IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 33 EXISTENCE OF SOLUTIONS OF FRACTIONAL INTEGRODIFFERENTIAL SYSTEM WITH NONLOCAL CONDITIONS KAMALENDRA KUMAR1 AND RAKESH KUMAR2 1Department of Mathematics, S R M S College of Engineering and Technology, Bareilly-243001, India. 2Department of Mathematics, Hindu College, Moradabad-244001, India. kamlendra.14kumar@gmail.com; rakeshainil@gmail.com (Received 6 May 2014; accepted 25 February 2015; published on line 29 May 2015) ABSTRACT:In the present paper we prove the existence and uniqueness of local solutions of a nonlocal Cauchy problem for a class of fractional integrodifferential equation. The results are obtained by using the theory of resolvent operators, the fractional powers of operators, fixed point techniques, and the Gelfand-Shilov principle. ABSTRAK:Menerusi kertas kerja ini, kehadiran dan keunikan penyelesaian lokal terhadap permasalahan tak setempat Cauchy untuk peringkat pecahan persamaan integrodifferential dibuktikan. Keputusan diperolehi menggunakan teori operator leraian, operator kuasa pecahan, teknik titik tetap dan prinsip Gelfand-Shilov. KEYWORDS: fractional integrodifferential equation; nonlocal conditions; fractional powers; mild and classical solutions; resolvent operators 1. INTRODUCTION The purpose of this paper is to prove the existence and uniqueness of local solutions for nonlocal fractional integrodifferential equations of the form: ( ) + ( ) ( ) = , ( ) + ℎ , , ( ), , , ( ) , (1.1) (0) + ( ) = , (1.2) in a Banach space , where 0 < ≤ 1, ≥ 0. Let = [0, ]. We assume that − ( ) is a closed linear operator defined on a dense domain ( ) in into such that ( ) is independent of . It is considered also that − ( ) generates an evolution operator in the Banach space . Let : × into , ℎ: × × × into , : × × into and : ( , ) → ( ) be given nonlinear operators. The differential equations involving fractional derivative in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economy and science. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, porous media, electromagnetic etc. (see [1-6]). They involve a wide area of applications by bringing into a broader paradigm concepts of physics and mathematics [7-9]. Hence, fractional differential equations have become an object of extensive study during recent years as cited in [10-14] and references therein. IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 34 On the other hand, integrodifferential equations arise in many fields such as electronic, fluid dynamics, biological models and chemical kinetics. The equations of basic electric circuit analysis are well-known examples of these equations. Hence, the study of integrodifferential equation is very important. Fractional integrodifferential equations have been studied by many authors [8, 15-18]. Fractional integro-differential equations arise in many fields of engineering such as optimal control problem and heat conduction of materials with memory, etc. The nonlocal condition, which is a generalization of the classical initial condition, was motivated by physical problems. The problem of existence of solutions of evolution equation with nonlocal conditions in Banach space was first studied by Byszewski [19]. As indicated in [19, 20] and the references therein, the nonlocal condition (0) + ( ) = can be applied in physics with better effect than the classical condition (0) = . In the present paper we have generalized the results given by Pazy [21], Balachandran and Chandrasekaran [22] and Debbouche [23]. The rest of this paper is organized as follows: In section 2, we give preliminary results and in section 3, we prove the results of existence and uniqueness of local solutions for the equations (1.1) − (1.2). 2. PRELIMINARIES In this section, we introduce some notations, definitions and results about fractional calculus and resolvent operators. Following Gelfand and Shilov [24], we define the fractional integral of order > 0 as ( ) = 1 Γ( ) ( − ) ( ) , and also, the fractional derivative of the function of order 0 < < 1 ta D  ( ) = ( ) ∫ ( )( − ) , where is an abstract continuous function on the interval [ , ] and Γ( ) is the Gamma function, see [25]. Definition 2.1: By a solution of (1.1) − (1.2), we mean a function with values in such that: (1) is continuous function on and ( ) ∈ ( ), (2) exists and is continuous on (0, ), 0 < < 1, and satisfies (1.1) on (0, )and the nonlocal condition (1.2). Definition 2.2: (See [26]) A resolvent operator for problem (1.1) − (1.2) is a bounded operator-valued function ( , ) ∈ ( ), 0 ≤ ≤ ≤ , having the following properties: (a) ( , ) is strongly continuous in and , ( , ) = , the identity operator on , 0 ≤ ≤ , and ‖ ( , )‖ ≤ ( ) for some constants and . (b) ( , ) ⊂ , ( , ) is strongly continuous in and on , and is the Banach space formed from ( ), the domain − ( ), endowed with the graph norm. (c) For every ∈ , ( , ) is continuously differentiable in ∈ and IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 35 ( , ) = ( , ) ( ) . (d) For every ∈ , and ∈ , ( , ) is continuously differentiable in ∈ and ( , ) = − ( ) ( , ) , with ( , ) and ( , ) are strongly continuous on 0 ≤ ≤ ≤ . Here, ( , ) can be deduced from the evolution operator of the generator − ( ). The resolvent operator is similar to the evolution operator for non-autonomous differential equations in Banach space. It will not, however, be an evolution operator because it will not satisfy an evolution or semigroup property. Because a number of results follow directly from the definition of resolvent operator. Definition 2.3: A continuous solution : → is said to be a mild solution of problem (1.1) − (1.2) on if for all ∈ , it satisfies the following integral equation ( ) = ( , 0)[ − ( )] + 1 Γ( ) ( − ) ( , ) , ( ) + ℎ , , ( ), , , ( ) . (2.1) We define the fractional power ( ) by ( ) = 1 Γ( ) ( , ) , > 0. For 0 < ≤ 1, ( ) is a closed linear operator whose domain ( ) ⊃ ( ) is dense in . The closedness of implies that ( ), endowed with the graph norm of , ‖ ‖ ( ) = ‖ ‖ + ‖ ‖, ∈ ( ), is a Banach space. Since 0 ∈ ( ), is invertible. Its graph norm |‖. ‖| is equivalent to the norm‖ ‖ = ‖ ‖. Thus ( ) equipped with the norm ‖. ‖ is a Banach space, which we denote by . 3. EXISTENCE THEOREM To prove the main results we state the following lemma: Lemma 3.1: (see [21, section 2.6]) Let ( ) be the infinitesimal generator of a resolvent operator ( , ). We denote by [ ( )] the resolvent set of ( ). If 0 ∈ [ ( )], then (a) ( , ): → ( ) for every 0 ≤ ≤ ≤ and ≥ 0, (b) For every ∈ ( ), we have ( , ) ( ) = ( ) ( , ) , (c) The operator ( , ) is bounded and‖ ( , )‖ ≤ , ( − ) . Theorem 1.1: Assume that (a) − ( ) is the infinitesimal generator of a resolvent operator ( , ), 0 ≤ ≤ ≤ , in . (b) 0 ∈ [− ( )], the resolvent set. (c) For ≥ 0, the fractional power satisfies ‖ ( , )‖ ≤ , ( − ) for 0 ≤ ≤ ≤ , where , is a real constant. IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 36 (d) For an open subset of × , : → satisfies the condition, if for every ( , ) ∈ there is a neighborhood ⊂ and constants ≥ 0, 0 < ≤ 1, such that ‖ ( , ) − ( , )‖ ≤ | − | + ‖ − ‖ (3.1) for all ( , ) ∈ , = 1, 2. (e) For an open subset Q of × × × , ℎ: → satisfies the condition, if for every ( , , , ) ∈ there is a neighborhood ⊂ and constants ≥ 0,0 < ≤ 1, such that ‖ℎ( , , , ) − ℎ( , , , )‖ ≤ | − | + | − | +‖ − ‖ +‖ − ‖ (3.2) for all ( , , , ) ∈ , = 1, 2. (f) For an open subset R of × × , : → satisfies the condition, if for every ( , , ) ∈ there is a neighborhood ⊂ and constants ≥ 0,0 < ≤ 1, such that ‖ ( , , ) − ( , , )‖ ≤ | − | + | − | +‖ − ‖ (3.3) for all ( , , ) ∈ , = 1, 2. (g) : → is continuous and there exists a number such that ‖ ( , 0)‖ < and ‖ ( ) − ( )‖ ≤ ‖ − ‖ (3.4) for all , ∈ . Note that if ∈ , then ∈ . Then the problem (1.1) – (1.2) has a unique local solution ∈ ([0, ): ) ∩ (0, ): . Proof: Choose ∗ > 0 and > 0 such that estimates (3.1) – (3.4) hold on the sets = {( , ): 0 ≤ ≤ ∗, ‖ − ‖ ≤ }, = {( , , , ): 0 ≤ , ≤ ∗, ‖ − ‖ ≤ , ‖ − ‖ ≤ }, and = {( , , ): 0 ≤ , ≤ ∗, ‖ − ‖ ≤ }, respectively. Let = max ‖ ( , )‖ and = max , ∗ ℎ , , , ( , , ) Set = sup ∈ ‖ ( )‖ and choose such that for 0 ≤ < , ‖ ( , 0) − ‖ ‖ ‖ + ≤ 2 , 0 ≤ < and 0 < < ∗, 2 , Γ( )( − )( + + + + ) (3.5) Let = (0, ]: be endowed with the supremum norm ‖ ‖ = sup ‖ ( )‖ , ∈ . IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 37 Then be a Banach space. Define a map : → ( ) = ( , 0) [ − ( )] + 1 Γ( ) ( − ) ( , ) , ( ) + 1 Γ( ) ( − ) ( , ) ℎ , , ( ), , , ( ) . (3.6) Obviously, (0) = [ − ( )]. Let be the nonempty closed and bounded subset of defined by = { : ∈ , (0) = [ − ( )], ‖ ( ) − [ − ( )]‖ ≤ }. For ∈ , we have ‖ ( ) − [ − ( )]‖ ≤ ‖ ( , 0) − ‖‖ [ − ( )]‖ + 1 Γ( ) ( − ) ‖ ( , )‖ , ( ) − ( , ) + 1 Γ( ) ( − ) ‖ ( , )‖‖ ( , )‖ + 1 Γ( ) ( − ) ‖ ( , )‖ ℎ , , ( ), , , ( ) − ℎ , , , ( , , ) + 1 Γ( ) ( − ) ‖ ( , )‖ ℎ , , , ( , , ) ≤ 2 + , Γ( ) ( − ) ( + ) + , Γ( ) ( − ) { ( + ) } + , Γ( ) ( − ) ≤ 2 + , Γ( ) ( − ) [ + + + + ] ≤ 2 + 2 = . Therefore, maps into itself. Moreover, if , ∈ , then ‖ ( ) − ( )‖ ≤ ‖ ( , 0)‖‖ ( ) − ( )‖ + 1 Γ( ) ( − ) ‖ ( , )‖ , ( ) − , ( ) + 1 Γ( ) ( − ) ‖ ( , )‖ ℎ , , ( ), , , ( ) − ℎ , , ( ), , , ( ) IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 38 ≤ ‖ ( , 0)‖‖ − ‖ + 1 Γ( ) , ( − ) ‖ − ‖ + 1 Γ( ) , ( − ) [(‖ − ‖ + ‖ − ‖ ) ] ≤ ‖ ( , 0)‖‖ − ‖ + 1 Γ( ) , ( − ) [ + (1 + ) ]‖ − ‖ ≤ 1 2 ‖ − ‖ + 1 2 ‖ − ‖ , which implies that‖ − ‖ ≤ ‖ − ‖ + ‖ − ‖ . By the contraction mapping theorem, mapping has a unique fixed point ∈ . This fixed point satisfies the integral equation ( ) = ( , 0) [ − ( )] + 1 Γ( ) ( − ) ( , ) , ( ) + 1 Γ( ) ( − ) ( , ) ℎ , , ( ), , , ( ) . (3.7) From (3.1), (3.2) and the continuity of it follow that → , ( ) and → ℎ , , ( ), , , ( ) are continuous on [0, ], and, hence, there exist constants and such that , ( ) ≤ (3.8) and ℎ , , ( ), , , ( ) ≤ (3.9) By using the same method as in [20, Theorem 3.2], we can prove that ( ) is locally H ̈lder continuous on(0, ]. Then there exist a constant such that for every > 0, we have ‖ ( ) − ( )‖ ≤ | − | , for all 0 ≤ ≤ , ≤ . The local H ̈lder continuity of → , ( ) follows from , ( ) − , ( ) ≤ | − | + ‖ ( ) − ( )‖ ≤ | − | + | − | for some > 0 and the local H ̈lder continuity of → ℎ , , ( ), , , ( ) follows from IIUM Engineering Journal, Vol. 16, No. 1, 2015 Kumar and Kumar 39 ℎ , , ( ), , , ( ) − ℎ , , ( ), , , (ϕ) ≤ | − | + ‖ ( ) − ( )‖ + | − | + | − | + ‖ ( ) − ( )‖ ≤ | − | + | − | + | − | + | − | + | − | for some > 0. Let be a solution of (3.7). Consider the inhomogeneous initial value problem ( ) + ( ) ( ) = , ( ) + ℎ , , ( ), , , ( ) (3.10) (0) + ( ) = (3.11) This problem has a unique solution ∈ (0, ]: [21], which is given by ( ) = ( , 0)[ − ( )] + 1 Γ( ) ( − ) ( , ) , ( ) + 1 Γ( ) ( − ) ( , ) ℎ , , ( ), , , ( ) . (3.12) for > 0, each term of (3.12) belongs to ( ) and a fortiori in ( ). Operating on both sides of (3.12) with we find that ( ) = ( , 0) [ − ( )] + 1 Γ( ) ( − ) ( , ) , ( ) + 1 Γ( ) ( − ) ( , ) ℎ , , ( ), , , ( ) . (3.13) From (3.7) the right hand side of (3.13) equals y(t) and therefore u(t) = A y(t) and by (3.12), u is a C (0, T]: X solution of (1.1) − (1.2). The uniqueness of u follows from the uniqueness of the solutions of (3.7) and (3.10) − (3.11). Hence, the theorem is proved. 4. CONCLUSION In this paper, the existence and uniqueness of mild and classical solutions for the nonlinear fractional integrodifferential equation with nonlocal condition are discussed. We applied the resolvant operators, the fractional powers of operaters, fixed point technique and Gelfand-Shilov principle to establish the existence results. The results presented in this paper may be useful in the field of engineering and physics. ACKNOWLEDGEMENT The authors wish to thank the referees for their valuable comments and suggestions. 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