SOME EXISTENCE RESULTS FOR BOUNDARY VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL INCLUSIONS WITH NON-SEPARATED BOUNDARY CONDITIONS

In this paper, we study the existence of solutions for a boundary value problem of differential inclusions of order q ∈ (1,2] with non-separated boundary conditions involving convex and non-convex multivalued maps. Our results are based on the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.


Introduction
Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in the modelling of dynamical systems associated with phenomena such as fractals and chaos.In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, fitting of experimental data, etc. [1][2][3][4].Recently, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions.For some recent work on fractional differential equations, see [5][6][7][8][9][10][11] and the references therein.EJQTDE, 2010 No. 71, p. 1 Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, etc. and are widely studied by many authors; see [12][13][14][15] and the references therein.For some recent development on differential inclusions of fractional order, we refer the reader to the references [16][17][18][19][20][21].
In this paper, we consider the following fractional differential inclusion with nonseparated boundary conditions c D q x(t) ∈ F (t, x(t)), t ∈ [0, T ], T > 0, 1 < q ≤ 2, where c D q denotes the Caputo fractional derivative of order q, F : [0, T ] × R → P(R) is a multivalued map, P(R) is the family of all subsets of R, and λ 1 , λ 2 , µ 1 , µ 2 ∈ R.

Preliminaries
Let C([0, T ]) denote a Banach space of continuous functions from [0, T ] into R with the norm x = sup t∈[0,T ] |x(t)|.Let L 1 ([0, T ], R) be the Banach space of measurable functions x : [0, T ] → R which are Lebesgue integrable and normed by x L 1 = T 0 |x(t)|dt.Now we recall some basic definitions on multi-valued maps [22,23].For a normed space (X, .), let P cl (X) = {Y ∈ P(X) : Y is closed}, P b (X) = {Y ∈ P(X) : Y is bounded}, P cp (X) = {Y ∈ P(X) : Y is compact}, and P cp,c (X) = {Y ∈ P(X) : Y is compact and convex}.A multi-valued map G : G is called upper semi-continuous (u.s.c.) on X if for each x 0 ∈ X, the set G(x 0 ) is a nonempty closed subset of X, and if for each open set N of X containing G(x 0 ), there exists an open neighborhood N 0 of x 0 such that G(N 0 ) ⊆ N. G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X).If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c.if and only if G has a closed graph, i.e., The fixed point set of the multivalued operator G will be denoted by Fix G.A multivalued map G : [0; 1] → P cl (R) is said to be measurable if for every y ∈ R, the function For each y ∈ C([0, T ], R), define the set of selections of F by Let X be a nonempty closed subset of a Banach space E and G : X → P(E) be a multivalued operator with nonempty closed values.G is lower semi-continuous (l.s.c.) if the set {y ∈ X : belongs to the σ−algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, T ] and D is Borel measurable in R. A subset A of L 1 ([0, T ], R) is decomposable if for all u, v ∈ A and measurable J ⊂ [0, T ] = J, the function χ J u + χ J−J v ∈ A, where χ J stands for the characteristic function of J .Definition 2.2.Let Y be a separable metric space and let N : Y → P(L 1 ([0, T ], R)) be a multivalued operator.We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Let F : [0, T ] × R → P(R) be a multivalued map with nonempty compact values.Define a multivalued operator Let (X, d) be a metric space induced from the normed space (X; .).Consider H d : P(X) × P(X) → R ∪ {∞} given by The following lemmas will be used in the sequel.
Lemma 2.1.( [25]) Let X be a Banach space.Let F : [0, T ] × R → P cp,c (X) be an L 1 − Carathéodory multivalued map and let Θ be a linear continuous mapping from Lemma 2.2.( [26]) Let Y be a separable metric space and let N : be a multivalued operator satisfying the property (BC).Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Let us recall some definitions on fractional calculus [1][2][3].Definition 2.5.For a function g : [0, ∞) → R, the Caputo derivative of fractional order q is defined as where [q] denotes the integer part of the real number q and Γ denotes the gamma function.
Definition 2.6.The Riemann-Liouville fractional integral of order q for a function g is defined as q > 0, provided the right hand side is pointwise defined on (0, ∞).
Definition 2.7.The Riemann-Liouville fractional derivative of order q for a function g is defined by provided the right hand side is pointwise defined on (0, ∞).
In order to define a solution of (1.1), we consider the following lemma.
Lemma 2.4.For a given ρ ∈ C[0, T ], the unique solution of the boundary value problem is given by , where G(t, s) is the Green's function given by , 0 ≤ t ≤ s ≤ T. (2.2) Proof.As argued in [8], for some constants c 0 , c 1 ∈ R, we have In view of the relations c D q I q x(t) = x(t) and I q I p x(t) = I q+p x(t) for q, p > 0, x ∈ L(0, T ), we obtain Applying the boundary conditions for (2.1), we find that .
Substituting the values of c 0 and c 1 in (2.3), we obtain the unique solution of (2.1) given by where G(t, s) is given by (2.2).This completes the proof.

