On boundary value problems for fractional integro-differential equations in Banach spaces

Transcription

1 Malaya J. Mat On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad - 434, Maharashtra, India. Abstract This paper aims to study the existence and uniqueness of solutions of fractional integro-differential equations in Banach spaces by applying a new generalized singular type Gronwall s inequality, fixed point theorems and Hölder inequality. Example is provided to illustrate the main results. Keywords: Fractional integro-differential equations, boundary value problems, existence and uniqueness, generalized singular type Gronwall s inequality, fixed point theorems. 2 MSC: 26A33, 34A8, 34B5. c 22 MJM. All rights reserved. Introduction This paper deals with the existence and uniqueness of solutions of boundary value problems for short BVP for fractional integro-differential equations given by { c D α xt = f t, xt, Sxt, t J = [, T], α, ], ax bxt = c,. where c D α is the Caputo fractional derivative of order α, f : J X X X is a given function satisfying some assumptions that will be specified later and a, b, c are real numbers with a b = and S is a nonlinear integral operator given by Sxt = t k t, s, xs ds, where k CJ J X, X. The ordinary differential equations is considered the basis of the fractional differential equations. In the last few decades, fractional order models are found to be more adequate than integer order models for some real world problems. For more details about fractional calculus and its applications we refer the reader to the monographs of Hilfer [6], Kilbas et al. [8], Miller and Ross [9], Podlubny [], Samko et al. [] and the references given therein. Recently, some fractional differential equations and optimal controls in Banach spaces were studied by Balachandran and Park [2], El-Borai [3], Henderson and Ouahab [4], Hernandez et al. [5], Wang et al. [4] and Wang et al. [5, 6]. Very recently, Karthikeyan and Trujillo [7] and Wang et al. [3] have extended the work in [] from real line R to the abstract Banach space X by using more general assumptions on the nonlinear function f. Our attempt is to generalize the results proved in [, 7, 3]. This paper is organized as follows. In Section 2, we set forth some preliminaries. Section 3 introduces a new generalized singular type Gronwall inequality to establish the estimate for priori bounds. In Section 4, we prove our main results by applying Banach contraction principle and Schaefer s fixed point theorem. Finally, in Section 5, application of the main results is exhibited. Corresponding author. address: th.sabri@yahoo.com Sabri T. M. Thabet, mbdhakne@yahoo.commachindra B. Dhakne.

2 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations Preliminaries Before proceeding to the statement of our main results, we set forth some preliminaries. Let the Banach space of all continuous functions from J into X with the supremum norm x := sup{ xt : t J} is denoted p by CJ, X. For measurable functions m : J R, define the norm m L p J,R = J dt mt p, p <, where L p J, R the Banach space of all Lebesgue measurable functions m with m L p J,R <. Definition 2.. The Riemann-Liouville fractional integral of order α > of a suitable function h is defined by where a R and Γ is the Gamma function. Iaht α = t t s α hsds, a Definition 2.2. For a suitable function h given on the interval [a, b], the Riemann-Liouville fractional derivative of order α > of h, is defined by D α aht = d n t t s n α hsds, Γn α dt a where n = [α], [α] denotes the integer part of α. Definition 2.3. For a suitable function h given on the interval [a, b], the Caputo fractional order derivative of order α > of h, is defined by c Daht α t = t s n α h n sds, Γn α a where n = [α], [α] denotes the integer part of α. Lemma 2.. [8, 7] Let α > ; then the differential equation c D α ht =, has the following general solution ht = c c t c 2 t 2 c n t n, where c i R, i =,, 2,..., n, where n = [α]. Lemma 2.2. [8, 7] Let α > ; then for some c i R, i =,, 2,..., n, where n = [α]. I α c D α ht = ht c c t c 2 t 2 c n t n, Definition 2.4. A function x C J, X is said to be a solution of the fractional BVP. if x satisfies the equation c D α xt = f t, xt, Sxt a.e. on J, and the condition ax bxt = c. For the existence of solutions for the fractional BVP., we need the following auxiliary lemma. Lemma 2.3. Let f : J X be continuous. A function x CJ, X is solution of the fractional integral equation xt = t t s α f sds [ b T ] T s α f sds c, 2.2 a b if and only if x is a solution of the following fractional BVP { c D α xt = f t, t J = [, T], α, ], ax bxt = c. 2.3 Proof. Assume that x satisfies fractional BVP 2.3; then by using Lemma 2.2 and Def. 2., we get xt c = t t s α f sds,

