Boundary Value Problems for Fractional Differential Equations and Inclusions in Banach Spaces

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1 Malaya J. Mat. 5(2)(27) Boundary Value Problems for Fractional Differential Equations and Inclusions in Banach Spaces Samira Hamani a and Johnny Henderson b, a Département de Mathématiques, Université de Mostaganem, B.P. 27, Mostaganem, Algérie. b Department of Mathematics, Baylor University, Waco, Texas , USA. Abstract In this paper, we are concerned with the existence of solutions for boundary value problems, first for a class of fractional differential equations and second for a class of fractional differential inclusions. The methods include techniques associated with measure of noncompactness in conjunction with fixed point theorems of Mönch type. Keywords: Fractional differential equation, fractional differential inclusion, boundary value problem, measure of noncompactness, fixed point. 2 MSC: 26A33, 34A8, 34A6, 34B5. c 22 MJM. All rights reserved. Boundary Value Problems for Fractional Differential Equations in Banach Spaces. Introduction In this section, we are concerned with the existence of solutions for boundary value problems (BVP for short), for a class of fractional order differential equations, when we apply the method associated with the technique of measure of noncompactness and a fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Banas and Goebel [] and subsequently developed and used in many papers; see, for example, Banas and Sadarangani [2], Guo et al. [25], Lakshimikantham and Leela [38], Mönch [42], and Szufla [47]..2 Preliminaries We introduce notations, definitions, and preliminary facts, many of which will be used throughout the remainder of this paper. Let C(J, E) be the Banach space of all continuous functions from J into E with the norm y = sup{ y(t) : t T}, and we let L (J, E) denote the Banach space of functions y : J E which are Bochner integrable with norm y L = y(t) dt. Corresponding author. address: hamani samira@yahoo.fr (Samira Hamani), Johnny Henderson@baylor.edu (Johnny Henderson),

2 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 347 Let L (J, E) be the Banach space of functions y : J E which are bounded and equipped with the norm y L = inf{c > : y(t) c : a.e. t J}. Let AC (J, E) is the space of functions y : J E, which are absolutely continuous whose first derivative y is absolutely continuous. For a given set V of functions v : J E, we define V(t) = {ϑ(t) : ϑ V}, t J, V(J) = {ϑ(t) : ϑ V, t J}. Definition.2.. ([37, 45]). The fractional (arbitrary) order integral of the function h L ([a, b], R + ) of order r R + is defined by t Iah(t) r (t s) = r h(s)ds, where Γ is the gamma function. When a =, we write I r h(t) = h(t) ϕ r (t), where ϕ r (t) = tr ϕ r (t) = for t, and ϕ r δ(t) as r, where δ is the delta function. a for t >, and Definition.2.2. ([37, 45]). For a function h given on the interval [a, b], the r Riemann-Liouville fractional-order derivative of h, is defined by (D r a+h)(t) = ( ) d n t (t s) nr h(s)ds. Γ(n r) dt a Here n = [r] + and [r] denotes the integer part of r. For convenience, we first recall the definition of the Kuratowski measure of noncompactness, and summarize the main properties of this measure. Definition.2.3. ([6, ]) Let E be a Banach space and let Ω E be the family of bounded subsets of E. The Kuratowski measure of noncompactness is the map α : Ω E [, ) defined by Properties: α(b) = inf{ɛ >, : B m B j and diam(b j ) ɛ} ; here B Ω E. j= () α(b) = B is compact (B is relatively compact). (2) α(b) = α(b). (3) A B α(a) α(b). (4) α(a + B) α(a) + α(b). (5) α(cb) = cα(b) ; c R. (6) α(conb) = α(b). Here B and conb denote the closure and the convex hull of the bounded set B, respectively. The details of α and its properties can be found in [6, ]. Definition.2.4. A multivalued map F : J E E is said to be Carathéodory if () t F(t, u) is measurable for each u E. (2) u F(t, u) is upper semicontinuous for almost all t J. Let us now recall Mönch s fixed point theorem and an important lemma.

