ON BEST PROXIMITY POINT APPROACH TO SOLVABILITY OF A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS

In this article, a class of cyclic (noncyclic) condensing operators is deﬁned on a Banach space using the notion of measure of noncompactness and C -class functions. For these newly deﬁned condensing operators, best proximity point (pair) results are manifested. Then the obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional diﬀerential equations involving ψ -Hilfer fractional derivatives.


Introduction and basic concepts
Various physical phenomena are represented mathematically using a mathematical model with the help of different forms of equations.Those equations may be differential, integral, integro differential or that of fractional type and functional equations.The existence of fixed point is equivalent to the existence of solution of an equation.This concept is very well utilised in demonstrating actuality of solutions of mathematical models.This fact made fixed point theory an indispensable tool in mathematics.In the absence of fixed point for a mapping, we search for the points which are most closed with the image of the point under a mapping.Such points are called best proximity points.In the last three decades there is significant development in the field of best proximity point (pair) results.The concept of best proximity point is applicable in establishing existence of optimum solutions for a system of equations.Let us recall the concept of best proximity points (pairs) in brief.Let X be a normed linear space (NLS) and G and H be its nonempty subsets.It is understood that a pair (G, H) holds a property, if both G and H individually hold that property.Let D(G, H) = inf{ g − h : g ∈ G, h ∈ H} defines distance between sets G and H.We define It is proved in [10] that if (G, H) is a pair of nonempty, convex and weakly compact subsets of a Banach space X then the pair (G 0 , H 0 ) retains these three properties.If G 0 is same as G and H 0 is same as H then the pair (G, H) is called proximinal pair in X.We consider a mapping f : G ∪ H → G ∪ H.The mapping f is said to be cyclic mapping if it maps G into H and H into G (i.e.f (G) ⊆ H and f (H) ⊆ G).Whereas f is said to be noncyclic if image of G under f lies in G while image of H under f lies in H.The mapping f is said to be relatively nonexpansive if it satisfies f (a)−f (b) ≤ a−b where a ∈ G and b ∈ H.A relatively nonexpansive mappings becomes nonexpansive mapping if G = H.Note that f is said to be compact mapping if (f (G), f (H)) is compact.A cyclic mapping f can possess a best proximity point and is mathematically defined as a point w * ∈ G ∪ H satisfying w * − T w * = D(G, H).In case of noncyclic mappings we consider existence of best proximity pair mathematically represented as pair (g, h) Eldred et al. in [10] introduced the cyclic and noncyclic versions of relatively nonexpansive mappings and proved the existence of at least one best proximity point and pair in the setting of Banach spaces using a concept of proximal normal structure (in short, PNS).Gabeleh [12] proved the existence of best proximity points (pairs) in Banach space without the concept of PNS and using some intrinsic properties of space in the following results.

Theorem 1.1 ( [12]
).A relatively nonexpansive cyclic mapping f : G ∪ H → G ∪ H possesses a best proximity point if f is compact and G 0 is nonempty, where (G, H) is a nonempty, bounded, closed and convex (in short, N BCC) pair in a Banach space X.
Following result establishes existence of best proximity pairs for noncyclic mappings on a Banach space which is strictly convex.A Banach space X is strictly convex if for g, h, α ∈ X and ∆ > 0, following holds 12]).Let (G, H) be an N BCC pair in a strictly convex Banach space X.A relatively nonexpansive noncyclic mapping f : G ∪ H → G ∪ H possesses a best proximity pair if f is compact and G 0 is nonempty.
Theorems 1.1 and 1.2 are extensions of Schauder fixed point theorem in case of best proximity point and pairs.It is very well known that the compactness condition on mapping f is very strong one.There is a technique called measure of noncompactness (MNC) which allows us to select classes of mappings that are more general than that of compact mappings.Realizing this fact, Darbo [9] and Sadovskii [26] genaralized the Schauder's fixed point theorem using the concept of MNC which is defined axiomatically as follows.