3 542 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... where c R, that is: By applying boundary condition ax bxt = c, we have c = Now, by substituting the value of c in 2.4, we obtain xt = t t s α f sds c. 2.4 b T T s α f sds c a b a b. xt = t t s α b T f sds T s α f sds c a b a b. Conversely, it is clear that if x satisfies fractional integral equation 2.2, then fractional BVP 2.3 is also satisfied. As a consequence of lemma 2.3, we have the following result which is useful in what follows. Lemma 2.4. Let f : J X X X be continuous function. Then, x CJ, X is a solution of the fractional integral equation xt = t t s α f s, xs, Sxs ds a b [ b T if and only if x is solution of the fractional BVP.. T s α f s, xs, Sxs ] ds c, Lemma 2.5. Bochner theorem A measurable function f : J X is Bochner integrable if f is Lebesgue integrable. Lemma 2.6. Mazur theorem, [2] Let X be a Banach space. If U X is relatively compact, then convu is relatively compact and convu is compact. Lemma 2.7. Ascoli-Arzela theorem Let S = {st} is a function family of continuous mappings s : [a, b] X. If S is uniformly bounded and equicontinuous, and for any t [a, b], the set {st } is relatively compact, then, there exists a uniformly convergent function sequence {s n t}n =, 2,..., t [a, b] in S. Lemma 2.8. Schaefer s fixed point theorem Let F : X X be a completely continuous operator. If the set EF = {x X : x = ηfx for some η [, ]} is bounded, then, F has fixed points A generalized singular type Gronwall s inequality Before dealing with the main reslts, we need to introduce a new generalized singular Gronwall type inequality with mixed type singular integral operator. We, first, state a generalized Gronwall inequality from [5]. Lemma 3.9. Lemma 3.2, [5] Let x CJ, X satisfies the following inequality: t xt a b e T T t xθ λ dθ c xθ λ 2 dθ d x θ λ 3 B dθ x θ λ 4 B dθ, t J, where λ, λ 3 [, ], λ 2, λ 4 [,, a, b, c, d, e are constants and x θ B = sup sθ xs. Then there exists a constant L > such that xt L. Using the above generalized Gronwall inequality, we can obtain the following new generalized singular type Gronwall inequality.

4 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations Lemma 3.. Let x CJ, X satisfies the following inequality: t T xt a b t s α xs λ ds c T s α xs λ ds t T d t s α x s λ B ds e T s α x s λ B ds, 3.6 where α, ], λ [, p for some < p < α, x s B constants.then, there exists a constant L >, such that = sup τs xτ and a, b, c, d, e are Proof. Let Using 3.6 and Hölder inequality, we get xt yt a b t d yt = t xt L. {, xt, xt, xt >. T t s α ys λ ds c T s α ys λ ds T t s α y s λ B ds e T s α y s λ B ds t a b t s pα p t ds ys λp p ds T c T s pα p T ds ys λp p ds t T p p d y s λ B t s α ds e y s λ B T s α ds t a pα p t b ys λp p ds pα T pα p T c ys λp p ds pα d y s λ t α B α e y s λ B α a d y s λ T α B α e y s λ T α B α T pα p t b ys λp p ds pα T α T pα p T c pα ys λp p ds, p p where < λp p <. Hence, by lemma 3.9 there exists a constant L >, such that xt L. 4 Main results For convenience, we list hypotheses that will be used in our further discussion.