3 348 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions Theorem.2.. ([42],[5]) Let D be a bounded, closed and convex subset of a Banach space E such that D, and let N be a continuous mapping of D into itself. if the implication, holds for every subset V of D, then N has a fixed point. V = con(v) orv = N(V) {} = α(v) =, (.) Lemma.2.. ([47]) Let D be a bounded, closed and convex subset of a Banach space C(J, E), G be a continuous function on J J, and f : J E E be a function satisfying the Carathéodory conditions, and suppose there exists p L (J, R + ) such that for each t J and each bounded set B E one has lim α( f (J k + t,k B)) α(b); where J t,k = [t k, t] J. (.2) If V is an equicontinuous subset of D, then α({ G(s, t) f (s, y(s))ds : y V}) G(t, s) p(s)α(v(s))ds. (.3) J.3 Boundary Value Problems of Order r (, 2] We consider the boundary value problem with nonlocal conditions J D r y(t) = f (t, y(t)), for a.e. t J = [, T], (.4) y() =, y(t) = g(y), (.5) where < r 2, D r is the Riemann-Liouville fractional derivative, f : J E E is a continuous function, g : E E is a continuous function and (E, ) denotes a Banach space. Later we will study another boundary value problem for another fractional differential equation with nonlocal conditions. In particular, we will consider the boundary value problem with nonlocal conditions, D r y(t) = f (t, y(t)), for a.e. t J = [, T], (.6) y() =, βy(η) = y(t), (.7) where < r 2, < βη r <, < η <, and D r, f, (E, ) are as in (.4)-(.5)..3. Main Results for (.4)-(.5) and (.6)-(.7) Let us start by defining what we mean by a solution of the problem (.4)-(.5). Definition.3.5. A function y C([, T], E) is said to be a solution of (.4)-(.5) if y satisfies the equation D r y(t) = f (t, y(t)) on J, and the conditions y() =, y(t) = g(y). For the existence of solutions for the problem (.4)-(.5), we need the following auxiliary lemma. Lemma.3.2. [] Let r >, and h C(, T) L(, T) then I α D α h(t) = h(t) + c r + c 2 t r c n t rn for some c i R, i =,, 2,..., n, where n is the smallest integer greater than or equal to r. Lemma.3.3. Let < α 2 and let h : [, T] R be continuous. A function y C([, T], E) is a solution of the fractional integral equation y(t) = + t r T r if and only if y is a solution of the fractional BVP (t s)r h(s)ds (T s)r h(s)ds tr g(y) T r (.8) D r y(t) = h(t), t [, T], (.9) y() =, y(t) = g(y). (.)

4 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 349 Proof: Assume y satisfies (.9). Then Lemma.3.2 implies that y(t) = c r + c 2 t r2 + t (t s) r h(s)ds. From (.), a simple calculation gives c 2 =, and c = T T r (T s) r h(s)ds + g(y). Tr Hence we get equation (.8). Conversely, it is clear that if y satisfies the integral equation (.8), then equations (.9)-(.) hold. Theorem.3.2. Assume the following hypotheses hold: (H) (H2) The function f : J E E satisfies the Carathéodory conditions. There exists p L (J, R + ), such that f (t, y) y for a.e. t J and each y E. (H3) There exists constant k > such that g(y) k y for each y E. (H4) For almost each t J and each bounded set B E we have lim k + α( f (J t,k B)) α(b). (H5) For almost each bounded set B E we have α(g(b)) k α(b). Then the BVP (.4)-(.5) has at least one solution on C(J, B), provided that T r + T 2r p L + k T r <. (.) Proof. Transform the problem (.4)-(.5) into a fixed point problem. Consider the operator (Ny)(t) = + tr T r (t s)r f (s, y(s))ds (T s)r f (s, y(s))ds tr g(y). T r Clearly, from Lemma.3.3, the fixed points of N are solutions to (.4)-(.5). Now, let R > and consider the set D R = {y C(J, E) : y R}. We shall show that N satisfies the assumptions of Mönch s fixed point theorem. The proof will be given in several steps. Step : N is continuous. Let {y n } be a sequence such that y n y in C(J, E). Then, for each t J, (Ny n )(t) (Ny)(t) + tr T r (t s)r f (s, y n (s) f (s, y(s)) ds (T s)r f (s, y n (s) f (s, y(s)) ds.

5 35 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions Let ρ > be such that y n ρ, y ρ. By (H2)-(H3) we have f (s, y n (s) f (s, y(s)) 2ρp(s) := σ(s); σ L (J, R + ). Since f is a Carathéodory function, the Lebesgue dominated convergence theorem implies that Step 2: N maps D R into itself. N(y n ) N(y) as n. For each y D R, by (H2) and (.) we have for each t J N(y)(t) t + r T r Tr +T 2r R. Step 3: N(D R ) is bounded and equicontinuous. (t s)r f (s, y(s)) ds (T s)r f (s, y(s)) ds + tr p L + k T r By Step 2, it is obvious that N(D R ) C(J, E) is bounded. T r g(y) For the equicontinuity of N(D R ), let t, t 2 J, t < t 2, and y D R. We have (Ny)(t 2 ) (Ny)(t ) = + t t2 t [(t 2 s) r (t s) r ] f (s, y(s))ds (t 2 s) r f (s, y(s))ds + (t t 2 ) r T r (T s) r f (s, y(s)) ds + (t t 2 ) r g(y) + T r t t2 [(t s) r (t 2 s) r ]ds t (t 2 s) r ds + (t 2 t ) r T r Γ(r + ) [(t 2 t ) r + t r tr 2 ] + + (t 2 t ) r T r Γ(r + ) (t 2 t ) r + + (t 2 t ) r T r (T s) r ds + k (t t 2 ) r T r + k (t t 2 ) r T r Γ(r + ) (tr tr 2 ) + k (t t 2 ) r T r. As t t 2, the right-hand side of the above inequality tends to zero. Γ(r + ) (t 2 t ) r Now let V be a subset of D R such that V co(n(v) {}). V is bounded and equicontinuous, and therefore the function ϑ ϑ = α(v(t)) is continuous on J. By (H3), Lemma.2., and the properties of the

