+1 , v n ) = σ (L, M) and σ (u n , v n-1 ) = σ (L, M) and (L, M) has the P-property From equation (2.4) and the above inequality we get which is contradiction, so we have Hence, the sequence {σ (u n , u n+1 )} is monotonic nonincreasing and bounded below. Thus, there exists r ≥ 0 such that Suppose that lim n→∞ σ (u n , u n+1 ) = r > 0. Using (2.6), inequality (2.5) becomes Continuing this process, we get We obtain that In view of (F 3 ) (by the definition of F-contraction) there exists k ∈ (0, 1) such that Letting n → ∞ in this, by (2.9) and (2.10), we get Now, let us observe that from (2.11) for given > 0 there exists n 1 ∈ N such that n(β n ) k -0 < for all n ≥ n 1 , for all n ≥ n 1 .
We claim that {u n } is a Cauchy sequence. Consider m, n ∈ N such that m > n > n 1 . Therefore, Since k ∈ (0, 1) then 1 k > 1. Therefore, by the P-series test, the series ∞ Since σ (u n , u n+1 ) = σ (v n-1 , v n ). The sequence {v n } in M is Cauchy and then is convergent.
Suppose that v n → v. By the relation σ (u n+1 , v n ) = σ (L, M) for all n.
Since {u n } is an increasing sequence in L and u n → u by hypothesis (3), u n ≤ u for all n. Suppose that v / ∈ Tu. Thus, we consider Taking n → ∞ in the above inequality, using u n → u, v n → v and σ (u, v) = σ (L, M), we get which is contraction. This implies that v ∈ Tu, and hence D(u, Tu) = σ (L, M). That is, u is the best proximity point of T.
As the consequence of our result, we deduce new best proximity point and fixed point results for multivalued and single valued mappings in the partially ordered metric spaces. Proof The proof is similar to Theorem 2.1.
As a consequence of Theorem 2.2, we find the following results by using T is a single valued mapping.  N(u, v) and ϕ is an altering distance function such that If we take L = M in Theorem 2.1 and Theorem 2.2, then we get the following results respectively.

Corollary 2.2
Let (X, ≤, σ ) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X and let T : L → CB(L) be a multivalued F-contraction with an altering distance function ϕ such that the following conditions are satisfied: 1. There exist two elements u 0 , u 1 in L and v 0 ∈ Tu 0 such that σ (u 1 , v 0 ) = 0 and u 0 ≤ u 1 = v 0 ; 2. For all u, v ∈ L, u ≤ v implies Tu ⊆ Tv; 3. If {u n } is a nondecreasing sequence in L such that u n → u, then u n ≤ u for all n. Then there exists an element u in L such that D(u, Tu) = 0 i.e. u is a fixed point of the mapping T.

Corollary 2.3
Let (X, ≤, σ ) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X, and let T : L → L be a single valued mapping and ϕ be an altering distance function such that the following conditions are satisfied: 1. There exist two elements u 0 , u 1 in L such that σ (u 1 , Tu 0 ) = 0 and u 0 ≤ u 1 ; 2. T satisfies F(ϕ(σ (Tu, Tv))) ≤ F(ϕ (N(u, v) } and ϕ is an altering distance If {u n } is a nondecreasing sequence in L such that u n → u, then u n ≤ u for all n. Then there exists an element u in L such that σ (u, Tu) = 0 i.e. u is a fixed point of the mapping T.
Also, if we revenue T is single valued mapping and ϕ is an identity function, then we realize the next result.

Corollary 2.4
Let (X, ≤, σ ) be a partially ordered complete metric space. Let L be a nonempty closed subset of the metric space X, and let T : L → L be a single valued mapping such that the following conditions are satisfied: 1. There exist two elements u 0 , u 1 ∈ L such that σ (u 1 , Tu 0 ) = 0 and u 0 ≤ u 1 ; 3. For all u, v ∈ L, u ≤ v implies Tu < Tv; 4. If {u n } is a nondecreasing sequence in L such that u n → u, then u n ≤ u for all n. Then there exists an element u in L such that σ (u, Tu) = 0 i.e. u is a fixed point of the mapping T.
Proof It follows from the same lines as earlier.
Example 2.1 Let W = R 2 and consider the order (x, y) ≤ (z, t) ⇐⇒ x ≤ z and y ≤ t, where ≤ is the usual order in R. Thus, (W , ≤) is a partially ordered set. Besides, (W , σ ) is a complete metric space where the metric is defined as Let F be defined by F(β) = log β + β and τ = 1.
Example 2.2 Let W = {0, 1, 2, 3, . . . } with the usual order ≤ be a partial order set, and let σ : W × W → R be given as Then (W , σ ) is a complete metric space. Let S : W → W be defined as Then we prove that S is an F-contraction with respect to F(β) = log β + β and τ = 1. We discuss the following cases for u, v ∈ W . Case I: If u > v and v = 0, then Case II: If v > u and u = 0, then Case III: If u > v and v = 0, then Case IV: If v > u and u = 0, then In all the above cases, the conditions of Corollary 2.4 are satisfied, and it is clear that 0 is a fixed point of T.

Application to equation of motion
We know that (D([0, 1]), σ ) is a complete metric space. Consider the following problem.

Problem.
A particle of mass m is at rest at x = 0, t = 0. A force f starts activity on it in Xdirection such that its velocity jumps from 0 to 1 immediately after t = 0. Find the position of the particle at time t. The equation of motion is m d 2 x dt 2 = f t, x(t) and (3.1) Green's function associated with (3.1) is defined by Let ϕ : R × R → R be a function with the following conditions: In this work, we present the existence of the solution of nonlinear fractional differential equation with u(0) = u(1) = 0 and f : [0, 1] × R → R being a continuous function, and Green's function associated with Problem (4.1) is given by Assume that the following conditions hold: 2. There exists u 0 ∈ C[0, 1] such that ϕ(u 0 (t), Tu 0 (t)) ≥ 0 for all t ∈ [0, 1]. Assume that the mapping T : Now, we have to prove the existence of a solution of fractional differential equation (4.1). Proof It is well known that u is a solution of (4.1) is equivalent to u ∈ X is a solution of the integral equation   Therefore, the mapping T is an F-contraction. It follows from Corollary 2.4 that T has a fixed point in C([0, 1]), which in turn is the solution of Problem (4.1).

Conclusion
In this paper, we introduced the concept of multivalued F-contraction on partially ordered complete metric spaces with the notion of altering distance function. Results endowed with a partial order have been obtained for aforesaid contractions. All examples agree with the theoretical results of this study. In the end, we applied our main results to provide solutions of differential equations (equation of motion) and also of fractional differential equations. We concluded that the new results are influential and effective in finding the solutions for a wide class of nonlinear equations dealing with science and engineering.