Definition 1.1.[3,5,6] Let B(X) be a collection of bounded subsets in metric space X.An MNC is a mapping κ : B(X) → [0, ∞) that satisfies the following axioms: An MNC κ on B(X) satisfies following properties: 0 for a nonincreasing sequence {G n } of nonempty, bounded and closed subsets of X, then G ∞ = ∩ n≥1 G n is nonempty and compact.
κ satisfies the following properties on a Banach space X: (i) κ(con(H)) = κ(H), for all H ∈ B(X); (ii) κ(λH) = |λ|κ(H) for any number λ and H ∈ B(X); In particular, if B(ξ, ρ) denotes closed ball of radius ρ with center ξ and diam(G) denotes diameter of the set G, then the numbers .., N }, assigned with a bounded subset G of a metric space X are called Kuratowski MNC and Hausdorff MNC, respectively.
MNC is used to generalize the Schauder fixed point theorem by Darbo [9] and Sadovskii [26].The combined statement of theorems of both the mathematicians is as follows: Theorem 1.3.Let G be a N BCC subset of a Banach space X and κ be an MNC on X.A mapping f on G has at least one fixed point in G if it is continuous and for every M ⊂ G it satisfies any one of the following: A mapping satisfying condition (D) is called L-set contraction (Darbo [9]) whereas satisfying (S) is called as κ-condensing (Sadovskii [26]).
Motivated by the Theorem 1.3, Gabeleh and Markin in [14] used the concept of MNC and generalized the Theorems 1.1 and 1.2 by relaxing the condition of compactness on mapping f .They also applied the obtained results to prove the existence of the optimum solutions of system of differential equations.Recently, the results of [14] have been generalized further in different directions in [13,[15][16][17][20][21][22][23][24]) in which best proximity point and pairs results are obtained using MNC.
In this article, we prove best proximity point and pair theorems for a new class of cyclic and non-cyclic operators facilitated by MNC and C-class functions.We apply the obtained results to prove existence of optimal solutions of system of fractional differential equation with initial value involving 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.

Main results
In this section, we prove our main results for best proximity points and pairs.We consider the following abstract function known as C-class function.
Then C 1 and C 2 are C-class functions.
Following theorem is our first main result.Some part of the proof and the concept of T -invariant pair is adopted from [16].
Theorem 2.1.Let (G, H) be a nonempty and convex pair in a Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive cyclic mapping f : G ∪ H → G ∪ H has at least one best proximity point if for every N BCC, proximinal and f invariant pair ) and proximinal pair considering conditions on f (for more details see [16]).For Next, to show that the pair (G n , H n ) is proximinal.We use mathematical induction for this purpose.Obviously for n = 0, the pair (G 0 , H 0 ) is proximinal.Suppose that (G k , H k ) is proximinal.We show that (G k+1 , H k+1 ) is also proximinal.Let u be an arbitrary member in This means that the pair (G k+1 , H k+1 ) is proximinal and induction does the rest to prove (G n , H n ) is proximinal for all n ∈ N.
Considering the construction of G 2n and H 2n , we have Keeping this fact in view and using (2.1), we have As κ is nondecreasing function, we get Thus, κ(G2n∪H2n) 0 ϕ(ξ)dξ is non-negative and non-increasing sequence bounded below.Therefore it converges to r ≥ 0.
Passing to the limit n → ∞, we get Property (C 3 ) of C yields κ(r) = 0. Hence by assumption of κ, r = 0.This gives us All this is sufficient to ensure that f admits a best proximity point.
Next result is an analogous of the above theorem for relatively nonexpansive noncyclic mapping which constitute second main result of the section.
Theorem 2.2.Let (G, H) be a nonempty and convex pair in a strictly convex Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive noncyclic mapping f : G ∪ H → G ∪ H has at least one best proximity pair if for every N BCC, proximinal and f invariant pair Proof.It is evident that (G 0 , H 0 ) is N BCC pair which is proximinal and f invariant (see [16] for more details on proof).