5 544 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... H The function f : J X X X is measurable with respect to t on J and is continuous with respect to x on X. α H2 There exists a constant α, α and real-valued functions m t, m 2 t L J, R, such that f t, xt, Sxt f t, yt, Syt m t xt yt Sxt Syt, for each s [, t], t J and all x, y X. k t, s, xs k t, s, ys m 2 t xs ys, α H3 There exists a constant α 2, α and real-valued function ht L 2 J, R, such that f t, xt, Sxt ht, for each t J, and all x X. For brevity, let M = m m m 2 T L α and H = h J,R L α. 2 J,R H4 There exist constants λ [, p for some < p < α and N f, N k >, such that f t, xt, Sxt N f xt λ Sxt, k t, s, xs N k xs λ, for each s [, t], t J and all x X. H5 For every t J, the set K = { t s α f s, xs, Sxs : x CJ, X, s [, t] } is relatively compact. Now, we are in position to deal with our main results. Theorem 4.. Assume that H-H3 hold. If Ω α,t = M T α α α α α α Then, the fractional BVP. has a unique solution on J. b <. 4.7 Proof. By making use of hypothesis H3 and Hölder inequality, for each t J, we have t t s α f s, xs, Sxs ds t t t s α f s, xs, Sxs ds t s α hsds t α t s α α2 t 2 ds hs α2 α 2 ds o α t α t s 2 H α α 2 α 2 H Tα α 2 α α. 2 α 2 α 2 Thus, t s α f s, xs, Sxs is Lebesgue integrable with respect to s [, t] for all t J and x CJ, X. Then, t s α f s, xs, Sxs is Bochner integrable with respect to s [, t] for all t J due to lemma 2.5. Hence, the fractional BVP. is equivalent to the following fractional integral equation xt = t a b t s α f s, xs, Sxs ds [ b T T s α f s, xs, Sxs ] ds c, 4.8 Now, let B r = {x CJ, X : x r}, where r HT α α 2 α α 2 α 2 α 2 b HT α α2 α α c 2 α 2 α

6 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations Define the operator F on B r as follows: Fxt = t a b t s α f s, xs, Sxs ds [ b T T s α f s, xs, Sxs ] ds c, t J. 4. Clearly, the solution of the fractional BVP. is the fixed point of the operator F on B r. We shall use the Banach contraction principle to prove that F has a fixed point. The proof is divided into two steps. Step. Fx B r for every x B r. For every x B r and δ >, by H3 and Hölder inequality, we have Fxt δ Fxt tδ t δ s α f s, xs, Sxs ds t t s α f s, xs, Sxs ds t [ t δ s α t s α ] f s, xs, Sxs ds tδ t t tδ t t t δ s α f s, xs, Sxs ds [t δ s α t s α ]hsds t tδ t α α2 H δ α 2 α α 2 α 2 t δ s α hsds α t δ s α α2 t 2 ds α hs α2 2 ds α t s α α2 t 2 α ds hs 2 ds t δ s α α 2 ds α α 2 α t δ 2 α α 2 α 2 α2 tδ α 2 t H α2 α2 α hs 2 ds α α2 t α 2 α α 2 α 2 α 2 H α α2 δ α 2 α α 2 α 2 α 2. It is obvious that the right-hand side of the above inequality tends to zero as δ. Therefore, F is continuous on J, that is, Fx CJ, X. Moreover, for x B r and all t J, by using 4.9, we have Fxt t t s α f s, xs, Sxs ds T s α f s, xs, Sxs ds c t t s α hsds T s α hsds c t b t s α α 2 ds T α2 t T s α α 2 ds hs α 2 ds α2 T α2 hs α 2 ds α2 c

7 546 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... r. HT α α 2 α α 2 α 2 α 2 b HT α α2 α α c 2 α 2 α 2 Thus, Fx r and we conclude that for all x B r, Fx B r, that is, F : B r B r. Step 2. F is contraction mapping on B r. For x, y B r and any t J, by using 4.7, H2 and Hölder inequality, we have Fxt Fyt t t s α f s, xs, Sxs f s, ys, Sys ds T s α f s, xs, Sxs f s, ys, Sys ds t t s α m s xs ys Sxs Sys ds T s α m s xs ys Sxs Sys ds t t s α m s s xs ys k s, τ, xτ ks, τ, yτ dτ ds T b T s α m s s xs ys k s, τ, xτ ks, τ, yτ dτ ds t t s α m s s xs ys m 2 s xτ yτ dτ ds T b T s α m s s xs ys m 2 s xτ yτ dτ ds t t s α m s x y m 2 st x y ds T s α m s x y m 2 st x y ds t x y b T T t s α α ds α t α T s α α ds m s m sm 2 st α ds α α M x y t α α α α [ M T α α α α α α = Ω α,t x y. α α M b x y b ] x y. m s m sm 2 st α α ds α α T α α α α α

