Global optimum solutions for a system of ( k , ψ )-Hilfer fractional di ﬀ erential equations: Best proximity point approach

: In this article, a class of cyclic (noncyclic) operators are de ﬁ ned on Banach spaces via concept of measure of noncompactness using some abstract functions. The best proximity point (pair) results are manifested for the said operators. The obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional di ﬀ erential equations involving k ψ , ( ) -Hilfer fractional derivatives.


Introduction and basic concepts
Fixed point theory serves a incontestable purpose in the development of different branches of sciences.With the advent of new results in the theory, the new applications have been coined out.One such all-time significant application is in proving the existence of solutions to various kinds of equations viz.differential equations, integral equations, fractional differential and integral equations, integro-differential equations, functional equations, etc.
When a mapping has no fixed points, we search for the points that are most close to the fixed points, which are called as best proximity points.In the last three decades, there has been significant development in the field of best proximity point (pair) results.The application of such results lies in establishing the existence of optimum solutions for a system of equations.Let us recall the concept of best proximity points (pairs) in brief (see [1][2][3] for more details).
Throughout this article the following notations are used: set of natural numbers, set of real numbers, )closed ball of radius ρ with center a, closure of the set , con ( ) -convex and closed hull of , diam ( ) -diameter of the set .
Let X ( ) be a collection of bounded subsets in a metric space X .Let us take two nonempty subsets P and Q of a normed linear space ( ) X .We consider that a pair P Q , ( ) satisfies a property, if both P and Q individually satisfy that property.For example, we say a pair , ( ) is nonempty if and only if P and Q are nonempty.For the pair P Q , ( ), we will define, We consider a best proximity point for a cyclic mapping T , which is defined as, a point In case of a noncyclic mapping T , we consider the existence of a pair ) for which = b Tb, = a Ta and ).Such pairs are called best proximity pairs.The notion of cyclic (noncyclic) relatively nonexpansive mappings is presented by Eldred et al. in [1] and the best proximity point (pair) results in Banach spaces are obtained.The existence of best proximity point is manifested in [1] using a concept of proximal normal structure (PNS).Gabeleh [4] proved that every nonempty, convex and compact pair in a Banach space has PNS.This fact enabled Gabeleh to prove the following results.