6 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 35 measure α, we have for each t J This means that ϑ(t) α(n(v)(t) {}) α(n(v)(t)) t T r +T 2r p(s)α(v(s))ds + k T r [ ] α(v(t)) ϑ L T r +T 2r p L + k T r. ϑ L ( Tr + T 2r p L + k T r ). By (.) it follows that ϑ =, that is, ϑ = for each t J, and then V(t) is relatively compact in E. In view of the Ascoli-Arzela theorem, V is relatively compact in D R. Applying now Theorem.2. we conclude that N has a fixed point which is a solution of the problem (.4)-(.5). Example. As an application of Theorem.3.2, we consider the fractional differential equation D r y(t) = 2 y(t), for a.e. t J = [, ], < r 2, (.2) 9 + et y() =, y() = n c i y(t i ), (.3) i= where < t < t 2 < < t n <, c i, i =,..., n, are given positive constants with i= n c i < 5 4. Set 2 f (t, x) = x, (t, x) J [, ), 9 + et Clearly, conditions (H) and (H2) hold with = e t. Condition (.) is satisfied with T = and k = 4 5. Indeed T r +T 2r p L + k T r 2 25 <, which is satisfied for each r (, 2]. Then by Theorem.2. (namely, Theorem.3.2), the problem (.2)-(.3) has a solution on [, ]. Now we study the fractional boundary value problem (.6)-(.7). Definition.3.6. A function y AC([, T], E) is said to be a solution of (.6)-(.7) if y satisfies the equation D r y(t) = f (t, y(t)) on J, and the conditions (.7). For the existence of solutions for the problem (.6)-(.7), we need the following auxiliary lemma. Lemma.3.4. Let < r 2 and let h : [, T] R be continuous. A function y is a solution of the fractional integral equation y(t) = t (t s)r h(s)ds + r T T r βη r (T s)r h(s)ds βt r η (η (.4) s)r h(s)ds T r βη r if and only if y is a solution of the fractional BVP D r y(t) = h(t), t [, T], (.5) y() =, βy(η) = y(t). (.6)

7 352 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions Proof: Assume y satisfies (.5), then Lemma.3.3 implies that for c, c 2 R. Consequently, the general solution is y(t) = I r h(t) + c r + c 2 t r2, y(t) = t (t s) r h(s)ds + c Γ(α) r + c 2 t r2. From y() =, a simple calculation gives c 2 =, and from βy(η) = y(t) combined with y(t) = (T s) r h(s)ds + c T r, we have y(η) = η (η s) r h(s)ds + c η r, c = T T r βη r (T s) r β η h(s)ds T r βη r (η s) r h(s)ds. Hence we get equation (.4). Conversely, it is clear that if y satisfies equation (.4), then equations (.5)- (.6) hold. Remark.3.. The problem (.5)-(.6) is equivalent to y(t) = G(t, s)h(s)ds. (.7) where G(t, s) = [t(ts)] r βt r (ηs) r T r βη r (ts) r T r βη r T r βη r [t(ts)] r (ts) r T r βη r T r βη r [t(ts)] r βt r (ηs) r T r βη r, s t T, s η,, η s t T,, t s η T, [t(ts)] r, t s T, η s. T r βη r (.8) Remark.3.2. The function t Theorem.3.3. Assume (H),(H2) and the following hypothesis: G(t, s) ds is continuous on [, T], and hence is bounded. (H6) For almost each t J and each bounded set B E we have lim α( f (J l + t,l B )) α(b ). Then the BVP (.6)-(.7) has at least one solution on C(J, E), provided that G T p L <. (.9) Proof: Transform the problem (.6)-(.7) into a fixed point problem. Consider the operator (N y)(t) = where the function G(t, s) is given by (.8). G(t, s) f (s, y(s))ds (.2)

8 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 353 (We remark that, from Lemma.3.4, the fixed points of N are solutions to (.6)-(.7).) Let R > and consider the set D R = {y C(J, E) : y R }. We shall show that N satisfies the assumptions of Mönch s fixed point theorem. The proof will be given in several steps. where Step : N is continuous. Let {y n } be a sequence such that y n y in C(J, E). Then, for each t J, Let ρ > be such that By (H)-(H2) we have (N y n )(t) (N y)(t) G(t, s) f (s, y n(s) f (s, y(s)) ds sup (t,s) J J G(t, s) f (s, y n(s) f (s, y(s)) ds. G f (s, y n (s) f (s, y(s)) ds, G = sup G(t, s). (t,s) J J y n ρ and y ρ. f (s, y n (s) f (s, y(s)) 2ρG p(s) := σ(s); σ L (J, R + ). Since f is a Carathéodory function, the Lebesgue dominated convergence theorem implies that N (y n ) N (y) as n. Step 2: N maps D R into itself. For each y D R, by (H2) and (.9) we have for each t J, N (y)(t) G(t, s) f (s, y(s)) ds T p L G p L R. Step 3: N (D R ) is bounded and equicontinuous. By Step 2, it is obvious that N (D R ) C(J, E) is bounded. For the equicontinuity of N (D R ). Let τ, τ 2 J, τ < τ 2, and y D R. We have N (y)(τ 2 ) N 2 (y)(τ ) = G(τ 2, s) G(τ, s) f (s, y(s)) ds p(s)r G(τ 2, s) G(τ, s) ds As τ τ 2, the right-hand side of the above inequality tends to zero. Now let V be a subset of D R such that V co(n (V) {}). V is bounded and equicontinuous, and therefore the function ϑ ϑ = α(v(t)) is continuous on J. By (H6), Lemma.2., and the properties of the measure α, we have for each t J, This implies that ϑ(t) α(n (V)(t) {}) α(n (V)(t)) p(s) G(t, s) α(v(s))ds ϑ L [TG p L ]. ϑ L ( [TG p L ]). By (.9) it follows that ϑ =, that is, ϑ = for each t J, and then V(t) is relatively compact in E. In view of the Ascoli-Arzela theorem, V is relatively compact in D R. Applying now Theorem.2. we conclude that N has a fixed point which is a solution of the problem (.6)-(.7). (.2)