Let us define a pair (G n , H n ) as G n = con(f (G n−1 )) and Continuing this pattern, we get H n ⊆ H n−1 by using induction.Similarly, G n+1 ⊆ G n for all n ∈ N. Hence we get a decreasing sequence {(G n , H n )} of nonempty, closed and convex pairs in All this is sufficient to ensures that f admits a best proximity pair.Now, we give some consequences of above theorems as corollaries.
Corollary 2.1.Let (G, H) be a nonempty and convex pair in a (strictly convex) Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive cyclic (noncyclic) mapping f : G ∪ H → G ∪ H has at least one best proximity point (pair) if for every N BCC, proximinal and f invariant pair where ψ : [0, ∞) → [0, 1) is a continuous function and κ : [0, ∞) → [0, ∞) is a nondecreasing and continuous function with κ(t) = 0 if and only if t = 0.
Corollary 2.2.Let (G, H) be a nonempty and convex pair in a (strictly convex) Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive cyclic (noncyclic) mapping f : G ∪ G → G ∪ H has at least one best proximity point (pair) if for every N BCC, proximinal and f invariant pair where ψ : [0, ∞) → [0, 1) is a continuous function and κ : [0, ∞) → [0, ∞) is a nondecreasing and continuous function with κ(t) = 0 if and only if t = 0.
Proof.If we take C(u, v) = vu in Corollary 2.1, we obtain the desired result.
Corollary 2.3.Let (G, H) be a nonempty and convex pair in a (strictly convex) Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive cyclic (noncyclic) mapping f : G ∪ H → G ∪ H has at least one best proximity point (pair) if for every N BCC, proximinal and f invariant pair where ψ : [0, ∞) → [0, 1) is a continuous function and κ : [0, ∞) → [0, ∞) is a nondecreasing and continuous function with κ(t) = 0 if and only if t = 0.
Proof.If we take C(u, v) = u − v in Corollary 2.1, we obtain the desired result.
The following result is the main result of [14].
Corollary 2.4.Let (G, H) be a nonempty and convex pair in a (strictly convex) Banach space X with G 0 being nonempty and κ being an MNC on X.A relatively nonexpansive cyclic (noncyclic) mapping f : G ∪ H → G ∪ H has at least one best proximity point (pair) if for every N BCC, proximinal and f invariant pair where λ ∈ [0, 1).

Application
In this section, we study the existence of an optimal solution of systems of ψ-Hilfer fractional differential equations with initial conditions using the best proximity point results which we have proved in section 2 of this article.
First we recall some concepts and outcomes from fractional calculus.Let −∞ < a < b < ∞.We denote the space of all continuous functions on be two functions such that ψ(x) is an increasing and ψ (t) = 0 for all t ∈ [a, b].The left-sided ψ-Hilfer fractional derivative of order p and type 0 ≤ q ≤ 1 of g is defined as the following expression provided the right-hand side exists.
It can be written as: with γ = p + q(n − p) and where D γ;ψ a + is the ψ-Riemamm-Liouville fractional derivative.
Remark 3.1.The ψ-Hilfer fractional derivative is considered to be most general and unified defintion of fractional derivative.In fact, by choosing different values of ψ(x), a and taking limit on parameters p, q in definition of ψ-Hilfer fractional derivative, we get a wide variety of fractional derivatives in the literature.See [27,28] for more information related to this.
We have following results for the fractional derivatives.
We consider the following system of right sided ψ-Hilfer fractional differential equations of arbitrary order with initial conditions: where H D ν,µ a + is the left sided Hilfer fractional differential operator of order 0 < µ < 1 and type 0 < ν ≤ 1; ‫ג‬ is the Riemann-Liouville fractional integral of order (1 − ν)(1 − µ); the state u(•) takes the values from Banach space E; f : I ×B 1 → E and g : I ×B 2 → E are given mappings satisfying some assumptions.
Lemma 3.3.The initial value problem (3.1) is equivalent to the following integral equation: Proof.The proof is similar to the proof of Lemma 3.1 in [28].So we skip the proof.
