Existence results for fractional order functional differential equations with infinite delay

Transcription

1 J. Math. Anal. Appl. 338 (28) Existence results for fractional order functional differential equations with infinite delay M. Benchohra a, J. Henderson b, S.K. Ntouyas c,,a.ouahab a a Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22 Sidi Bel Abbès, Algeria b Department of Mathematics, Baylor University, Waco, TX , USA c Department of Mathematics, University of Ioannina, Ioannina, Greece Received 9 January 26 Available online 22 June 27 Submitted by I. Podlubny Abstract The Banach fixed point theorem and the nonlinear alternative of Leray Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay. 27 Elsevier Inc. All rights reserved. Keywords: Functional differential equations; Fractional derivative; Fractional integral; Existence; Fixed point 1. Introduction This paper is concerned with the existence of solutions for initial value problems (IVP for short) of fractional order functional differential equations with infinite delay. In Section 3 we will consider the IVP of the form D α y(t) = f(t,y t ), for each t J =[,b], <α<1, (1) y(t) = φ(t), t (, ], (2) where D α is the standard Riemman Liouville fractional derivative, f : J B R is a given function satisfying some assumptions that will be specified later, φ B, φ() = and B is called a phase space that will be defined later (see Section 2). For any function y defined on (,b] and any t J, we denote by y t the element of B defined by y t (θ) = y(t + θ), θ (, ]. * Corresponding author. addresses: benchohra@yahoo.com (M. Benchohra), Johnny_Henderson@baylor.edu (J. Henderson), sntouyas@cc.uoi.gr (S.K. Ntouyas), ouahab@univ-sba.dz (A. Ouahab) X/$ see front matter 27 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Here y t ( ) represents the history of the state from time up to the present time t. Section 4 is devoted to fractional neutral functional differential equations, D α[ y(t) g(t,y t ) ] = f(t,y t ), for each t J, (3) y(t) = φ(t), t (, ], where f and φ are as in problem (1) (2), and g : J B R is a given function such that g(,φ)=. The notion of the phase space B plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato [9] (see also Kappel and Schappacher [12] and Schumacher [22]). For a detailed discussion on this topic we refer the reader to the book by Hino et al. [11]. In fact, for the case where α = 1, in addition to [9,11], an extensive theory has been developed for the problems (1) (2) and (3) (4) by Corduneanu and Lakshmikantham [1], Lakshmikantham et al. [13] and Shin [23]. Fractional differential equations have been of great interest recently. In cause, in part to both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, etc. For details, see the monographs of Miller and Ross [16], Podlubny [19] and Samko et al. [21], and the papers of Delboso and Rodino [3], Diethelm et al. [2,4,5], Gaul et al. [6], Glockle and Nonnenmacher [7], Mainardi [14], Metzler et al. [15], Momani and Hadid [17], Momani et al. [18], Podlubny et al. [2], Yu and Gao [24] and the references therein. Our approach is based on the Banach fixed point theorem and on the nonlinear alternative of Leray Schauder type [8]. These results can be considered as a contribution to this emerging field. 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. We consider R + =x R: x>}, C (R + ) the space of all continuous functions on R +. Consider also the space C (R + ) of all continuous real functions on (4) R + =x R: x }, which later identify by abuse of notation, with the class of all f C (R + ) such that lim t + f(t)= f( + ) R. By C(J,R) we denote the Banach space of all continuous functions from J into R with the norm y := sup y(t) : t J }, where denotes a suitable complete norm on R. Definition 2.1. The fractional primitive of order α> of a function h : R + R of order α R + is defined by I α h(t) = t (t s) α 1 h(s) ds, provided the right side exists pointwise on R +. Ɣ is the gamma function. For instance, I α h exists for all α>, when h C (R + ) L 1 loc (R+ ); note also that when h C (R + ) then I α h C (R + ) and moreover I α h() =. Definition 2.2. The fractional derivative of order α> of a continuous function h : R + R is given by d α h(t) dt α = 1 d Ɣ(1 α) dt a (t s) α h(s) ds = d dt I a 1 α h(t). In this paper, we assume that the state space (B, B ) is a seminormed linear space of functions mapping (, ] into R, and satisfying the following fundamental axioms which were introduced by Hale and Kato in [9].