8 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations Thus, we have Fx Fy Ω α,t x y. Since Ω α,t <, F is contraction. By Banach contraction principle, we can deduce that F has a unique fixed point which is the unique solution of the fractional BVP.. Our second main result is based on the well known Schaefer s fixed point theorem. Theorem 4.2. Assume thath, H4 and H5 hold. Then the fractional BVP. has at least one solution on J. Proof. Transform the fractional BVP. into a fixed point problem. Consider the operator F : CJ, X CJ, X defined as 4.. It is obvious that F is well defined due to H, Hölder inequality and the lemma 2.5. For the sake of convenience, we subdivide the proof into several steps. Step. F is continuous operator. Let {x n } be a sequence such that x n x in CJ,X. Then for each t J, we have Fx n t Fxt t t s α f s, xn s, Sx n s f s, xs, Sxs ds T s α f s, xn s, Sx n s f s, xs, Sxs ds t f., x n., Sx n. f., x., Sx. t s α ds f., xn., Sx n. f., x., Sx. T s α ds f., x n., Sx n. f., x., Sx. t α α f., x n., Sx n. f., x., Sx. α α T α b f., xn., Sx n. f., x., Sx. Γα. Taking supremum, we get Fx n Fx T α Γα b f., xn., Sx n. f., x., Sx., since f is continuous, we have Therefore, F is continuous operator. Fx n Fx as n. Step 2. F maps bounded sets into bounded sets in CJ, X. Indeed, it is enough to show that for any η >, there exists a l > such that for each

9 548 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... x B η = {x CJ, X : x η }, we have Fx l. For each t J, by H4, we get Fxt t t s α f s, xs, Sxs ds T s α f s, xs, Sxs ds c t t s α N f xs λ Sxs ds T s α N f xs λ Sxs ds c t b t s α N f xs λ T N f xs λ T s α s t t s α N f xs λ b T N f xs λ T s α s s k s, τ, xτ dτ ds k s, τ, xτ dτ ds c s N k xτ λ dτ ds N k xτ λ dτ ds t t s α N f x λ N k x λ T ds T s α N f x λ N k x λ T ds c N f η λ N k T t N f η λ N k T b N f η λ N k T N f η λ N k T b b l, t α α t s α ds T c T s α ds c T α α c T α Γα N f η λ N k T c where l := b T α Γα N f η λ N k T c. Thus, we have Fxt l and hence Fx l. Step 3. F maps bounded sets into equicontinuous sets of CJ, X. Let t t 2 T, x B η. Using H4, again we have

10 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations Fxt 2 Fxt t t2 t t t2 t t t2 t [ t 2 s α t s α ] f s, xs, Sxs ds t 2 s α f s, xs, Sxs ds [ t 2 s α t s α ] N f xs λ Sxs ds t 2 s α N f xs λ Sxs ds [ t 2 s α t s α ] N f x λ N k x λ T ds t 2 s α N f x λ N k x λ T ds N f η λ N k T N f η λ N k T N f η λ N k T Γα t t2 t [ t 2 s α t s α ] ds t α 2 tα t 2 s α ds As t 2 t, the right-hand side of the above inequality tends to zero and since x is an arbitrary in B η, F is equicontinuous. where Now, let {x n }, n =, 2,... be a sequence on B η, and Fx n t = F x n t F 2 x n T, t J, F x n t = t t s α f s, x n s, Sx n s ds, t J, b F 2 x n T = a b T T s α f s, x n s, Sx n s ds c a b. In view of hypothesis H5 and lemma 2.6, the set convk is compact. For any t J, F x n t = t t s α f s, x n s, Sx n s ds = lim k = t ζ n, k i= t k α t it it f k k, x n it k, Sx n it k where k k i= ζ n = lim k α t it it f k k, x n it k, Sx n it k. Now, we have {F x n t} is a function family of continuous mappings F x n : J X, which is uniformly bounded and equicontinuous. As convk is convex and compact, we know ζ n convk. Hence, for any