9 354 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions.4 Boundary Value Problems of Order r (2, 3] In this section, we will consider the boundary value problem D r y(t) = f (t, y(t)), for a.e. t J = [, T], (.22) y() =, y (T) =, y () =, (.23) where 2 < r 3, D r is the Riemann-Liouville fractional derivative and f and (E, ) are as in (.4)-(.5). We will make use of some of the hypotheses (H) - (H5) of Theorem.3.2 in this section..4. Main Results for (.22)-(.23) Let us start by defining what we mean by a solution of the problem (.22) (.23). Definition.4.7. A function y AC 2 ([, T], E) is said to be a solution of (.22)-(.23) if it satisfies D r y(t) = f (t, y(t)) on J, and the conditions y() =, y (T) = y () =. For the existence of solutions for the problem (.22)-(.23), we need the following auxiliary lemma. Lemma.4.5. Let 2 < r 3 and let h : [, T] E be continuous. A function y is a solution of the fractional integral equation y(t) = (t s)r h(s)ds tr (T (.24) s)r2 h(s)ds. if and only if y is a solution of the fractional BVP T r2 Proof: Assume y satisfies (.25), then Lemma.3.2 implies that From (.26), a simple calculation gives and D r y(t) = h(t), t [, T], (.25) y() =, y (T) =, y () =. (.26) y(t) = c r + c 2 t r2 + c 3 t r3 + t (t s) r h(s)ds. Γ(α) c = T r2 c 2 =, c 3 = (T s) r2 h(s)ds. Hence we get equation (.24). Conversely, it is clear that if y satisfies equation (.24), then equations (.25)- (.26) hold. Theorem.4.4. Assume (H), (H2) and the following hypothesis: (H7) For almost each t J and each bounded set B 2 E we have lim α( f (J k + t,k B 2 )) α(b 2 ). Then the BVP (.22)-(.23) has at least one solution on C(J, B), provided that [ T r Γ(r + ) + T 2r ] p (r ) L <. (.27) Proof. Transform the problem (.22)-(.23) into a fixed point problem. Consider the operator (N 2 y)(t) = (t s)r f (s, y(s))ds tr T (T s)r2 f (s, y(s))ds. T r2

10 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 355 Remark.4.3. Clearly, from Lemma.4.5, the fixed points of N 2 are solutions to (.22)-(.23). Let R 2 > and consider the set D R2 = {y C(J, E) : y R 2 }. We shall show that N 2 satisfies the assumptions of Mönch s fixed point theorem. The proof will be given in several steps. Step : N 2 is continuous. Let {y n } be a sequence such that y n y in C(J, E). Then, for each t J, Let ρ > be such that By (H2) we have (N 2 y n )(t) (N 2 y)(t) t (t s)r f (s, y n (s) f (s, y(s)) ds (T s)r2 f (s, y n (s) f (s, y(s)) ds. tr T r2 y n ρ, y ρ. f (s, y n (s) f (s, y(s)) 2ρp(s) := σ(s); σ L (J, R + ). Since f is a Carathéodory function, the Lebesgue dominated convergence theorem implies that N 2 (y n ) N 2 (y) as n. Step 2: N 2 maps D R2 into itself. For each y D R2, by (H2) and (.27) we have for each t J t t r T T r2 T r Γ(r+) + N 2 (y)(t) (t s)r f (s, y(s)) ds (T s)r2 f (s, y(s)) ds [[ ] ] R 2 p L R 2. Step 3: N 2 (D R2 ) is bounded and equicontinuous. T2r (r) By Step 2, it is obvious that N 2 (D R2 ) C(J, E) is bounded. For the equicontinuity of N 2 (D R2 ). Let t, t 2 J, t < t 2, and y D R2. we have (N 2 y)(t 2 ) (N 2 y)(t ) = t [(t 2 s) r (t s) r ] f (s, y(s))ds + t2 (t 2 s) r f (s, y(s))ds t + (t 2 t ) r T r2 (T s) r2 f (s, y(s)) ds t [(t s) r (t 2 s) r ]ds + t2 (t 2 s) r ds t + (t 2 t ) r T r2 (T s) r2 ds Γ(r ) Γ(r + ) [(t 2 t ) r + t r tr 2 ] + Γ(r + ) (t 2 t ) r + (t 2 t ) r T 2r Γ(r + ) (t 2 t ) r + Γ(r + ) (tr tr 2 ) + T2r (t 2 t ) r.