Let J ⊆ I and S = C(J , E) be a Banach space of continuous mappings from J into E endowed with supremum norm.Let So (S 1 , S 2 ) is a nonempty, bounded, closed and convex pair in S × S. Now for every u ∈ S 1 and v ∈ S 2 , we have u − v = sup u(s) − v(s) ≥ α a − β a .Therefore dist(S 1 , S 2 ) = α a − β a which ensures that (S 1 ) 0 is nonempty.Now, let us define the operator : S 1 ∪ S 2 → S as follows Proof.Let u ∈ S 1 and set γ = µ + ν − µν.We have Applying ‫ג‬ 1−γ;ψ a + on both sides and applying Lemma 3.1 and 3.2, we get Here ‫ג[‬ 1−ν(1−µ);ψ a + f (s, u(s))](t) → 0 as t → a. Therefore ‫ג‬ 1−γ;ψ a + u(a) = β a which means that u(t) ∈ S 2 .Similarly one can show that u(t) ∈ S 1 if u ∈ S 2 .Thus is cyclic operator.
We say that z ∈ S 1 ∪ S 2 is an optimal solution for the system (3.1) and (3.2) provided that z − z = dist(S 1 , S 2 ), that is z is a best proximity point of the operator defined in (3.3).
Assumptions.We consider the following hypotheses to prove our existence of optimal solutions.
(A 1 ) Let κ be a measure of noncompactness on E such that for any bounded pair Following result is the Mean-Value Theorem for fractional differential.Proof.It is clear that, system (3.1)-(3.2) has an optimal solution if the operator defined in (3.3) has a best proximity point.From Lemma 3.4, is a cyclic operator.It follows trivially that (S 1 ) is a bounded subset of S 2 .We prove that (S 1 ) is also an equicontinuous subset of S 2 .For t 1 , t 2 ∈ J with t 1 < t 2 and u ∈ S 1 , we observe that As t 2 → t 1 , right hand side tends to 0. Thus u(t 2 ) − u(t 1 ) → 0 as t 2 → t 1 .Thus (S 1 ) is equicontinuous.With the similar argument we can prove that (S 2 ) is bounded and equicontinuous subset of S 1 .Thus with the application of Arzela-Ascoli theorem we can conclude that (S 1 , S 2 ) is relatively compact.Next we show that is relatively nonexpansive.For any (x, y) ∈ S 1 × S 2 , with assumption (A 2 ) we have and thereby, x − y ≤ x − y .Therefore is relatively nonexpansive.At last, let (K 1 , K 2 ) ⊆ (S 1 , S 2 ) be nonempty, closed, convex and proximinal pair which is -invariant and such that dist(K 1 , K 2 ) = dist(S 1 , S 2 )(= α a − β a ).By using a generalized version of Arzela-Ascoli theorem (see Ambrosetti [4]) and assumption (A 1 ), we get Choosing λ = (ψ(τ )−ψ(a)) µ Γ(µ+1) , we get that κ( (K 1 ) ∪ (K 2 )) < λκ(K 1 ∪ K 2 ) and 0 ≤ λ < 1.Therefore, we conclude that satisfies all the hypotheses of Corollary 2.4 and so the operator has a best proximity point z ∈ S 1 ∪S 2 which is an optimal solution for the system (3.1) and (3.2).

Conclusions
The fixed point theory serves as an indipensable tool in nonlinear analysis.When an operator do not possess a fixed point, the case becomes significant from the point of view of existence and leads to the study of best approximation.In the work [7], one can study the characterizations of nearly strongly convex and very convex spaces in terms of best approximation theoretic properties of Banach spaces.The theory further extends to the concept which fairly resembles with the notion of fixed point and called as best proximity point.In this article, we have studied the existence of best proximity points (pairs) by considering a new family of cyclic (noncyclic) condensing operators by using an appropriate measure of noncompactness.To this end we have used C-class functions to introduce such class of mappings.As an application of our main existence result, we have surveyed the existence of an optimal solution of a system of fractional differential equation with initial value involving ψ-Hilfer fractional derivative.We further refer to the articles [8,19] for possible applications of the theory of best proximity points (pairs).