Research Article On the Dimension of the Solution Set for Semilinear Fractional Differential Inclusions

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1 Abstract and Applied Analysis Volume 212, Article ID 35924, 1 pages doi:1.1155/212/35924 Research Article On the Dimension of the Solution Set for Semilinear Fractional Differential Inclusions Ravi P. Agarwal, 1, 2 Bashir Ahmad, 2 Ahmed Alsaedi, 2 and Naseer Shahzad 2 1 Department of Mathematics, Texas A&M University-Kingsville, 7 University Boulvard Kingsville, TX , USA 2 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 823, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Naseer Shahzad, nshahzad@kau.edu.sa Received 24 January 212; Accepted 11 March 212 Academic Editor: Ngai-Ching Wong Copyright q 212 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the existence and dimension of the solution set for a nonlocal problem of semilinear fractional differential inclusions. The main tools of our study include some well-known results on multivalued maps. 1. Introduction The subject of fractional calculus has recently emerged as an important and popular field of research due to its numerous applications in science and engineering. Examples can be found in various disciplines such as mechanics, electricity, signal and image processing, thermodynamics, biophysics, blood flow phenomena, aerodynamics, economics, and fitting of experimental data 1 4 whereas the theoretical development of the subject can be found in 5, 6. For some recent results on fractional differential equations and inclusions, see In this paper, we study the following problem for semilinear fractional differential inclusion with nonlocal condition: c D q xt Atxt Ft, xt, x gx x, x R n, t,tt>, 1.1

2 2 Abstract and Applied Analysis where c D q denote the Caputo fractional derivative of order q, 1 5, At is a bounded linear operator on,tthe function t At is continuous in the uniform operator topology, g : C,T, R n R n,andf :,T R n PR n, where PR n is the family of all nonempty subsets of R n. 2. Terminology and Preliminary Results In this section, we discuss some basic concepts of multivalued analysis and recall some results involving multivalued maps. Let C,T, R n denote the Banach space of continuous functions from,t into R n with the norm x sup t,t xt. LetL 1,T, R n be the Banach space of measureable functions x :,T R n that are Lebesgue integrable and normed by x L 1 T xt dt. For a nonempty subset C of a complete metric space X, letpc Y C : Y / }, P cl C Y PC : Y is closed}, P b C Y PC : Y is bounded}, P b,cl C Y PC : Y is bounded and closed}, andp cp C Y PC : Y is compact}. IfC is a nonempty subset of a Banach space X, then we set P c,cl C Y PC : Y is closed and convex}, and P c,cp CY PC : Y is compact and convex}. A multivalued map F : C PX is closed resp., compact valued if Fx is closed resp., compact for all x C. The map F is bounded on bounded sets if FB x B Fx is bounded in X for all B P b C i.e., sup x B sup y : y Fx}} <. The map F is called upper semicontinuous u.s.c. if x C : Fx V } is open in C whenever V X is open. F is called lower semi-continuous l.s.c. if the set y C : Fy V / } is open for any open set V X. F is called continuous if it is both l.s.c. and u.s.c. F is said to be completely continuous if FB is relatively compact for every B P b C. A mapping f : C X is called a selection of F : C PX if fx Fxfor every x C. We say that the mapping F has a fixed point if there is x X such that x Fx. The fixed points set of the multivalued operator F will be denoted by FixF. A multivalued map F :,T P cl R n is said to be measurable if, for every y 1 R n, the function t d ( y 1, Ft ) inf y1 y 2 : y2 Ft } 2.1 is measurable. Definition 2.1. Let X, d be a metric space. Consider H : PX PX R }given by HA, B max sup a A da, B, sup db, A b B }, 2.2 where da, B inf b B da, b. H is the generalized Pompeiu-Hausdorff functional. It is known that P b,cl X,H is a metric space and P cl X,H is a generalized metric space see 17. Definition 2.2. A multivalued operator F : X P cl X is called a k-contraction if there exists <k<1 such that H ( Fx, F ( y )) kd ( x, y ), for each x, y X. 2.3