11 356 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions As t t 2, the right-hand side of the above inequality tends to zero. Now let V 2 be a subset of D R2 such that V 2 co(n 2 (V 2 ) {}). V 2 is bounded and equicontinuous, and therefore the function ϑ ϑ = α(v 2 (t)) is continuous on J. By (H3), Lemma.2., and the properties of the measure α, we have for each t J This means that ϑ(t) α(n 2 (V 2 )(t) {}) α(n 2 (V 2 )(t)) [ ] t T r Γ(r+) + T2r p(s)α(v 2 (s))ds [[ ] ] ϑ L p L. ϑ L ( [ T r Γ(r+) + T2r T r Γ(r + ) + T2r ] ) p L. By (.27) it follows that ϑ =, that is, ϑ = for each t J, and then V 2 (t) is relatively compact in E. In view of the Ascoli-Arzela theorem, V 2 is relatively compact in D R2. Applying now Theorem.2., we conclude that N 2 has a fixed point which is a solution of the problem (.22) (.23). 2 Boundary Value Problems for Fractional Differential Inclusions in Banach Spaces 2. Introduction In this section, we are concerned with the existence of solutions for boundary value problems, for a class of fractional order differential inclusions, when the right hand side is convex valued. This result relies on the setvalued analog of Mönch s fixed point theorem combined with the technique of measure of noncompactness. Recently, this has proved to be a valuable tool in solving fractional differential equations and inclusions in Banach spaces; for details, see the papers of Lasota et al. [39], Agarwal et al. [4] and Benchohra et al. [8], [9], [2]. This result extends to the multivalued case some previous results in the literature, and constitutes an interesting contribution to this emerging field. 2.2 Preliminaries We introduce notations, definitions, and preliminary facts that will be used in the remainder of this section. Let (E, ) be a Banach space. Let P cl (E) = {A P(E) : A closed}, P c (E) = {A P(E) : A convex}, P cp,c (E) = {A P(E) : A compact and convex}. A multivalued mapping G : E P(E) has a fixed point if there is x E such that x G(E). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map G : J P cl (R) is said to be measurable if for every y R, the function t d(y, G(t)) = inf{ y z : z G(t)} is measurable. Let X, Y be two sets, and N : X P(Y) be a set-valued map. We define graph(n) = {(x, y) : x X, y N(X)}. For more details on multi-valued maps see the books of Deimling [23], Aubin et al. Papageorgiou [34]. [7, 8] and Hu and and Let R >, and let B = {x E : x R}, U = {x C(J, E) : x R},

12 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 357 Clearly U = C(J, B). For each y C(J, E), define the set of selections of F by S F,y = {v L (J, E) : v(t) F(t, y(t)) a.e. t J}. Theorem ([32]) Let E be a Banach space and C L (J, E) be countable with u(t) h(t) for a.e. t J, and every u C, where h L (J, R + ). Then the function φ(t) = α(c(t)) belongs to L (J, R + ) and satisfies where α is the Kuratowski measure of non compactness. ({ }) α u(s)ds : u C 2 α(c(s))ds, Let us now recall the set-valued analog of Mönch s fixed point theorem. Theorem ([44]) Let K be a closed, convex subset of a Banach space E, U a relatively open subset of K, and N : U P c (K). Assume that graph(n) is closed, that N maps compact sets into relatively compact sets, and that, for some x U, the following two conditions are satisfied : { M U, M conv(x N(M)) and M = U with C Mcountable Mcompact, (2.28) Then there exists x U with x N(x). x ( λ)x + λn(x) for allx U U, λ (, ). (2.29) Lemma ([39]) Let I be a compact real interval. Let F be a Carathéodory multivalued map and let Θ be a linear continuous map from L (I, E) C(I, E). Then the operator is a closed graph operator in C(I, E) C(I, E). Θ S F,y : C(I, E) P cp,c (C(I, E)), y (Θ S F,y )(y) = Θ(S F,y ) 2.3 Boundary Value Problems of Order r (, 2] We consider the boundary value problem D r y(t) F(t, y(t)), for a.e. t J = [, T], (2.3) y() =, βy(η) = y(t), (2.3) where < r 2, < βη α <, < η <, D r is as in (.6)-(.7), (E, ) is a Banach space, F : J E P(E) is a multivalued map, and P(E) is the family of all nonempty subsets of E Main Results for (2.3)-(2.3) Let us start by defining what we mean by a solution of the problem (2.3)-(2.3). Definition A function y AC ([, T], E) is said to be a solution of (2.3)-(2.3) if there exists a function v L (J, E) with v(t) F(t, y(t)), for a.e. t J, such that D r y(t) = v(t) on J, and the condition (2.3) is satisfied. For the existence of solutions for the problem (2.3)-(2.3), we make use of the auxiliary Lemma.3.2 and Lemma.3.4. Theorem Assume the following hypotheses hold: (H ) F : J R P cp,c (R) is a Carathéodory multi-valued map.

13 358 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions (H 2) For each R >, there exists a function p L (J, E) and such that for each (t, y) J E with y R, and F(t, u) P = sup{ v, v(t) F(t, y)} lim inf R + dt = δ <. R (H 3) There exists a Carathéodory function ψ, : J [, 2R] R + such that α(f(t, M)) ψ(t, α(m)), a.e. t J, and each M B, and the unique solution ϕ C(J, [, 2R]) of the inequality, [ t ] φ(t) 2 G(t, s)ϕ(s, φ(s))ds is φ. (2.32) Then the BVP (2.3)-(2.3) has at least one solution on C(J, B), provided that where G = TG p L <, (2.33) sup G(t, s). (t,s) J J Proof. Transform the problem (2.3)-(2.3) into a fixed point problem. Consider the multivalued operator { t } Q(y) = h C(J, E) : h(t) = G(t, s)v(s)ds, v S F,y. We shall show that Q satisfies the assumptions of the set-valued analog of Mönch s fixed point theorem. The proof will be given in several steps. Step : Q(y) is convex for each y C(J, E). Indeed, if h, h 2 belong to Q(y), then there exist v, v 2 S F,y such that for each t J we have h i (t) = Let d. Then, for each t J, we have t (dh + ( d)h 2 )(t) = Since S F,y is convex (because F has convex values), we have G(t, s)v i (s)ds, i =, 2. t G(t, s)[dv (s) + ( d)v 2 (s)]ds. dh + ( d)h 2 Q(y). Step 2: Q(M) is relatively compact for each compact M U. Let M U be a compact set and let {h n } by any sequence of elements of Q(M). We show that {h n } has a convergent subsequence by using the Ascoli-Arzela criterion of compactness in C(J, E). Since h n Q(M) there exist y n M and v n S F,yn such that h n (t) = t G(t, s)v n (s)ds. Using Theorem and the properties of the measure of noncompactness of Kuratowski, α, we have [ t ] α({h n (t)}) 2 G α({(v n (s)})ds. (2.34)

14 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 359 On the other hand, since M(s) is compact in E, the set {v n (s) ; n } is compact. Consequently, α({v n (s) ; n }) = for a.e. s J. Now (2.32) implies that{h n (t) ; n } is relatively compact in E, for each t J. In addition, for each τ, τ 2 J, τ < τ 2, and y D R, we have Q(y)(τ 2 ) Q(y)(τ ) = G(τ 2, s) G(τ, s) v(s) ds p(s) G(τ 2, s) G(τ, s) ds p(s)ds G(τ 2, s) G(τ, s) ds (2.35) As τ τ 2, the right-hand side of the above inequality tends to zero. This shows that {h n ; n } is equicontinuous. Consequently, {h n ; n } is relatively compact in C(J, E). Step 3: Q has a closed graph. Let (y n, h n ) graph(q), n, with y n y, h n h, as n. We must show that (y, h) graph(q). (y n, h n ) graph(q) means that h n Q(y n ), which means that there exists v n S F,yn, such that for each t J, Consider the continuous linear operator h n (t) = t G(t, s)v n (s)ds. Θ : L (J, E) C(J, E), Clearly, Θ(v)(t) h n (t) = t G(t, s)v n (s)ds h n (t) h(t), as, n. From Lemma it follows that Θ S F is a closed graph operator. Moreover, we have Since y n y, Lemma implies that for some v S F,y. h n (t) Θ(S F,yn ). h(t) = t G(t, s)v(s)ds Step 4: Suppose M U, M conv({} Q(M)), and M = C for some countable set C M. Using an estimation of type (2.35), we see that Q(M) is equicontinuous. Then from M conv({} Q(M)), We deduce that M is equicontinuous, too. In order to apply the Ascoli-Arzela theorem, it remains to show that M(t) is relatively compact in E for each t J. Since C M conv({} Q(M)) and C is countable, we can find a countable set H = {h n : n } Q(M) with C conv({} H). Then, there exist y n M and v n S F,yn such that h n (t) = t G(t, s)v n (s)ds From M C conv({} H)), and according to Theorem 2.2.5, we have α(m(t)) (α(c(t)) α(h(t)) = α({h n ((t) : n }). Using(2.34), we obtain [ t ] α(m(t)) 2 G α({v n (s)})ds.

15 36 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions Now, since v n (s) M(s), we have [ t ] α(m(t)) 2 G α({v n (s); n })ds Also, since v n (s) M(s), we have It follows that α({v n (s); n }) = α(m(s)) [ ] t α(m(t)) 2 [ G α(m(s))ds ] t 2 G ψ(s, α(m(s)))ds. Also, the function φ given by φ(t) = α(m(t)) belongs to C(J, [, 2R]). Consequently by (H 3), φ ; that is, α(m(t)) = for all t J. Now, by the Ascoli-Arzela theorem, M is relatively compact in C(J, E). Step 5: Let h Q(y) with y U. Since y(s) R and by (H 2), we have Q(U) U, because if it were not true, then there exists a function y U, but Q(y) P > R and for some v S F,y. On the other hand, we have h(t) = t G(t, s)v(s)ds R Q(y) P t G(t, s) v(s) ds TG p L. Dividing both sides by R and taking the lower limits as R, we conclude that [TG p L ]δ which contradicts (2.33). Hence Q(U) U. As a consequence of Steps -5 together with Theorem 2.2.6, we can conclude that Q has a fixed point y C(J, B) which is a solution of the problem (2.3)-(2.3). 2.4 Boundary Value Problems of Order r (2, 3] In this section,we are concerned with the existence of solutions for the boundary value problem for a fractional differential inclusion, D r y(t) F(t, y(t)), for a.e. t J = [, T], (2.36) y() =, y () =, y (T) =, (2.37) where 2 < r 3, D r is the Riemann-Liouville fractional derivative, F and (E, ) as are in (2.3)-(2.3) Main Results for (2.36)-(2.37) Let us start by defining what we mean by a solution of the problem (2.36) (2.37). Definition A function y AC 2 ([, T], E) is said to be a solution of (2.36)-(2.37) if there exist a function v L (J, E) with v(t) F(t, y(t)), for a.e. t J, such that D r y(t) = v(t) on J, and the condition y() =, y () = y (T) =. For the existence of solutions for the problem (2.36)-(2.37), we will make use of the auxiliary Lemma.3.2 and Lemma.4.5. Theorem Assume (H )-(H 2) and the following hypothesis hold:

16 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 36 (H 4) There exists a Carathéodory function ψ : J [, 2R] R + such that α(f(t, M)) ψ(t, α(m)), a.e. t J, and each M B, and the unique solution ϕ C(J, [, 2R]) of the inequality φ(t) 2[ (t s)r ϕ(s, φ(s))ds t r (r)(r2)γ(r2) (T s)r3 ϕ(s, φ(s))ds], t J (2.38) is φ. Then the BVP (2.36)-(2.37) has at least one solution on C(J, B), provided that [ δ < T Γ(r + ) + T 2 (r )(r 2)Γ(r ) ]. (2.39) Proof. Transform the problem (2.36)-(2.37) into a fixed point problem. Consider the multivalued operator Q 2 (y) = h C(J, E) : h(t) = (t s)r v(s)ds t r (r)(r2)γ(r2) T (T s)r3 v(s)ds, v S F,y. We shall show that Q 2 satisfies the assumptions of the set-valued analog of Mönch s fixed point theorem. The proof will be given in several steps. Step : Q 2 (y) is convex for each y C(J, E). Indeed, if h, h 2 belong to Q 2 (y), then there exist v, v 2 S F,y such that for each t J we have h i (t) = Let d. Then, for each t J, we have (t s)r v i (s)ds t r (r)(r2)γ(r2) (T s)r3 v i (s)ds, i =, 2. (dh + ( d)h 2 )(t) = (t s)r [dv (s) + ( d)v 2 (s)]ds+ t r (r)(r2)γ(r2) T (T s)r3 [dv (s) + ( d)v 2 (s)]ds. Since S F,y is convex (because F has convex values), we have dh + ( d)h 2 N(y). Step 2: Q 2 (M) is relatively compact for each compact M U. Let M U be a compact set and let {h n } by any sequence of elements of Q 2 (M). We show that{h n } has a convergent subsequence by using the Ascoli-Arzela criterion of compactness in C(J, E). Since {h n } Q 2 (M) there exist {y n } M and v n S F,yn such that h n (t) = (t s)r v n (s)ds t r (r)(r2)γ(r2) (T s)r3 v n (s)ds. Using Theorem and the properties of the measure of Kuratowski α, we have α({h n (t)}) 2[ t α({(t s)r v n (s)})ds t α (r)(r2)γ(r2) α({(t s)r3 v n (s)})ds ]. (2.4)

17 362 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions On the other hand, since M(s) is compact in E, the set {v n (s) ; n } is compact. Consequently, α({v n (s) ; n }) = for a.e. s J. Furthermore α({(t s) r v n (s)}) = (t s) r α({v n (s) ; n }) = α({(t s) r v n (s)}) = (T s) r α({v n (s) ; n }) = for a.e. t, s J. Now (2.4) implies that{h n (t) ; n } is relatively compact in E, for each t J. In addition for each t and t 2 from J, t < t 2, we have h n (t 2 ) h n (t ) = t [(t 2 s) r (t s) r ]v n (s)ds + t2 t (t 2 s) r v n (s)ds + (t 2t ) r (T s)r3 v n (s) ds (r)(r2)γ(r2) t [(t s) r (t 2 s) r ]ds t2 + + (t 2t ) r t (t 2 s) r ds (r)(r2)γ(r2) (T s)r3 ds Γ(r+) [(t 2 t ) r + t r tr 2 ] + Γ(r+) (t 2 t ) r + (t 2t ) r Γ(r) Γ(r+) (t 2 t ) r + + Tr (t 2 t ) r. Γ(r) Γ(r+) (tr tr 2 ) (2.4) As t t 2, the right-hand side of the above inequality tends to zero.this shows that {h n ; n } is equicontinuous. Consequently,{h n ; n } is relatively compact in C(J, E). Step 3: Q 2 has a closed graph. Let (y n, h n ) graph(q 2 ), n, with y n y, h n h, as n. We must show that (y, h) graph(q 2 ). (y n, h n ) graph(q 2 ) means that h n Q 2 (y n ), which means that there exists v n S F,yn, such that for each t J, h n (t) = Consider the continuous linear operator Clearly, (t s)r v n (s)ds t r (r)(r2)γ(r2) Θ(v)(t) h n (t) = Θ : L (J, E) C(J, E) (T s)r3 v n (s)ds. (t s)r v n (s)ds t r (r)(r2)γ(r2) h n (t) h(t), as n. (T s)r3 v n (s)ds. From Lemma it follows that Θ S F is a closed graph operator. Moreover, we have h n (t) Θ(S F,yn ). Since y n y, Lemma implies that h(t) = (t s)r v(s)ds t r T (T s)r3 v(s)ds, for some v S F,y. (r)(r2)γ(r2) Step 4: Suppose M U, M conv({} Q 2 (M)), and M = C for some countable set C M. Using an estimation of type (2.4), we see that Q 2 (M) is equicontinuous. Then from M conv({} Q 2 (M)), we