Theorem 1.2. [4] A relatively nonexpansive cyclic mapping
∪ → ∪ T P Q P Q : has a best proximity point if T is compact and P 0 is nonempty, where P Q , ( ) is a nonempty, bounded, closed and convex ( ) pair in a Banach space X .
Next result is for noncyclic mappings on a strictly convex Banach space.A Banach space X is strictly convex if for ∈ a b x X , , and > Λ 0, the following holds The L p space ( < < ∞ p 1 ) and Hilbert space are examples of strictly convex Banach spaces.
Theorem 1.3.[4] Let X be a strictly convex Banach space and P Q , ( ) is an pair in X .Then, a relatively nonexpansive noncyclic mapping ∪ → ∪ T P Q P Q : admits a best proximity pair, provided it is compact and P 0 is nonempty.
These results (Theorems 1.2 and 1.3) can be considered as extensions of Schauder fixed point theorem for best proximity point (pair).The condition of compactness on the mapping T is a strong one.The Schauder's fixed point theorem is generalized by Darbo [5] and Sadovskii [6] using the concept of measure of noncompactness (MNC) which is defined axiomatically as follows (see Definition 1.4).One of the important aspects about MNC is that it facilitates to choose a class of mappings which are more general than compact operators.
An MNC μ on X ( ) satisfies the following properties.(a for a nonincreasing sequence P n { } of nonempty, bounded and closed subsets of X , then is nonempty and compact.
On a Banach space X , μ has the following properties. (i) Example 1.5.[8] The non-negative numbers , assigned with a bounded subset of a metric space X are called Kuratowski MNC (K-MNC) and Hausdorff MNC (H-MNC), respectively.
Schauder fixed point theorem is generalized using MNC by Darbo [5] and Sadovskii [6].We present the combined statement of both the theorems as follows: Theorem 1.6.Let T be a continuous self-mapping on a subset of a Banach space X , for every ⊂ M satisfying one of the followings: Then T has at least one fixed point.
On the line of Darbo fixed point theorem, Gabeleh and Markin in [10] generalized Theorems 1.2 and 1.3 by relaxing the condition of compactness on the operator T by using the concept MNC and applied the obtained results to actualize the optimum solutions of a system of differential equations.Recently, the results of [10] have been generalized further in different directions in [11][12][13][14][15][16] in which best proximity point (pairs) results are obtained using MNC.
In this article, we prove best proximity point (pair) theorems for a new class of cyclic (non-cyclic) operators facilitated by MNC and some abstract functions.We apply the obtained results to prove the existence of optimal solutions of system of fractional differential equation (FDE) with initial value involving k ψ , ( )-Hilfer fractional derivative.This is achieved by means of defining an operator from integral equations equivalent to the system of differential equations and proving that this operator has at least one best proximity point.
In this section, we present our main results for the existence of best proximity point (pair) for new classes of cyclic and noncyclic operators.We consider the following class of mappings introduced in [11], which will be used to define the new classes of condensing operators.
Let us denote by the collection of all functions ) that satisfy the following conditions: For example, if we take ) , then it is clear that ∈ k .The following theorem is our first main result.Some part of the proof and the concept of T -invariant pair is adopted from [12].
) be a nonempty and convex pair in a Banach space E with P 0 being nonempty and μ an MNC on E. A relatively nonexpansive cyclic mapping ∪ → ∪ T P Q P Q : has at least one best proximity point if for every , proximinal and T invariant pair ) and for continuous mappings where ) is a nondecreasing and continuous function. Proof.
) is convex, closed, T -invariant and proximinal pair considering the conditions on T (for more details see [12]).For ∈ a P 0 , there is a ).Since T is relatively nonexpansive cyclic mapping, . Continuing this pattern, we obtain by using induction.Similarly, we can see that 1 for all ∈ n .Hence, we obtain a decreasing sequence G H , {( ) }of nonempty, closed and convex pairs in

(
) is also proximinal.Let x be an arbitrary member in This means that the pair {( ) }is a decreasing sequence, holds for all ∈ n .Therefore, the sequence { ( ( ) ( ( )))} is nonnegative, bounded below and decreasing.Thus, there exists ≥ a 0 such that Now if possible, let > a 0. From the assumed hypothesis on T , we have ( ) ( ) ( ), which is a contradiction to the assumption on mapping Δ , Δ 1 2 and Δ 3 .This means that = a 0. Therefore, By definition of mapping κ, we deduce that That is, ) is nonempty, convex, compact and , which is a cyclic, relatively nonexpansive mapping on a nonempty, convex, compact and ).Therefore, Theorem 1.2 ensures that T admits a best proximity point.□ Next result is an analogous of the above theorem for relatively nonexpansive noncyclic mapping which constitutes second main result of the section.
) be a nonempty and convex pair in a strictly convex Banach space E with P 0 nonempty and μ an MNC on E. A relatively nonexpansive noncyclic mapping ∪ → ∪ T P Q P Q : has at least one best proximity pair if for every , proximinal and T invariant pair ) is pair which is proximinal and T -invariant (see [12] for more details on proof).

T H T H T H H con
) is nonempty, convex, compact and ).Thus, application of Theorem 1.3 ensures that T admits a best proximity pair.
□ Now, we give some consequences of the aforementioned theorems as corollaries.
) be a nonempty and convex pair in a (strictly convex) Banach space E with P 0 nonempty and μ an MNC on E. A relatively nonexpansive cyclic (noncyclic) mapping ∪ → ∪ T P Q P Q : has at least one best proximity point (pair) if for every , proximinal and T -invariant pair ) and for continuous mappings where ) is a nondecreasing and continuous function.
) and for continuous mappings Proof.If we take φ as a constant zero function in Corollary 2.3, we obtain the desired result.□ The following corollary is the main result of [10].
) be a nonempty and convex pair in a (strictly convex) Banach space E with P 0 is nonempty and μ an MNC on E. A relatively nonexpansive cyclic (noncyclic) mapping ∪ → ∪ T P Q P Q : has at least one best proximity point (pair) if for every , proximinal and T -invariant pair where Proof.If we take and φ as a constant zero function in Corollary 2.3, we obtain the required result.□

