Nonlocal Fractional Semilinear Delay Differential Equations in Separable Banach spaces

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: x. Volume 10, Issue 1 Ver. IV. (Feb. 2014), PP Nonlocal Fractional Semilinear Delay Differential Equations in Separable Banach spaces V.Dhanapalan 1, M.Thamilselvan 2, M.Chandrasekaran 3 1 (Department of Mathematics, Government College of Technology, Coimbatore Tamilnadu, India.) 2 (Department of Physics, Thanthai Periyar Government Institute of Technology, Vellore Tamilnadu, India.) 3 (Department of Mathematics, Higher College of Technology, Muscat, Sultanate of Oman.) Abstract : In this article, we study the existence of mild solutions for fractional semilinear differential equations with nonlocal conditions in separable Banach spaces. The result is obtained. The result is obtained using the Hausdorff measure of noncompactness and the Schauder fixed point theorem. Keywords: Fractional differential equation, nonlocal conditions, Housdorff measure of noncompactness, mild solution.. I. INTRODUCTION Let be a separable Banach space endowed with norm, the infinitesimal generator of a strongly continuous semigroup of bounded linear operators { } the domain of. We consider the nonlocal fractional semilinear differential equation C [ ] (1.1) C where is the -order Caputo fractional derivative operator, are functions to be specified later. Recently, the theory of fractional differential equations has attracted much interest due to their many applications in physics, chemistry, biology, finance and so on. We refer to the books of Podlubny [17], Samko et al [18], Kilbas et al [9] and the papers of Nigmatullin [14],Orisingher and Beghin [16], Meerschaert et al [13], Hahn et al [7], Li et al [10]. The semilinear evolution nonlocal Cauchy problem was initiated by Byszewski [4].The nonlocal condition can be applied in physics with better effect in applications than the classical initial condition since nonlocal conditions are usually more precise for physical measurements than the classical initial condition. Lin and Liu [11] studied semilinear integrodifferential equations with nonlocal Cauchy problems under Lipchitz type conditions. Ntouyas and Tsamatos [15] studied the global existence of solutions semilinear evolution equations with nonlocal conditions via a fixed point analysis approach. Fu and Ezzinbi [6] studied the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions by using fractional power of operators and Sadovskii s fixed point theorem. Xue [19] studied the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in separable Banach spaces. Xue [20] discussed the semilinear nonlocal differential equations when the semigroup generated by the coefficient operator is compact and the nonlocal term is not compact.fan and Li [5] discussed the existence for impulsive semilinear differential equations with nonlocal conditions by using Sadovskii s fixed point theorem and Schauder s fixed point theorem. In this article, we shall study the existence of mild solutions of (1.1) by using the Hausdorff measure of compactness and fixed point theorems. We assume that the semigroup generated by 49 Page

2 the coefficient operator is equicontinuous. The compactness of of and the Lipchitz condition of are the special cases our conditions. Therefore, the results in this paper generalize and improve some of previous ones in this field. The article is organized as follows. Section 2 contains some preliminaries about fractional calculus and the Hausdorff s measure of compactness. In Section 3 the existence result is given. II. PRELIMINARIES Let be a separable Banach space Banach space and [ we denote by [ ] the space of - valued continuous functions on [0, b] with the norm and by [ ] we denote the space of - valued Bochner integrable functions [ ] with the norm [ ] [ ] 2.1. Definition The Riemann- Liouville fractional integral of [ ] of order is defined by The Riemann-Liouville fractional derivative of [ ] of order is defined by The Caputo fractional derivative of [ ] of order is defined by C ( ) We recall the Hausdorff measure of noncompactness defined on a bounded subset of of Banach space by inf{ has a finite - net in }. Some basic properties of are presented in the following lemma Lemma ([1]). Let be real Banach space and be bounded, the following properties are satisfied: (1) B is precompact if and only if (2), where and mean the closure and convex hull of respectively; (3) and (4) where { } (5) { } (6) for any (7) if the mapping is Lipchitz continuous with constant, then for any bounded subset, where is a Banach space; (8) If { } is a decreasing of bounded closed nonempty subsets of and then is nonempty and compact in. The map is said to be a -contraction if there exists appositive constant such that for any bounded closed subset where is a Banach space Lemma (Darbo-Sadovskii). 50 Page

3 If is bounded closed and convex, the continuous map is a -contraction, then the map Q has at least one fixed point in W. In this article, without loss of generality, we denote the Hausdorff measure of noncompactness of and [ ] 2.4. Lemma ([1]). If [ ] is bounded and equicontinuous, then the set is continuous on [0,b] and [ ] ( ) 2.5. Lemma ([3]). If { } [ ] satisfies a.e. on [0,b] for all with some [ ], then ({ }) { } 2.6. Definition A semigroup is said to be equicontinuous if the mapping { } is equicontinuous at for all bounded set in Banach space Definition ([17]). The Mainardi s function is defined by (2.1) where is a Gamma function. It is known that satisfies the following equality (see [12, (F.333)]) (2.2) III. MAIN RESULTS In this section we prove the existence of a mild solution of (1.1) by using the Hausdorff measure of noncompactness. The function is assumed to be compact. A function [ ] is called a mild solution of the equation (1.1) if where (3.2) (3.3) where is the Mainardi s function. (3.1) 3.1.Remark If T(t) is equicontinuous, by (2.2), it is easy to show that, are equicontinuous Lemma Let Let the semigroup be equicontinuous and [ ]. Then the set { a.e. [ ]} is equicontinuous for [ ]. 51 Page

4 Proof: For we have From this inequality it follows that as The proof is complete Lemma ([8]). Suppose and is a nonnegative function locally integrable on (some and suppose is nonnegative and locally integrable on with on this interval. Then where ( ) If constant, then To prove the main results, we use the following assumptions: (A1) The semigroup is equicontinous, and there exists a constant such that 52 Page

5 (3.4) (A2) [ ] is continuous and compact, and there exists and a constant such that [ ] (A3) [ ] satisfied the Caratheodory condition; i.e. is measurable for all and is continuous for a.e. [ ] (A4) [ ] [ ] is continuous function such that [ ] (A5) There exists a function [ ] such that [ ] for all is continuous and increasing for a.e. [ ], and for a.e. [ ] and all moreover, there exists at least one solution to the following scalar equation: ( ) [ ] (3.5) (A6) There exists a constant such that ( ) ( ) (3.6) for a.e. [ ] and every bounded [ ] 3.4. Lemma Assume that conditions (A1) to (A6) are satisfied. Then the nonlocal problem (1.1) has at least one mild solution on [ ] Proof: Define the mapping [ ] [ ] by ( ) [ ] (3.7) It is obvious that the fixed point of is the mild solution of (1.1) and it is easy to show that is continuous on [ ]. From (3.2), (2.2) and (3.4), it follows that (3.8) From(3.3) and (2.2), it follows that (3.9) Set { [ ] [ ]} Then [ ] is bounded and convex. Define where means the closure of the convex hull in [ ]. From Remark 3.1, Lemma 3.2, (A4), the equicontinuity of compactness of and [ ], it follows that [ ] is bounded closed convex and equicontinuous on [0,b]. For every, ( ) This implies We denote It is easy to show that { } is a decreasing sequence of equicontinuous on [0,b], and is bounded closed convex subsets of [ ] Since is separable, then [ ] is separable, hence there exists a dense subset { } of From Lemma 2.2, it follows that ( ) { } Since is compact, by Lemma 2.5, we have ( ) { } 53 Page

6 By (3.6) and (3.10), we have ( ) ( ) (3.11) Since (3.12) Taking { } is decreasing with respect to, we define [ ]. ( ) (3.10) to both sides of (3.11), we have (3.13) ( ) By Lemma 3.3, we obtain [ ]. By Lemma 2.3, By Lemma 2.2, it follows that is convex compact in [ ] and From the Schauder fixed point theorem, there exists at least one fixed point which is mild solution of (1.1). REFERENCES [1] J. Banas, K.Goebel, Measure of Noncompactness in Banach spaces, Lect. Notes Pure Appl. Math., vol.60, Marcel Dekker, New York, [2] E. Bazhlekova, Fractional Evolution Equations in Banach spaces, Ph.D. Thesis, Eindohoven University of Technology, [3] D.Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108, 1998, [4] L.Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J.Math. Anal. Appl. 162, 1991, [5] Z. Fan, G. Li, Existence results for semiliinear differential equations with nonlocal and impulsive conditions, J.Funct. Anal. 258, 2010, [6] X.Fu, K. Ezzinbi, Existence of solutions of a semilinear functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54, 2003, [7] M.Hahn, K.Kobayashi, S.Umarov, Fokker Planck-Kolmogorov equations associated with time changed fractional Brownian motion, Proc.Amer.Math.Soc.13, 2011, [8] D.Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol.840, Springer-verlag, New York, Berlin, [9] A.A.Kilbas, H.M. Srivastava, J.J.Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol.204, Elsevier Science B.V., Amsterdam, [10] K.Li, J.Peng, J. Gao, Nonlocal Fractional Semilinear differential equations in separable Banach spaces, Elec. J. Diff. Eq., vol.2013, no.7, 2013, 1-7. [11] Y.Lin, J.Liu, Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal. 26, 1996, [12] F. M.Mainardi, Functional Calculus and Waves in Linear Viscoelasticity, An introduction to Mathematical Models ( Imperial College Press, 2010.) [13] M. M. Meerschaert, E. Nane, E. Vellaisamy, Fractional Cauchy problems on bounded domains, Ann. Probab. 37,2009, [14] R.R. Nigmatullin, To the theoretical explanation of the universal response. Phys. Stat. Sol. (b), 123, 1984, [15] S.Ntouyas, P.Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl , Page

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