3 1342 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) (A) If y : (,b] R, and y B, then for every t [,b] the following conditions hold: (i) y t is in B, (ii) y t B K(t)sup y(s) : s t}+m(t) y B, (iii) y(t) H y t B, where H is a constant, K : [,b] [, ) is continuous, M : [, ) [, ) is locally bounded and H,K,M are independent of y( ). (A-1) For the function y( ) in (A), y t is a B-valued continuous function on [,b]. (A-2) The space B is complete. 3. FDEs of fractional order Let us start by defining what we mean by a solution of problem (1) (2). Let the space Ω = y : (,b] R: y (,] B and y [,b] is continuous }. Definition 3.1. A function y Ω is said to be a solution of (1) (2) if y satisfies the equation D α y(t) = f(t,y t ) on J, and the condition y(t) = φ(t) on (, ]. For the existence results on the problem (1) (2) we need the following auxiliary lemma. Lemma 3.2. (See [3].) Let <α<1 and let h : (,b] R be continuous and lim t + h(t) = h( + ) R. Then y is a solution of the fractional integral equation y(t) = 1 (t s) α 1 h(s) ds, if and only if y is a solution of the initial value problem for the fractional differential equation D α y(t) = h(t), y() =. t (,b], Our first existence result for the IVP (1) (2) is based on the Banach contraction principle. Theorem 3.3. Let f : J B R. Assume (H) there exists l> such that f(t,u) f(t,v) l u v B, for t J and every u, v B. If b α K b l Ɣ(α+1) < 1, where K b = sup } K(t) : t [,b], then there exists a unique solution for the IVP (1) (2) on the interval (,b]. Proof. Transform the problem (1) (2) into a fixed point problem. Consider the operator N : Ω Ω defined by φ(t), t (, ], N(y)(t) = 1 t (t s)α 1 f(s,y s )ds, t [,b]. Let x( ) : (,b] R be the function defined by, if t [,b], x(t) = φ(t), if t (, ].

4 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Then x = φ. For each z C([,b], R) with z() =, we denote by z the function defined by z(t), if t [,b], z(t) =, if t (, ]. If y( ) satisfies the integral equation y(t) = 1 (t s) α 1 f(s,y s )ds, we can decompose y( ) as y(t) = z(t) + x(t), t b, which implies y t = z t + x t, for every t b, and the function z( ) satisfies z(t) = 1 (t s) α 1 f(s, z s + x s )ds. Set C = z C ( [,b], R ) : z = }, and let b be the seminorm in C defined by z b = z B + sup z(t) : t b } = sup z(t) : t b }, z C. C is a Banach space with norm b. Let the operator P : C C be defined by (P z)(t) = 1 (t s) α 1 f(s, z s + x s )ds, t [,b]. (5) That the operator N has a fixed point is equivalent to P has a fixed point, and so we turn to proving that P has a fixed point. We shall show that P : C C is a contraction map. Indeed, consider z, z C. Then we have for each t [,b] P (z)(t) P(z )(t) K b (t s) α 1 f(s, z s + x s ) f ( s, z s + x s) ds (t s) α 1 l zs z s B ds (t s) α 1 lk b sup z(s) z (s) ds s [,t] (t s) α 1 lds z z b. Therefore P(z) P(z ) b bα K b l z z b, and hence P is a contraction. Therefore, P has a unique fixed point by Banach s contraction principle. Now we give an existence result based on the nonlinear alternative of Leray Schauder type. For this, we state the following generalization of Gronwall s lemma for singular kernels, whose proof can be found in [1, Lemma 7.1.1], which will be essential for the main result of this section.

5 1344 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Lemma 3.4. Let v : [,b] [, ) be a real function and w( ) is a nonnegative, locally integrable function on [,b] and there are constants a> and <α<1 such that v(t) w(t) + a v(s) (t s) α ds. Then there exists a constant K = K(α) such that v(t) w(t) + Ka for every t [,b]. w(s) (t s) α ds, Theorem 3.5. Assume that the following hypotheses hold: (H1) f is a continuous function; (H2) there exist p,q C(J,R + ) such that f(t,u) p(t) + q(t) u B for t J and each u B, and I α p < +. Then the IVP (1) (2) has at least one solution on (,b]. Proof. Let P : C C be defined as in (5). We shall show that the operator P is continuous and completely continuous. Step 1. P is continuous. Let z n } be a sequence such that z n z in C. Then (P z n )(t) (P z)(t) 1 b (t s) α 1 f(s, z ns + x s ) f(s, z s + x s ) ds. Since f is a continuous function, we have P(zn ) P(z) b α b f(, zn( ) + x ( ) ) f(, z ( ) + x ( ) ) Step 2. P maps bounded sets into bounded sets in C. asn. Indeed, it is enough to show that for any η>, there exists a positive constant l such that for each z B η =z C : z b η} one has P(z) l.letz B η. Since f is a continuous function, we have for each t [,b], (P z)(t) 1 1 b b (t s) α 1 f(s, z s + x s )ds (t s) α 1[ p(s) + q(s) z s + x s B ] ds bα p + bα q η =: l, where z s + x s B z s B + x s B K b η + M b φ B := η,