11 55 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... t J = [, T], the set {F x n t }, is relatively compact. Therefore by lemma 2.7, every {F x n t} contains a uniformly convergent subsequence {F x nk t}, k =, 2,..., on J. Thus, {F x : x B η } is relatively compact. Similarly, one can obtain {F 2 x n t} contains a uniformly convergent subsequence {F 2 x nk t}, k =, 2,..., on J. Thus, {F 2 x : x B η } is relatively compact. As a result, the set {Fx : x B η } is relatively compact. As a consequence of steps -3, we can conclude that F is continuous and completely continuous. Step 4. A priori bounds. Now it remains to show that the set EF = {x CJ, X : x = ηfx for some η [, ]}, is bounded Let x EF, then x = ηfx for some η [, ]. Thus, for each t J, we have t xt = η t s α f s, xs, Sxs ds Using H4, for each t J, we have xt Fxt b a b t b T T s α f s, xs, Sxs ds c. a b t s α N f xs λ T N f xs λ T s α s s N k xτ λ dτ ds N k xτ λ dτ ds c t t s α N f xs λ N k x s λ B T ds T s α N f xs λ N k x s λ B T ds c N f t t s α ds N f t t s α xs λ ds N f N k T t t s α ds N f N k T t t s α x s λ B ds b N f b N f b N f N k T b N f N k T T T T T T s α ds T s α xs λ ds T s α ds T s α x s λ c Bds N f T α Γα N f N k T α Γα b N f T α Γα b N f N k T α Γα c N f t t s α xs λ ds b N f T T s α xs λ ds

12 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations N f N k T t t s α x s λ B ds b N f N k T T T s α x s λ B ds. By lemma 3., there exists a N > such that xt N, t J. Thus for every t J, we have x N. This show that the set EF is bounded. As a consequence of Schaefer s fixed point theorem, we deduce that F has a fixed point that is solution of fractional BVP.. 5 Examples In this section, we give one example to illustrate the usefulness of our main results. Example 5.. c D 2 xt = e σt xt e t xt t s xs 2t 2 xs ds, t J, α, ], x xt =, 5. where σ > is constant. Set Take X = [,, J = [, ] and so T =. f t, xt, Sxt = e σt e t Let x, x 2 CJ, X and t [, ], we have and xt xt Sxt, k t, s, xs = k t, s, x s k t, s, x 2 s s x s x 2 s 2 t 2 x s x 2 s 2 t 2 x s x 2 s 4 x s x 2 s, f t, x t, Sx t f t, x2 t, Sx 2 t e σt x t e t x t x 2t x 2 t Sx t Sx 2 t e σt x t x 2 t 2 x t x 2 t Sx t Sx 2 t e σt x t x 2 t Sx 2 t Sx 2 t. s xs 2 t 2 xs. Also, for all x CJ, X and each t J, we have e σt xt t f t, xt, Sxt e t xt s xs 2 t 2 xs ds e σt 5 e σt

13 552 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations... For t J, β, 2, we have m t = e σt 2 L β J, R, m 2 t = 4 L β J, R, ht = 5 e σt 4 2 L β J, R and M = 5 e σt 4 Choosing some σ > large enough and β = 4, 2, one can arrive at the following inequality Ω 2, 4 = M Γ <. 2 2 L β J,R All the assumptions in Theorem 4. are satisfied, and therefore, the fractional BVP 5. has a unique solution on J.. References [] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 92, [2] K. Balachandran and J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal., 729, [3] M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractls, 422, [4] J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal., 729, [5] E. Hernandez, D. O Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 732, [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [7] K. Karthikeyan and J. J. Trujillo, Existenes and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 722, [8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 26. [9] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 993. [] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 999. [] S. K. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Switzerland, 993. [2] I. Stakgold and M. Holst, Green s Functions and Boundary Value Problems, 3rd ed. Wiley, Canada, 2. [3] J. R. Wang, L. Lv and Y. Zhou, Boundary value problems for fractional differential equations involoving Caputa derivative in Banach spaces, J. Appl. Math. Comput., 3822, [4] J. R. Wang, W. Wei and Y. L. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opusc. Math., 32, [5] J. R. Wang, X. Xiang, W. Wei and Q. Chen, The generalized Grownwall inequality and its applications to periodic solutions of integrodifferential impulsive periodic system on Banach space, J. Inequal. Appl., 2828, 22 pages. [6] J. R. Wang, Y. Yang and W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opusc. Math., 32,

14 Sabri T. M. Thabet and M. B. Dhakne / On boundary value problems for fractional integro-differential equations [7] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ., 26 26, 2. Received: June 25, 25; Accepted: August 23, 25 UNIVERSITY PRESS Website: