An Existence Theorem for Fractional Hybrid Differential Inclusions of Hadamard Type

This paper studies the existence of solutions for fractional hybrid differential inclusions of Hadamard type by using a fixed point theorem due to Dhage. The main result is illustrated with the aid of an example.


Introduction
Fractional calculus, in view of its numerous applications in technical and applied sciences, has attracted the attention of many researchers.The nonlocal nature of a fractional-order operator together with its ability to trace the hereditary properties of the underlying process/phenomea has helped to improve the B. Ahmad and S.K. Ntouyas mathematical modelling of many real world phenomena involving integer-order operators.Examples include signal processing, control theory, bioengineering and biomedical, viscoelasticity, finance, stochastic processes, wave and diffusion phenomena, plasma physics, social sciences, etc. ( [1]- [5]).Much of the work [6]- [19] on the topic deals with the governing equations involving Riemann-Liouville and Caputo type fractional derivatives.Another kind of fractional derivative is Hadamard type which was introduced in 1892 [20].This derivative differs from aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent.A detailed description of Hadamard fractional derivative and integral can be found in [2,21,22,23,24,25].
Hybrid fractional differential equations constitutes another interesting class of problems.For some recent work on the topic, we refer to [26]- [31] and the references cited therein.
In this paper, we introduce a new concept of fractional hybrid differential inclusions of Hadamard type.Precisely we investigate the existence of solutions for the following problem where is the family of all nonempty subsets of R, H J (.) is the Hadamard fractional integral and η ∈ R.
The paper is organized as follows: Section 2 contains some preliminary facts that we need in the sequel.In Section 3, we present the main existence result for the given problem whose proof is based on a fixed point theorem due to Dhage.

Preliminaries
Let C([1, T ], R) denote the Banach space of all continuous real valued functions defined on [1, T ] Let 0 ≤ γ < 1 and C γ,log [a, b] denote the weighted space of continuous functions defined by In the following we denote y C γ,log by y C .

IVP for fractional hybrid differential inclusions
satisfies the following Volterra integral equation: i.e., y(t) ∈ C n−α,log [a, b] satisfies the relations (2)-( 3) if and only if it satisfies the Volterra integral equation (4).
In particular, if 0 < α ≤ 1, the problem (2)-( 3) is equivalent to the following equation: Details can be found in [2].Some of propositions with the Hadamard calculus (derivative/integral) are formed as follows ( [32]).Proposition 2. If 0 < α < 1 the following relations hold: From Theorem 1 we have: B. Ahmad and S.K. Ntouyas is given by Let us recall some basic definitions on multi-valued maps [33,34].For a normed space (X, • ), let The fixed point set of the multivalued operator G will be denoted by Fix G.A multivalued map G : [1, T ] → P cl (R) is said to be measurable if for every y ∈ R, the function for all x ∈ R and for a.e.t ∈ [1, T ].
For each y ∈ C([1, T ], R), define the set of selections of F by The following lemma is used in the sequel.

Lemma 4 ([35]
).Let X be a Banach space.Let F : [1, T ] × R → P cp,cv (X) be an L 1 -Carathéodory multivalued map and let Θ be a linear continuous mapping from

Main result
In the forthcoming analysis, we consider the space The following fixed point theorem due to Dhage [36] is fundamental in the proof of our main result.
Lemma 5. Let X be a Banach algebra and let A : X → X be a single valued and B : X → P cp,cv (X) be a multi-valued operator satisfying: Then either (i) the operator inclusion x ∈ AxBx has a solution, or Then the boundary value problem (1) has at least one solution on [1, T ].
Transform the problem (1) into a fixed point problem.Consider the operator N : X → P(X) defined by

Now we define two operators
and B 1 : X → P(X) by Observe that N (x) = A 1 xB 1 x.We shall show that the operators A 1 and B 1 satisfy all the conditions of Lemma 5.For the sake of convenience, we split the proof into several steps.
Step 1.A 1 is a Lipschitz on X, i.e., (a) of Lemma 5 holds.
Let x, y ∈ X.Then by (H 1 ) we have Taking the supremum over the interval [1, T ], we obtain for all x, y ∈ X.So A 1 is a Lipschitz on X with Lipschitz constant φ .
Step 2. The multi-valued operator B 1 is compact and upper semicontinuous on X, i.e., (b) of Lemma 5 holds.
First we show that B 1 has convex values.Let u 1 , u 2 ∈ B 1 x.Then there are x and consequently B 1 x is convex for each x ∈ X.As a result B 1 defines a multi valued operator B 1 : X → P cv (X).
Next we show that B 1 maps bounded sets into bounded sets in X.To see this, let Q be a bounded set in X.Then there exists a real number r > 0 such that x ≤ r, ∀x ∈ Q.Now for each h ∈ B 1 x, there exists a v ∈ S F,x such that Then for each t ∈ [1, T ], using (H 2 ) we have s ds This further implies that s ds, and so B 1 (X) is uniformly bounded.
Next we show that B 1 maps bounded sets into equicontinuous sets.Let Q be, as above, a bounded set and h ∈ B 1 x for some x ∈ Q.Then there exists a v ∈ S F,x such that Then for any τ 1 , τ 2 ∈ [1, T ] we have Obviously the right hand side of the above inequality tends to zero independently of x ∈ Q as t 2 − t 1 → 0. Therefore it follows by the Arzelá-Ascoli theorem that B 1 : X → P(X) is completely continuous.
In our next step, we show that B 1 has a closed graph.Let x n → x * , h n ∈ B 1 (x n ) and h n → h * .Then we need to show that h * ∈ B 1 .Associated with h n ∈ B 1 (x n ), there exists v n ∈ S F,xn such that for each t ∈ [1, T ], Thus it suffices to show that there exists v * ∈ S F,x * such that for each t ∈ [1, T ], Let us consider the linear operator Θ : Observe that Thus, it follows by Lemma 4 that Θ • S F is a closed graph operator.Further, we have h n (t) ∈ Θ(S F,xn ).Since x n → x * , therefore, we have for some v * ∈ S F,x * .
As a result we have that the operator B 1 is compact and upper semicontinuous operator on X.

This is obvious by (H
s ds and k = φ .Thus all the conditions of Lemma 5 are satisfied and a direct application of it yields that either the conclusion (i) or the conclusion (ii) holds.We show that the conclusion (ii) is not possible.

209 Theorem 1 .
Let α > 0, n = −[−α] and 0 ≤ γ < 1.Let G be an open set in R and let f : (a, b] → R be a function such that: f (x, y) ∈ C γ,log [a, b] for any y ∈ G, then the following problem (a) A is single-valued Lipschitz with a Lipschitz constant k, (b) B is compact and upper semi-continuous, (c) 2M k < 1, where M = B(X) .

Example 7 .
for all t ∈ [1, T ].Then we have (log t) 1−α |u(t)| ≤ λ −1 |f (t, u(t)|Thus the condition (ii) of Thorem 5 does not hold.Therefore the operator equation A 1 xB 1 x and consequently problem (1) has a solution on[1, T ].This completes the proof.Consider the initial value problem

3 7 =
g(t), x ∈ R. With the given values, the condition (H 3 ) is clearly satisfied, that is,