3 Abstract and Applied Analysis 3 It is known that F : X P cp X is continuous on X if and only if F is continuous on X with respect to the Hausdorff metric. Also, if F : X P cp X is a k-contraction, then F is continuous with respect to Hausdorff metric. Further details of multivalued maps can be found in 18, 19. For the forthcoming analysis, we need the following results on multivalued maps. Lemma 2.3 Covitz and Nadler 2. Let X, d be a complete metric space. If Φ : X P cl X is a k-contraction, then, FixΦ /. Lemma 2.4 Dzedzej and Gelman 21. Let F :,α P c,cp R n be a measurable map such that the Lebesgue measure μ of the set t :dimft < 1} is zero. Then there are arbitrarily many linearly independent measurable selections x 1,x 2,...,x m of F. Lemma 2.5 Saint-Raymond 22. Let K be a compact metric space with dim K<n, X a Banach space, and Ω : K P c,cp X a lower semicontinuous map such that Ωx and dim Ωx n for every x K. Then, there exists a continuous selection f of Ω such that fx / for each x K. Lemma 2.6 Michael s selection theorem 23. Let C be a metric space, X a Banach space and Ω : C P c,cl C a lower semicontinuous map. Then, there exists a continuous selection f : C X of Ω. Lemma 2.7 see Dzedzej and Gelman 21 and Petrusel 24. Let C be a nonempty closed convex subset of a Banach space X. Suppose that Ω : C P c,cp C is a k-contraction. If f : C C is a continuous selection of Ω,thenFixf is nonempty. 3. Main Results Definition 3.1. A function x C,T, R n is a solution of the problem 1.1 if there exists a function f L 1,T, R n such that ft Ft, xt a.e. on,t and xt x gx Asxsds Γ ( ) fsds. 3.1 q Let S x,α denote the set of all solutions of 1.1 on the interval,α, where < α T. Lemma 3.2. Assume that H 1 F :,T R n P cp R n is such that F,x :,T P cp R n is measurable for each x R n, H 2 HFt, x,ft, x κ 1 t x x for almost all t,t and x, x R n with κ 1 C,T, R and Ft, x sup v : v Ft, x} κ 1 t for almost all t,t and x R n, H 3 g : C,T, R n R n is continuous and gx gy κ 2 x y for all x, y C,T, R n and some κ 2 >.

4 4 Abstract and Applied Analysis Then, the Cauchy problem 1.1 has at least one solution on,t if κ 2 T q Γ ( q 1 )A 1 κ 1 < 1, 3.2 where A 1 max t,t At. Proof. For each y C,T, R n, define the set of selections of F by S F,y : v L 1,T, R n : vt F ( t, yt ) } for a.e. t,t. 3.3 Observe that, by assumptions H 1 and H 2, F,x is measurable and has a measureable selection v see 25, Theorem III.6. Alsoκ 1 C,T, R and vt Ft, xt κ 1 t. 3.4 Thus, the set S F,x is nonempty for each x C,T, R n. Now we show that the operator Ω defined by Ωx h C,T, R n : ht x gx Asxsds } 3.5 fsds, f S F,x satisfies the assumptions of Lemma 2.3. To show that Ωx P cl C,T, R n for each x C,T, R n,letu n } n Ωx be such that u n un in C,T, R n. Then, u C,T, R n and there exists v n S F,x such that, for each t,t, u n t x gx Asxsds v nsds. 3.6 As F has compact values, we pass to a subsequence to obtain that v n converges to v in L 1,T, R n.thus,v S F,x and, for each t,t, u n t ut x gx Asxsds Γ ( ) vsds. 3.7 q Hence, u Ωx. Next we show that there exists k> such that HΩx, Ωx k x x for each x, x C,T, R n. 3.8

