EXISTENCE AND UNIQUENESS OF SOLUTIONS TO FUNCTIONAL INTEGRO-DIFFERENTIAL FRACTIONAL EQUATIONS

Electronic Journal of Differential Equations, Vol. 212 212, No. 13, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS TO FUNCTIONAL INTEGRO-DIFFERENTIAL FRACTIONAL EQUATIONS MOULAY RCHID SIDI AMMI, EL HASSAN EL KINANI, DELFIM F. M. TORRES Abstract. Using a fixed point theorem in a Banach algebra, we prove an existence result for a fractional functional differential equation in the Riemann- Liouville sense. Dependence of solutions with respect to initial data and an uniqueness result are also obtained. 1. Introduction Fractional Calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject has its origin in the 16s. During three centuries, the theory of fractional calculus developed as a pure theoretical field, useful only for mathematicians. In the last few decades, however, fractional differentiation proved very useful in various fields of applied sciences and engineering: physics classic and quantum mechanics, chemistry, biology, economics, signal and image processing, calculus of variations, control theory, electrochemistry, viscoelasticity, feedback amplifiers, and electrical circuits [4, 6, 8, 9, 15, 18, 2, 22, 23]. The bible of fractional calculus is the book of Samko, Kilbas and Marichev [2]. Several definitions of fractional derivatives are available in the literature, including the Riemann-Liouville, Grunwald-Letnikov, Caputo, Riesz, Riesz-Caputo, Weyl, Hadamard, and Chen derivatives [11, 12, 13, 16, 17, 2]. The most common used fractional derivative is the Riemann-Liouville [1, 17, 2, 21], which we adopt here. It is worth to mention that functions that have no first order derivative might have Riemann-Liouville fractional derivatives of all orders less than one [17]. Recently, the physical meaning of the initial conditions to fractional differential equations with Riemann-Liouville derivatives has been discussed [7, 14, 16]. Using a fixed point theorem, like Schauder s fixed point theorem, and the Banach contraction mapping principle, several results of existence have been obtained in the literature to linear and nonlinear equations, and recently also to fractional differential equations. The interested reader is referred to [1, 2, 3, 19]. Let I = [ δ, ] and I = [, T ] be two closed and bounded intervals in R; BI, R be the space of bounded real-valued functions on I; and C = CI, R be the space 2 Mathematics Subject Classification. 26A33, 47H1. Key words and phrases. Riemann-Liouville operator; fractional calculus; fixed point theorem; Lipschitz condition. c 212 Texas State University - San Marcos. Submitted January 27, 212. Published June 2, 212. 1

2 M. R. SIDI AMMI, E. H. EL KINANI, D. F. M. TORRES EJDE-212/13 of continuous real valued functions on I. Given a function φ C, we consider the functional integro-differential fractional equation d α [ xt ] dt α = ks, x s ds a.e., t I, 1.1 ft, xt subject to xt = φt, t I, 1.2 where d α /dt α denotes the Riemann-Liouville derivative of order α, < α < 1, and x t : I C is the continuous function defined by x t θ = xt + θ for all θ I, under suitable mixed Lipschitz and other conditions on the nonlinearities f and g. For a motivation to study such type of problems we refer to [1]. Here we just mention that problems of type 1.1 1.2 seem important in the study of dynamics of biological systems [1]. Our main aim is to prove existence of solutions for 1.1 1.2. This is done in Section 3 Theorem 3.1. Our main tool is a fixed point theorem that is often useful in proving existence results for integral equations of mixed type in Banach algebras [5], and which we recall in Section 2 Theorem 2.2. We end with Section 4, by proving dependence of the solutions with respect to their initial values Theorem 4.1 and, consequently, uniqueness to 1.1 1.2 Corollary 4.2. 2. Preliminaries In this section we give the notations, definitions, hypotheses and preliminary tools, which will be used in the sequel. We deal with the left Riemann-Liouville fractional derivative, which is defined in the following way. Definition 2.1 [2]. The fractional integral of order α, 1 of a function f L 1 [, T ] is defined by I α ft = t s α 1 fs ds, Γα t [, T ], where Γ is the Euler gamma function. The Riemann-Liouville fractional derivative operator of order α is then defined by d α dt α := d dt I1 α. In this article, X denotes a Banach algebra with norm. The space CI, R of all continuous functions endowed with the norm x = sup t I xt is a Banach algebra. To prove the existence result for 1.1 1.2, we shall use the following fixed point theorem. Theorem 2.2 [5]. Let B r and B r be, respectively, open and closed balls in a Banach algebra X centered at origin and of radius r. Let A, B : B r X be two operators satisfying: a A is Lipschitz with Lipschitz constant L A, b B is compact and continuous, and c L A M < 1, where M = BB r := sup{ Bx ; x B r }. Then, either i the equation λ[axbx] = x has a solution for λ = 1, or ii there exists x X such that x = r, λ[axbx] = x for some < λ < 1.

EJDE-212/13 EXISTENCE AND UNIQUENESS OF SOLUTIONS 3 Throughout this article, we assume the following hypotheses: H1 The function f : I R R \ {} is Lipschitz and there exists a positive constant L such that for all x, y R, ft, x ft, y L x y a.e., t I. H2 The function k : I C R is continuous and there exists a function β L 1 I, R + such that ks, y βs y a.e., s I, y C. H3 There exists a continuous function γ L I, R + and a contraction ψ : R + R + with contraction constant < 1 and ψ = such that for x C and y R, gt, x, y γtψ x + y a.e., t I. H4 The function k is Lipschitz with respect to the second variable with Lipschitz constant L k : ks, x ks, y L k x y for s I, x, y C. H5 There exist L 1 and L 2 such that for x 1, x 2 C, y 1, y 2 R and s I, gs, x 1, y 1 gs, x 2, y 2 L 1 x 1 x 2 + L 2 y 1 y 2. 3. Existence of solutions We prove existence of a solution to 1.1 1.2 under hypotheses H1 H5. Theorem 3.1. Suppose that hypotheses H1 H5 hold. Assume there exists a real number r > such that F Γα+1 r > sup s,t γs1 + β L 1ψr 1 L Γα+1 sup s,t γs1 + β L 1ψr, 3.1 where 1 L sup γs1 + β L 1ψr >, F = sup ft,. Γα + 1 s,t t [,T ] Then 1.1 1.2 has a solution on I. Proof. Let X = CI, R. Define an open ball B r centered at origin and of radius r >, which satisfies 3.1. It is easy to see that x is a solution to 1.1 1.2 if and only if it is a solution of the integral equation xt = ft, xti α ks, x s ds. In other terms, xt = ft, xt t s α 1 Γα g s, x s, kτ, x τ dτ ds. 3.2 The integral equation 3.2 is equivalent to the operator equation AxtBxt = xt, t I, where A, B : B r X are defined by Axt = ft, xt, Bxt = I α ks, x s ds. We need to prove that the operators A and B verify the hypotheses of Theorem 2.2. To continue the proof of Theorem 3.1, we use two technical lemmas. Lemma 3.2. The operator A is Lipschitz on X. Proof. Let x, y X and t I. By H1 we have Axt Ayt = ft, x ft, y L xt yt L x y. Then, Ax Ay L x y and it follows that A is Lipschitz on X with Lipschitz constant L.

4 M. R. SIDI AMMI, E. H. EL KINANI, D. F. M. TORRES EJDE-212/13 Lemma 3.3. The operator B is completely continuous on X. Proof. We prove that BB r is an uniformly bounded and equicontinuous set in X. Let x be arbitrary in B r. By hypotheses H2 H4 we have Bxt I α g t, x t, ks, x s ds 1 Γα 1 Γα 1 Γα sup s [,T ] γs Γα sup s [,T ] γs Γα + 1 t s α 1 g s, x s, t s α 1 γsψ x s + kτ, x τ dτ ds t s α 1 γs1 + β L 1ψrds 1 + β L 1 ψr 1 + β L 1 ψr. βτ x τ dτ ds t s α 1 ds Taking the supremum over t, we get Bx M for all x B r, where M = sup s [,T ] γs 1 + β L 1ψr. Γα + 1 It results that BB r is an uniformly bounded set in X. Now, we shall prove that BB r is an equicontinuous set in X. For t 1 t 2 T we have Bxt 2 Bxt 1 1 { 2 t 2 s α 1 g s, x s, kτ, x τ dτ ds Γα 1 } t 1 s α 1 g s, x s, kτ, x τ dτ ds 1 { 1 t 2 s α 1 g s, x s, Γα 2 + t 2 s α 1 g s, x s, t1 1 1 Γα + 1 Γα t 1 s α 1 g s, x s, 1 On the other hand, g s, x s, 2 kτ, x τ dτ kτ, x τ dτ ds kτ, x τ dτ ds } ds t 2 s α 1 t 1 s α 1 g s, x s, t 1 t 2 s α 1 g s, x s, kτ, x τ dτ ds γsψ x + γsψ x + kτ, x τ dτ ds. kτ, x τ dτ ds kτ, x τ dτ βτ x dτ

EJDE-212/13 EXISTENCE AND UNIQUENESS OF SOLUTIONS 5 where c is a positive constant. Then Bxt 2 Bxt 1 c Γα 1 γs x 1 + β L 1ψs t 2 s α 1 t 1 s α 1 ds + c Γα + 1 tα 2 t α 1 2t 2 t 1 α. supγsψs x 1 + β L 1 c, s c t2 t 2 s α 1 ds Γα t 1 Because the right hand side of the above inequality doesn t depend on x and tends to zero when t 2 t 1, we conclude that BB r is relatively compact. Hence, B is compact by the Arzela-Ascoli theorem. It remains to prove that B is continuous. For that, let us consider a sequence x n converging to x. Then, F x n t F xt ft, x n I α g t, x n t, ft, x n ft, x I α g t, x n t, + ft, x I α g t, x n t, ks, x n s ds ft, xi α ks, x n s ds L x n x + ft, x I α g t, x n t, On the other hand, I α g t, x n t, ks, x n s ds I α ks, x n s ds ks, x n s ds I α L 1 x n t x t + L 2 ks, x n s ds ks, x s ds ks, x s ds ks, x s ds ks, x s ds I α L 1 x n t x t + L 2 ks, x n s ks, x s ds I α L 1 x n t x t + L 2 L k x n s x s ds I α L 1 x n t x t + L 2 L k T x n s x s L 1 + L 2 L k T x n t s α 1 t x t ds Γα 1 Γα + 1 L 1 + L 2 L k T x n t x t. ks, x s ds. Taking the norm, F x n F x L x n x + 1 Γα+1 ft, x L 1 + L 2 L k T x n t x t. Hence, the right hand side of the above inequality tends to zero whenever x n x. Therefore, F x n F x. This proves the continuity of F. Using Theorem 2.2, we obtain that either the conclusion i or ii holds. We show that item ii of Theorem 2.2 cannot be realizable. Let x X be such that

6 M. R. SIDI AMMI, E. H. EL KINANI, D. F. M. TORRES EJDE-212/13 x = r and xt = λft, xti g α t, x t, t ks, x sds t I. It follows that xt λ ft, xt ft, + ft, I α λ L xt + F I α λ L x + F I α L x + F I α γtψ x + ks, x s ds ks, x s ds L x + F I α γtψ x + β L 1 x L x + F Γα L x + F Γα L x + F Γα + 1 LT α Γα + 1 for any λ, 1 and ks, x s ds γs 1 + β L 1 ψ x t s α 1 ds sup γs 1 + β L 1 ψ x s,t sup γs 1 + β L 1 ψ x s,t sup γs 1 + β L 1 ψ x s,t + F sup γs 1 + β L 1 ψ x. Γα + 1 s,t Passing to the supremum in the above inequality, we obtain x If we replace x = r in 3.3, we have r x ks, x s ds t s α 1 ds F Γα+1 sup s,t γs 1 + β L 1 ψ x 1 L Γα+1 sup s,t γs 1 + β L 1 ψ x. 3.3 F Γα+1 sup s,t γs1 + β L 1ψr 1 L Γα+1 sup s,t γs1 + β L 1ψr, which is in contradiction to 3.1. Then the conclusion ii of Theorem 2.2 is not possible. Therefore, the operator equation AxBx = x and, consequently, problem 1.1 1.2, has a solution on I. This ends the proof of Theorem 3.1. Let us see an example of application of our Theorem 3.1. Let I = [ π, ] and I = [, π]. Consider the integro-differential fractional equation d α xt dt α 1 + sin t 12 xt = g t, x t t 2, ks, x s ds a.e., t I, 3.4 where α = 1/2 and f : I R R + \ {}, g : I C R R and k : I C R are given by ft, x = 1 + sin t x xt, gt, x, y = γt x + y, kt, x = 12 4π,

EJDE-212/13 EXISTENCE AND UNIQUENESS OF SOLUTIONS 7 with γt = t, βt = 1 4π, T = π, L = 1 12, F = sup t ft, = 1, β L 1 = 1 4 and sup t [,π] γt = π. It is easy to see that all hypotheses H1 H5 are satisfied with ψr = r 12 for all r R+. By Theorem 3.1, r satisfies 12 F B, 26 r 12 18, 34, LB LB where B = Γα+1 sup s,t γs1 + β L 1. We conclude that if r = 2, then 3.4 has a solution in B 2. 4. Dependence on the data and uniqueness of solution In this section we derive uniqueness of solution to 1.1 1.2. Theorem 4.1. Let x and y be two solutions to the nonlocal fractional equation 1.1 subject to 1.2 with φ = φ 1 and φ = φ 2, respectively. Then, we have x y L y + F L 1 + L 2 L k T 1 L α T Γα+1 φ 1 φ 2. Proof. Let x and y be two solutions of 1.1. Then, from 3.2, one has xt yt ft, xi α ft, x ft, y I α g t, x t, + ft, y I α ks, x s ds ft, yi α g t, y t, L x y + ft, y I α g t, x t, ks, x s ds ks, x s ds for t >. On the other hand, we have ks, x s ds Then I α g t, y t, ks, x s ds g t, y t, g t, y t, ks, y s ds ks, y s ds ks, y s ds L 1 x t y t + L 2 ks, x s ks, y s ds L 1 φ 1 t φ 2 t + L 2 L k T φ 1 φ 2 L 1 + L 2 L k T φ 1 φ 2. ft, y I α g t, x t, ks, x s ds g t, y t, t s α 1 ft, y L 1 + L 2 L k T φ 1 φ 2 ds Γα ft, y L 1 + L 2 L k T φ 1 φ 2 Γα + 1. ks, y s ds ks, y s ds

8 M. R. SIDI AMMI, E. H. EL KINANI, D. F. M. TORRES EJDE-212/13 Taking the supremum, we conclude that x y L x y + ft, y L 1 + L 2 L k T φ 1 φ 2 Γα L x y + ft, y ft, + ft, L 1 + L 2 L k T φ 1 φ 2 Γα + 1 L x y + L y + sup ft, L 1 + L 2 L k T t Γα + 1 φ 1 φ 2 L x y + L y + F L 1 + L 2 L k T Γα + 1 φ 1 φ 2. Therefore, x y L y + F L 1 + L 2 L k T 1 L α T Γα+1 φ 1 φ 2. Corollary 4.2. The solution obtained in Theorem 3.1 is unique. Acknowledgments. This work was supported by FEDER through COMPETE Operational Programme Factors of Competitiveness Programa Operacional Factores de Competitividade and by Portuguese funds through the Center for Research and Development in Mathematics and Applications University of Aveiro and the Portuguese Foundation for Science and Technology FCT Fundação para a Ciência e a Tecnologia, within project PEst-C/MAT/UI416/211 with COMPETE number FCOMP-1-124-FEDER-2269. The authors were also supported by the project New Explorations in Control Theory Through Advanced Research NECTAR cofinanced by FCT, Portugal, and the Centre National de la Recherche Scientifique et Technique CNRST, Morocco. References [1] S. Abbas; Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations 211 211, no. 9, 11pp. [2] R. P. Agarwal, M. Benchohra, S. Hamani; Boundary value problems for fractional differential equations, Georgian Math. J. 16 29, no. 3, 41 411. [3] R. P. Agarwal, Y. Zhou, Y. He; Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 21, no. 3, 195 11. [4] R. Almeida, D. F. M. Torres; Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett. 22 29, no. 12, 1816 182. [5] B. C. Dhage, Nonlinear functional boundary value problems in Banach algebras involving Carathéodories, Kyungpook Math. J. 46 26, no. 4, 527 541. [6] L. Gaul, P. Klein, S. Kempfle; Damping description involving fractional operators, Mech. Syst. Signal Process. 5 1991, no. 2, 81 88. [7] C. Giannantoni; The problem of the initial conditions and their physical meaning in linear differential equations of fractional order, Appl. Math. Comput. 141 23, no. 1, 87 12. [8] R. Hilfer; Applications of fractional calculus in physics, World Sci. Publishing, River Edge, NJ, 2. [9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 24, Elsevier, Amsterdam, 26. [1] M. Kirane, S. A. Malik; Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. Comput. 218 211, no. 1, 163 17.

EJDE-212/13 EXISTENCE AND UNIQUENESS OF SOLUTIONS 9 [11] M. Klimek; Stationarity-conservation laws for fractional differential equations with variable coefficients, J. Phys. A 35 22, no. 31, 6675 6693. [12] A. B. Malinowska, M. R. Sidi Ammi, D. F. M. Torres; Composition functionals in fractional calculus of variations, Commun. Frac. Calc. 1 21, no. 1, 32 4. [13] K. S. Miller, B. Ross; An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, Wiley, New York, 1993. [14] D. Mozyrska, D. F. M. Torres; Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Process. 91 211, no. 3, 379 385. [15] M. D. Ortigueira; Fractional calculus for scientists and engineers, Springer, New York, 211. [16] I. Podlubny; Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999. [17] B. Ross, S. G. Samko, E. R. Love; Functions that have no first order derivative might have fractional derivatives of all orders less than one, Real Anal. Exchange 2 1994/95, no. 1, 14 157. [18] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado; Advances in fractional calculus, Springer, Dordrecht, 27. [19] S. N. Salunkhe, M. G. Timol; Existence theorem for first order ordinary functional differential equations with periodic boundary conditions, Gen. Math. Notes 1 21, no. 2, 185 194. [2] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional integrals and derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993. [21] S. G. Samko, M. N. Yakhshiboev; A modification of Riemann-Liouville fractional integrodifferentiation applied to functions on R 1 with arbitrary behavior at infinity, Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 4, 96 99. [22] H. M. Srivastava, R. K. Saxena; Operators of fractional integration and their applications, Appl. Math. Comput. 118 21, no. 1, 1 52. [23] J. A. Tenreiro Machado, V. Kiryakova, F. Mainardi; Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 211, no. 3, 114 1153. Moulay Rchid Sidi Ammi Faculty of Sciences and Techniques, Moulay Ismail University, AMNEA Group, Errachidia, Morocco E-mail address: sidiammi@ua.pt El Hassan El Kinani Faculty of Sciences and Techniques, Moulay Ismail University, GAA Group, Errachidia, Morocco E-mail address: elkinani 67@yahoo.com Delfim F. M. Torres Center for Research and Development in Mathematics and Applications CIDMA, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 381-193 Aveiro, Portugal E-mail address: delfim@ua.pt