18 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions 363 deduce that M is equicontinuous, too. In order to apply the Ascoli-Arzela theorem, it remains to show that M(t) is relatively compact in E for each t J. Since C M conv({} N(M)) and C is countable, we can find a countable set H = {h n : n } Q 2 (M) with C conv({} H). Then, there exists y n M and v n S F,yn such that h n (t) = (t s)r v n (s)ds t r (r)(r2)γ(r2) (T s)r3 v n (s)ds. From M C conv({} H)), and according to Theorem 2.2.5, we have Using (2.4), we obtain Now, since v n M(s) we have α(m(t)) (α(c(t)) α(h(t)) = α({h n ((t) : n }). α(m(t)) 2[ α(m(t)) 2[ Also, since v n (s) M(s) we have and It follows that α({(t s)r v n (s)})ds t α (r)(r2)γ(r2) t α({(t s)r3 v n (s)})ds]. α({(t s)r v n (s); n })ds t α T α({(t s)r3 v n (s); n })ds]. (r)(r2)γ(r2) α({(t s) r v n (s); n }) = (t s) r α(m(s)) α({(t s) r v n (s); n }) = (T s) r α(m(s)). α(m(t)) 2[ 2[ (t s)r α(m(s))ds t α (r)(r2)γ(r2) t (T s)r3 α(m(s))ds] (t s)r ψ(s, α(m(s)))ds t α (r)(r2)γ(r2) (T s)r3 ψ(s, α(m(s)))ds]. Also, the function φ given by φ(t) = α(m(t)) belong to C(J, [, 2R]). Consequently by (H 4), φ, that is α(m(t)) = for all t J. Now, by the Ascoli-Arzela theorem, M is relatively compact in C(J, E). Step 5: Let h Q 2 (y) with y U. Since y(s) R and (H 2), we have N(U) U, because if it is not true, then there exists a function y U but Q 2 (y) P > R and h(t) = for some v S F,y. On the other hand we have t R Q 2 (y) P (t s)r v(s)ds t r T (T s)r3 v(s)ds, (r)(r2)γ(r2) t (t s)r v(s) ds t r (r)(r2)γ(r2) T Γ(r+) T 2 [ t p(s)ds (T s)r3 v(s) ds (r)(r2)γ(r) T Γ(r+) + T 2 (r)(r2)γ(r) p(s)ds ] p(s)ds.

19 364 Samira Hamani and Johnny Henderson / Fractional Differential Equations and Inclusions Dividing [ both sides by ] R and taking the lower limits as R, we conclude that δ which contradicts (2.39). Hence Q 2 (U) U. T Γ(r+) + T 2 (r)(r2)γ(r) As a consequence of Steps -5 together with Theorem 2.2.6, we can conclude that Q 2 has a fixed point y C(J, B) which is a solution of the problem (2.36)-(2.37). References [] Agarwal, R.P., Benchohra, M., Hamani, S.: Boundary value problems for fractional differential inclusions. Adv. Stud. Contemp. Math. 6 (28) [2] Agarwal, R.P., Benchohra, M., Hamani, S., Pinelas, S.: Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half line. Dynamics of Continuous, Discrete and Impulsive Systems 8 (2), [3] Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems for nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematics 9 (2), [4] Agarwal, R.P., Benchohra, M., Seba, D.: An the application of measure of noncompactness to the existence of solutions for fractional differential equations. Results Math. 55 (29), [5] Agarwal, R.P., Meehan, M., and O Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics, vol. 4, Cambridge University Press, Cambridge, (2). [6] Akhmerov, R.R., Kamenski, M.I., Patapov, A.S., Rodkina, A.E., Sadovski, B.N.: Measure of Noncompactness and Condensing Operators. Translated from the 986 Russian original by A. Iacop. Operator theory. Advances and Applications, vol. 55, Birhhauser Verlag, Bassel, (992). [7] Aubin, J.P., Cellina, A.: Differential Inclusions. Springer-Verlag, Berlin-Heidelberg, New York, (984). [8] Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston, (99). [9] Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72 (2), [] Bai, Z., Lu, H.S.: Positive solution of a boundary value problem of a nonlinear fractional differential equation. J. Math. Anal. Appl. 3 (25), [] Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 6, Marcel Dekker, New York, (98). [2] Banas, J., Sadarangani, K.: On some measure of noncompactness in the space of continuous functions. Nonlinear Anal. 6 (28), [3] Benchohra M., Djebali, S., Hamani, S.: Boundary value problems of differential inclusions with Riemann- Liouville fractional derivative. Nonlinear Oscillations 4 (2), 7 2. [4] Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions. Appl. Anal. 87 (28), [5] Benchohra, M., Hamani, S.: Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative. Topol. Meth. Nonlinear Anal. 32 (28), 5 3. [6] Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order. Diss. Math. Diff. Inclusions. Control and Optimization 28 (28), [7] Benchohra, M., Hamani, S.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 7 (29),

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