5 Abstract and Applied Analysis 5 Let x, x C,T, R n and h 1 t,t, Ωx. Then, there exists v 1 t S F,x such that, for each By H 2, we have h 1 t x gx Asxsds v 1sds. 3.9 HFt, x,ft, x κ 1 t xt xt. 3.1 So, there exists w Ft, xt such that v 1 t w κ 1 t xt xt, t,t Define V :,T PR n by V t w R n : v 1 t w κ 1 t xt xt } Since the nonempty closed valued operator V t Ft, xt is measurable 25, Proposition III.4, there exists a function v 2 t that is a measurable selection for V t Ft, xt.sov 2 t Ft, xt and, for each t,t, we have v 1 t v 2 t κ 1 t xt xt. For each t,t,letusdefine Thus, h 2 t x gx Asxsds v 2sds h 1 t h 2 t t gx gx t s q 1 v 1s v 2 s ds. t s q 1 Asx xs ds 3.14 Hence, T q h 1 h 2 κ 2 x x Γ ( q 1 )A 1 κ 1 x x ( ) T q κ 2 Γ ( q 1 )A 1 κ 1 x x Analogously, interchanging the roles of x and x, weobtain HΩx, Ωx k x x, for each x, x C,T, R n, 3.16

6 6 Abstract and Applied Analysis where k κ 2 T q /Γq 1A 1 κ 1 < 1. Since Ω is a contraction, it follows by Lemma 2.3 that Ω has a fixed point x that is a solution of 1.1. This completes the proof. Lemma 3.3. Let F :,T R n P c,cp R n satisfy H 1, H 2, and H 3 and suppose that Ω : C,T, R n PC,T, R n is defined by Ωx h C,T, R n : ht x gx } fsds, f S F,x. Asxsds 3.17 Then, Ωx P c,cp C,T, R n for each x C,T, R n. Proof. First we show that Ωx is convex for each x C,T, R n. For that, let h 1,h 2 Ωx. Then. there exist f 1,f 2 S F,x such that, for each t,t, we have h i t x gx Asxsds f isds, i 1, Let λ 1. Then, for each t,t, we have λh 1 1 λh 2 t x gx Asxsds [ λf1 s 1 λf 2 s ] ds Since S F,x is convex F has convex values, it follows that λh 1 1 λh 2 Ωx. Next, we show that Ω maps bounded sets into bounded sets in C,T, R n. For a positive number r, letb r x C,T, R n : x r} be a bounded set in C,T, R n. Then, for each h Ωx,x B r, there exists f S F,x such that ht x gx Asxsds Γ ( ) fsds, 3.2 q

7 Abstract and Applied Analysis 7 and, in view of H 1, we have t ht x sup gx x B r t s q 1 Asxs ds t s q 1 fs ds x sup x B r gx T q Γ ( q 1 )A 1r κ Thus, h x sup gx T q x B r Γ ( q 1 )A 1r κ Now we show that Ω maps bounded sets into equicontinuous sets in C,T, R n.lett,t,t with t <t and x B r, where B r is a bounded set in C,T, R n. For each h Ωx, we obtain h ( t ) h ( t ) t s q 1 ( ) t t s q 1 Asxs fs ds ( ) Asxs fs ds [t s q 1 t s q 1] ( ) Asxs fs ds t t q 1 s ( ) Asxs fs ds Obviously the right-hand side of the above inequality tends to zero independently of x B r as t t. By the Arzela-Ascoli theorem, Ω : C,T, R n PC,T, R n is completely continuous. As in Lemma 3.2, Ω is closed valued. Consequently, Ωx P c,cp C,T, R n for each x C,T, R n. For <α T, let us consider the operator Ωx h C,α, R n : ht x gx } fsds, f S F,x. Asxsds 3.24