Main results
Theorem 3.1.Assume that (H 3 ) there exists a number M > 0 such that where Proof.Define an operator EJQTDE, 2010 No. 71, p. 6 We will show that Ω satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.The proof consists of several steps.As a first step, we show that Ω(x) is convex for each x ∈ C([0, T ], R).For that, let h 1 , h 2 ∈ Ω(x).Then there exist f 1 , f 2 ∈ S F,x such that for each t ∈ [0, T ], we have Let 0 ≤ ω ≤ 1.Then, for each t ∈ [0, T ], we have .
Next, we show that Ω(x) maps bounded sets into bounded sets in C([0, T ], R).For a positive number r, let B r = {x ∈ C([0, T ], R) : x ∞ ≤ r} be a bounded set in C([0, T ], R).Then, for each h ∈ Ω(x), x ∈ B r , there exists f ∈ S F,x such that and Now we show that Ω maps bounded sets into equicontinuous sets of C([0, T ], R).Let t ′ , t ′′ ∈ [0, T ] with t ′ < t ′′ and x ∈ B r , where B r is a bounded set of C([0, T ], R).For each h ∈ Ω(x), we obtain Obviously the right hand side of the above inequality tends to zero independently of x ∈ B r ′ as t ′′ − t ′ → 0. As Ω satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that Ω : In our next step, we show that Ω has a closed graph.Let x n → x * , h n ∈ Ω(x n ) and h n → h * .Then we need to show that h * ∈ Ω(x * ).Associated with h n ∈ Ω(x n ), there exists f n ∈ S F,xn such that for each t ∈ [0, T ], .
EJQTDE, 2010 No. 71, p. 8 Thus we have to show that there exists f * ∈ S F,x * such that for each t ∈ [0, T ], .
Let us consider the continuous linear operator Θ : .
Observe that Thus, it follows by Lemma 2.1 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 f * ∈ S F,x * .
In view of (H 2 ), for each t ∈ [0, T ], we obtain Consequently, by virtue of (3.1), we have In view of (H 3 ), there exists M such that x ∞ = M. Let us set Note that the operator Ω : U → P(C([0, T ], R)) is upper semicontinuous and completely continuous.From the choice of U, there is no x ∈ ∂U such that x ∈ µΩ(x) for some µ ∈ (0, 1).Consequently, by the nonlinear alternative of Leray-Schauder type [28], we deduce that Ω has a fixed point x ∈ U which is a solution of the problem (1.1).This completes the proof.
As a next result, we study the case when F is not necessarily convex valued.Our strategy to deal with this problems is based on the nonlinear alternative of Leray Schauder type together with the selection theorem of Bressan and Colombo [26] for lower semi-continuous maps with decomposable values.Theorem 3.2.Assume that (H 2 ), (H 3 ) and the following conditions hold: EJQTDE, 2010 No. 71, p. 10 (H 5 ) for each σ > 0, there exists ϕ σ ∈ L 1 ([0, T ], R + ) such that Proof.It follows from (H 4 ) and (H 5 ) that F is of l.s.c.type.Then from Lemma 2.1, there exists a continuous function f : Consider the problem Observe that if x ∈ C 2 ([0, T ]) is a solution of (3.2), then x is a solution to the problem (1.1).In order to transform the problem (3.2) into a fixed point problem, we define the operator Ω as .
It can easily be shown that Ω is continuous and completely continuous.The remaining part of the proof is similar to that of Theorem 3.1.So we omit it.This completes the proof.
Now we prove the existence of solutions for the problem (1.1) with a nonconvex valued right hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [27].
As F has compact values, we pass onto a subsequence to obtain that v n converges to v in L 1 ([0, T ], R).Thus, v ∈ S F,x and for each t ∈ [0, T ], .
Next we show that there exists γ < 1 such that .
By (H For each t ∈ [0, T ], let us define . Thus, where ν 1 is given by (3.1).Hence, Analogously, interchanging the roles of x and x, we obtain Since Ω is a contraction, it follows by Lemma 2.2 that Ω has a fixed point x which is a solution of (1.1).This completes the proof.EJQTDE, 2010 No. 71, p. 13 In this paper, we have presented some existence results for fractional differential inclusions of order q ∈ (1, 2] involving convex and non-convex multivalued maps with non-separated boundary conditions.Our results give rise to various interesting situations.Some of them are listed below: (i) The results for an anti-periodic boundary value problem of fractional differential inclusions of order q ∈ (1, 2] follow as a special case by taking λ 1 = −1 = λ 2 , µ 1 = 0 = µ 2 in the results of this paper.In this case, the operator Ω(x) takes the form T 0 (T − s) q−1 Γ(q) f (s)ds + 1 4 (T − 2t) T 0 (T − s) q−2 Γ(q − 1) f (s)ds, f ∈ S F,x , and the condition (H 3 ) becomes 4MΓ(q) (5 + q)T q−1 ψ(M) p L 1 > 1, while the condition ensuring the existence of at least one solution of the problem (1.1) in Theorem (3.3) reduces to T q−1 (5 + q) m L 1 4Γ(q) < 1.
(ii) For q = 2, we obtain new results for second order differential inclusions with non-separated boundary conditions.In this case, the Green's function G(t, s) is , 0 ≤ t < s ≤ T, (iii) The results for an initial value problem of differential inclusions of fractional order q ∈ (1, 2] can be obtained by taking λ 1 = 0 = λ 2 in the present results with the operator Ω(x) taking the form Ω(x) = h ∈ C([0, T ], R) : h(t) = t 0 (t − s) q−1 Γ(q) f (s)ds + µ 2 t + µ 1 , f ∈ S F,x .

. 1 )
Then the boundary value problem (1.1) has at least one solution on [0, T ].
for all x ∞ ≤ σ and for a.e.t ∈ [0, T ].Then the boundary value problem (1.1) has at least one solution on [0, T ].