6 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) and M b = sup } M(t) : t [,b]. Hence (P z) l. Step 3. P maps bounded sets into equicontinuous sets of C. Let t 1,t 2 [,b], t 1 <t 2, and let B η be a bounded set of C as in Step 2. Let z B η. Then for each t [,b], we have (P z)(t 2 ) (P z)(t 1 ) = 1 1 ( (t2 s) α 1 (t 1 s) α 1) f(s, z s + x s )ds+ 1 2 (t 2 s) α 1 f(s, z s + x s )ds p + q η p + q η 1 [ (t1 s) α 1 (t 2 s) α 1] ds + p + q η t 1 [ (t2 t 1 ) α + t1 α ] p + q η tα 2 + (t 2 t 1 ) α 2 t 1 (t 2 s) α 1 ds 2[ p + q η ] (t 2 t 1 ) α. As t 1 t 2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t 1 <t 2 and t 1 t 2 is obvious. As a consequence of Steps 1 3, together with the Arzela Ascoli theorem, we can conclude that P : C C is continuous and completely continuous. Step 4 (A priori bounds). We now show there exists an open set U C with z λp (z) for λ (, 1) and z U. Let z C and z = λp (z) for some <λ<1. Then for each t [,b] we have [ 1 t ] z(t) = λ (t s) α 1 f(s, z s + x s )ds. This implies by (H2) But z(t) 1 (t s) α 1 q(s) z s + x s B ds + bα p, t [,b]. z s + x s B z s B + x s B K(t)sup } z(s) : s t + M(t) z B + K(t)sup x(s) : s t } + M(t) x B K b sup } z(s) : s t + Mb φ B. (6) If we name w(t) the right-hand side of (6), then we have z s + x s B w(t), and therefore z(t) 1 (t s) α 1 q(s)w(s)ds + bα p, t [,b].

7 1346 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Using the above inequality and the definition of w we have that w(t) M b φ B + K bb α p + K b q Then from Lemma 3.4, there exists K = K(α) such that we have where w(t) M b φ B + K bb α p + K(α) K b q R = M b φ B + K bb α p. Hence w R + RK(α)bα K b := M. Then Set z M I α q + bα p := M. U = z C : z b <M + 1 }. (t s) α 1 w(s)ds, t [,b]. (t s) α 1 Rds, P : U C is continuous and completely continuous. From the choice of U, there is no z U such that z = λp (z), for λ (, 1). As a consequence of the nonlinear alternative of Leray Schauder type [8], we deduce that P has a fixed point z in U. 4. NFDEs of fractional order In this section we give existence results for the IVP (3) (4). Definition 4.1. A function y Ω is said to be a solution of (3) (4) if y satisfies the equation D α [y(t) g(t,y t )]= f(t,y t ) on J, and y(t) = φ(t) on (, ]. Our first existence result for the IVP (3) (4) is also based on the Banach contraction principle. Theorem 4.2. Assume that (H) holds and moreover (A) there exists a nonnegative constant c 1 such that g(t,u) g(t,v) c 1 u v B, for every u, v B. If K b [c 1 + bα l Ɣ(α+1) ] < 1, then there exists a unique solution for the IVP (3) (4) on the interval (,b]. Proof. Consider the operator N 1 : Ω Ω defined by φ(t), if t (, ], N 1 (y)(t) = g(t,y t ) + 1 (t s)α 1 f(s,y s )ds, if t [,b]. In analogy to Theorem 3.5, we consider the operator P 1 : C C defined by, t, (P 1 z)(t) = g(t, z t + x t ) + 1 (t s)α 1 f(s, z s + x s )ds, t (,b].

8 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) We shall show that the operator P 1 is a contraction. Let z, z Ω. Then following the steps of Theorem 3.3, we have P 1 (z)(t) P 1 (z )(t) g(t, z t + x t ) g(t, z t + x t ) + 1 f(s, z s + x s ) f(s, z s + x s ) (t s) α 1 ds c 1 z t z t B + 1 (t s) α 1 l z s z s B ds c 1 K b sup z(s) z (s) : s [,t] } + 1 (t s) α 1 lk b sup z(s) z (s) : s [,t] } ds. Consequently P1 (z) P 1 (z ) [ ] b K b c 1 + lbα z z b, which implies that P 1 is a contraction. Hence P 1 has a unique fixed point by Banach s contraction principle. Our second existence result for the IVP (3) (4) is based on the nonlinear alternative of Leray Schauder. Theorem 4.3. Assume (H1) (H2) and the following condition: (H3) the function g is continuous and completely continuous, and for any bounded set B in Ω, thesett g(t,y t ): y B} is equicontinuous in C([,b], R), and there exist constants K b d 1 < 1, d 2 such that g(t,u) d 1 u B + d 2, t [,b], u B. Then the IVP (3) (4) has at least one solution on (,b]. Proof. Let P 1 : C C be defined as in Theorem 4.2. We shall show that the operator P 1 is continuous and completely continuous. Using (H3) it suffices to show that the operator P 2 : C C defined by P 2 (z)(t) = g(t, z s + x s ) + 1 (t s) α 1 f(s, z s + x s )ds, t [,b], is continuous and completely continuous. This was proved in Theorem 3.5. We now show there exists an open set U C with z λp 1 (z) for λ (, 1) and z U. Let z C and z = λn 1 (z) for some <λ<1. Then and [ z(t) = λ g(t, z t + x t ) + 1 z(t) d1 z t + x t B + d 2 + bα p + 1 ] (t s) α 1 f(s, z s + x s )ds, t [,b], (t s) α 1 q(s) z s + x s B ds, t (,b].

9 1348 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Thus [ 1 w(t) 2K b d 2 + K bb α p 1 K b d 1 and consequently where Then w R 1 + q = bα R 1 K b q := L, (1 K b d 1 ) q 1 K b d 1, and R 1 = 1 1 K b d 1 + K b q ] (t s) α 1 w(s)ds, t (,b], [ 2K b d 2 + K bb α ] p. z d 1 φ B + 2d 2 + Ld 1 + bα p + L I α q := L. Set U 1 = y C : y b <L + 1 }. From the choice of U there is no y U 1 such that y = λp 2 (y) for λ (, 1). As a consequence of the nonlinear alternative of Leray Schauder type [8], we deduce that P 2 has a fixed point z in U 1. Then N 1 has a fixed point, which is a solution of the IVP (3) (4). 5. An example In this section we give an example to illustrate the usefulness of our main results. Let us consider the fractional functional differential equation, D α y(t) = ce γt+t y t ( e t + e t), t J := [,b], α (, 1), (7) (1 + y t ) y(t) = φ(t), t (, ], b where c = c sα 1 e s ds, and c > 1 fixed. Let γ be a positive real constant and B γ = y C ( (, ], R ) } : lim θ eγθ y(θ), exists in R. The norm of B γ is given by y γ = sup <θ e γθ y(θ). Let y : (,b] R be such that y B γ. Then lim θ eγθ y t (θ) = lim θ eγθ y(t + θ)= lim θ eγ(θ t) y(θ) = e γt lim θ eγθ y (θ) <. Hence y t B γ. Finally we prove that y t γ K(t)sup y(s) : s t } + M(t) y γ, where K = M = 1 and H = 1. We have y t (θ) = y(t + θ). If θ + t, we get yt (θ) } sup y(s) : <s. (8) For t + θ, then we have yt (θ) sup y(s) :<s t }.

10 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) Thus for all t + θ [,b], we get y t (θ) sup y(s) : <s } + sup y(s) : s t }. Then y t γ y γ + sup } y(s) : s t. It is clear that (B γ, γ ) is a Banach space. We can conclude that B γ is a phase space. Set e γt+t x f(t,x)= c(e t + e t )(1 + x), (t,x) [,b] B γ. Let x,y B γ. Then we have f(t,x) f(t,y) = e γt+t x c(e t + e t ) 1 + x y 1 + y = e t γt x y c(e t + e t )(1 + x)(1 + y) e t e γt x y c(e t + e t )(1 + x)(1 + y) et x y Bγ c(e t + e t ) 1 c x y B γ. b Hence the condition (H) holds. Assume that α cɣ(α+1) < 1. Since K = 1, then b α c < 1. Then by Theorem 3.3 the problem (7) (8) has a unique solution on (,b]. Acknowledgment The authors are grateful to the referee for her/his comments and remarks. References [1] C. Corduneanu, V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal. 4 (198) [2] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp [3] D. Delboso, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 24 (1996) [4] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (22) [5] K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997) [6] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991) [7] W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995) [8] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 23. [9] J. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978) [1] D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin, [11] Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, [12] F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (198) [13] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Math. Appl., Kluwer Academic Publishers, Dordrecht, [14] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp [15] F. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 13 (1995) [16] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, [17] S.M. Momani, S.B. Hadid, Some comparison results for integro-fractional differential inequalities, J. Fract. Calc. 24 (23) [18] S.M. Momani, S.B. Hadid, Z.M. Alawenh, Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. Math. Sci. 24 (24) [19] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, [2] I. Podlubny, I. Petraš, B.M. Vinagre, P. O Leary, L. Dorčk, Analogue realizations of fractional-order controllers, in: Fractional Order Calculus and Its Applications, Nonlinear Dynam. 29 (22)

11 135 M. Benchohra et al. / J. Math. Anal. Appl. 338 (28) [21] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, [22] K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay, Arch. Ration. Mech. Anal. 64 (1978) [23] J.S. Shin, An existence theorem of functional differential equations with infinite delay in a Banach space, Funkcial. Ekvac. 3 (1987) [24] C. Yu, G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 31 (25)