8 8 Abstract and Applied Analysis It is well known that FixΩ S x,α and, in view of Lemma 3.2, it is nonempty for each <α T. Theorem 3.4. Suppose that F :,α R n P c,cp R n satisfies H 1, H 2, and H 3 and that the Lebesgue measure μ of the sett :dimft, x < 1 for some x R n } is zero. Then, for each α, <α<min1 κ 2 Γq 1/A 1 κ 1 1/q,T}, the set S x,α of solutions of 1.1 has an infinite dimension for any x. Proof. Let the operator Ω be defined by Ωx h C,α, R n : ht x gx } fsds, f S F,x. Asxsds 3.25 Lemma 3.3 guarantees that Ωx P c,cp C,α, R n for each x C,α, R n and as in the proof of Lemma 3.2, it is a contraction if κ 2 α q /Γq 1A 1 κ 1 < 1orα<1 κ 2 Γq1/A 1 κ 1 1/q,T}. We shall show that dim Ωx m for any x C,α, R n and arbitrary m N. Consider Gt Ft, xt. ByLemma 2.4, there exist linearly independent measurable selections x 1,x 2,...,x m of G. Set y i t x gx Asx isds x isds Ωx Assume that m i1 a iy i t a.e.in,α. Taking the Caputo derivatives a.e. in,α, we have m i1 a ix i t a.e.in,α and hence a i for all i. Asaresult,y i are linearly independent. Thus, Ωx contains an m-dimensional simplex. So, dim Ωx m. Asin Lemma 3.2, FixΩ is nonempty. It is known that every multivalued k-contraction having compact values is condensing with respect to the Hausdorff measure of noncompactness χ 26. Since FixΩ ΩFixΩ, we have χfixω χωfixω Since Ω is χ-condensing, FixΩ is compact. Consider a map I Ω :FixΩ P c,cp R n, where I is the identity operator. Assume that dim FixΩ <n. Then, Lemma 2.5 guarantees that there is a continuous selection g of I Ω such that gx / for each x FixΩ. This implies that there exists a continuous selection h of F :FixF P c,cp R n without fixed points. Define Λ : R n P c,cp R n by Ωx, Λx hx, x R n \ FixΩ, x FixΩ Since Λ is lower semicontinuous, in view of Michael s selection result Lemma 2.6, Λ admits a continuous selection f : R n R n.thusf : R n R n is a continuous selection of Ω with no

9 Abstract and Applied Analysis 9 fixed points and f h on FixΩ contradicting Lemma 2.7. As a result, FixΩ S x,α is infinite dimensional. Definition 3.5. A metric space X is said to be an AR-space if, whenever it is nonempty closed subset of another metric space Y, there exists a continuous retraction r : Y X, rx x for x X. In particular, it is contractible and hence connected. Theorem 3.6 see 27. Let C be a nonempty closed convex subset of a Banach space X and F : C P c,cp C a contraction. Then FixF is a nonempty AR-space. The following result is a consequence of Theorems 3.4 and 3.6. Corollary 3.7. Suppose that F :,α R n P c,cp R n satisfies H 1, H 2, and H 3 and that the Lebesgue measure μ of the set t :dimft, x < 1 for some x R n } is zero. Then, for each α, <α<min1 κ 2 Γq 1/A 1 κ 1 1/q,T}, the set S x,α of solutions of 1.1 is a compact and infinite dimensional AR-space. Acknowledgment This project was funded by the Deanship of Scientifc Research DSR, King Abdulaziz University, Jeddah, under grant no. 8/31/Gr. The authors, therefore, acknowledge with thanks DSR s technical and financial support. References 1 G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Ridgefield, Conn, USA, J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 24 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, R. P. Agarwal, M. Belmekki, and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations, vol. 29, Article ID , 47 pages, Y.-K. Chang and J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Mathematical and Computer Modelling, vol. 49, no. 3-4, pp , B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications, vol. 58, no. 9, pp , A. Cernea, On the existence of solutions for nonconvex fractional hyperbolic differential inclusions, Communications in Mathematical Analysis, vol. 9, no. 1, pp , B. Ahmad, Existence results for fractional differential inclusions with separated boundary conditions, Bulletin of the Korean Mathematical Society, vol. 47, no. 4, pp , B. Ahmad and S. K. Ntouyas, Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions, Electronic Qualitative Theory of Differential Equations, no. 71, pp. 1 17, 21.

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