Application
In the last decade, from the view point of the numerous applications, the study of FDEs gained significant importance.The number of characteristics of physical events arising in the area of biology, medicines, and branches of engineering such as mechanics and electrical engineering are expressible in the form of mathematical model using impulsive FDEs.These applications lead to the development of theory of solutions of impulsive FDEs in various aspects.The interested readers can refer to the articles [17][18][19][20][21][22][23] for more details on FDEs and their applications.
In this section, we survey an application of the best proximity point results proved in Section 2 of this article.The existence of an optimal solution of systems of FDEs involving k ψ , ( )-Hilfer fractional derivative using the said result is established.
Let a b , and r be positive real numbers, = a b .We present some concepts and outcomes from fractional calculus which will be used in this section of the article.
] denote the space of all continuous functions on a b , [ ].We denote by L a b , m ( ), ≥ m 1, the spaces of Lebesgue integrable functions on a b , ( ). See [24] for more details.Let a b , ( ) be a finite or infinite interval of the real line .Let ψ x ( ) be an increasing and positive monotone function on a b , [ ], having a continuous derivative ′ ≠ ψ t 0 ( ) on a b , ( ).
) be two functions such that ψ x ( ) is increasing and )-Hilfer fractional derivative of order η and type ν of function f is defined as the following expression: provided the right-hand side exists.
Remark 3.4.The k ψ , ( )-Hilfer fractional derivative is considered to be the most general and unified definition of fractional derivative.In fact, by choosing different values of ψ x k a , , ( ) and taking limits on parameters η ν , in definition of k ψ , ( )-Hilfer fractional derivative, we obtain a wide variety of fractional derivatives in the literature.See [27][28][29] for more information related to this.
We have the following results for the fractional derivatives. .Then, we have ), we have We consider the following system of nonlinear FDEs involving k ψ , ( )-Hilfer fractional derivatives of arbitrary order with initial conditions of the form where )-Hilfer fractional differential operator of order p and type q; 2 are given mappings satisfying some assumptions.The following result establishes the equivalence of (2) with the fractional integral equation.Let Lemma 3.7.[27] The initial value problem (2) is equivalent to the following integral equation: ) be a Banach space of continuous mappings from into E endowed with supremum norm.Let ) is an NBCC pair in × S S. Now for every ∈ u S 1 and ∈ v S 2 , we have Lemma 3.8.The operator ∪ → T S S S : 1 2 defined by (4) is cyclic if f and g are bounded and continuous such that ∈ f g L a b , , Proof.For ∈ x S 1 and set = + − ξ p q k p k ( ), we have ;

‖ ‖ (
), that is z is a best proximity point of the operator T defined in (4).
Assumptions: We consider the following hypotheses to prove existence of optimal solutions.(A1) Let μ be an MNC on E and > Λ 0 such that for any bounded pair Following result is the Mean-Value Theorem for FDEs. ) such that Then, we give the following result.
Proof.It is clear that systems (2) and ( 3) have an optimal solution if the operator T defined in (4) ‖).By using a generalized version of Arzela-Ascoli theorem (see Ambrosetti [32]) and assumption  So, in view of Theorem (3.9), it follows that

( ( ( ) ( ))) ( ( )) ( ( ))
Therefore, we conclude that T satisfies all the hypotheses of Corollary 2.4 and so the operator T has a best proximity point ∈ ∪ z S S 1 2 , which is an optimal solution for systems (2) and (3).□ Global optimum solutions for a system of (k, ψ)-Hilfer FDEs: Best proximity point approach  5 n n ( ) is T -invariant.From the proof of Theorem 2.1, we have G H , has a best proximity point.From Lemma 3.8, T is a cyclic operator.It follows trivially that T S 1 ( ) is a bounded subset of S 2 .prove that T S 1 ( ) is also an equicontinuous subset of S 2 .For