Dynamic Systems and Applications 8 (29) 539-55 FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH STATE-DEPENDENT DELAY MOHAMED ABDALLA DARWISH AND SOTIRIS K. NTOUYAS Department of Mathematics, Faculty of Science Alexandria University at Damanhour, 225 Damanhour, Egypt Department of Mathematics, University of Ioannina, 45 Ioannina, Greece ABSTRACT. In this paper we study the existence of solutions for the initial value problem for functional differential equations, as well as, for neutral functional differential equations of fractional order with state-dependent delay. The nonlinear alternative of Leray-Schauder type is the main tool in our analysis. AMS (MOS) Subject Classification. 26A33, 26A42, 34K3. INTRODUCTION The purpose of this paper is to study the existence of solutions for initial value problems (IVP for short) for both functional differential equations of fractional order with state-dependent delay (.) D β y(t) = f(t, y ρ(t,yt)), for each t J = [, b], < β <, (.2) y(t) = ϕ(t), t (, ] as well as for neutral functional differential equations of fractional order with statedependent delay (.3) D β [y(t) g(t, y ρ(t,yt))] = f(t, y ρ(t,yt)), for each t J, (.4) y(t) = ϕ(t), t (, ], where D β is the standard Riemman-Liouville fractional derivative. Here, f : J B R, g : J B R and ρ : J B (, b] are appropriate given functions, ϕ B, ϕ() =, g(, 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 (, ]. Received July 5, 28 56-276 $5. c Dynamic Publishers, Inc.
54 M. A. DARWISH AND S. K. NTOUYAS 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 semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [2] (see also Kappel and Schappacher [9] and Schumacher [3]). For a detailed discussion on this topic we refer the reader to the book by Hino et al [8]. Functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equation has received a significant amount of attention in the last years, we refer to [2, 3, 5, 7, 3, 4, 5] and the references therein. On the other hand, the first serious attempt to give a logical definition of a fractional derivative is due to Liouville. Now, the fractional calculus topic is enjoying growing interest among scientists and engineers, see [9, 7, 2, 22, 24, 25, 26, 27] and references therein. Differential equations of fractional order play a very important role in describing some real world problems. For example some problems in physics, mechanics and other fields can be described with the help of fractional differential equations, see [8,, 7, 23, 27, 28, 29] and references therein. The theory of differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to fractional differential equations, for example see [, 2, 2, 26, 3]. Our approach is based on the nonlinear alternative of Leray-Schauder type []. 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. Let R + = {x R : x > }, and let C (R + ) be the space of all continuous functions on R +. Consider also the space C (R + ) of all continuous real functions on R + = {x R : x }, which later identify by abuse of notation, with the class of all f C (R + ) such that lim f(t) = f(+ ) R. t + 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. Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary orders, see [2, 25, 26, 27].
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 54 Definition 2.. The fractional primitive of order β > of a function h : R + R of order β R + is defined by I β h(t) = (t s) β h(s)ds, provided the right hand side exists pointwise on R +. Γ is the gamma function. For instance, I β h exists for all β >, when h C (R + ) L loc (R+ ); note also that when h C (R + ) then I β h C (R + ) and moreover I β h() =. Definition 2.2. The fractional derivative of order β > of a continuous function h : R + R is given by d β h(t) dt β = d Γ( β) dt = d dt I β a h(t). a (t s) β h(s)ds For our existence results we will need the following lemma. Lemma 2.3. [6] Let < β < and let h : (, b] R be continuous and lim h(t) = t + h( + ) R. Then y is a solution of the fractional integral equation y(t) = (t s) β h(y(s))ds, if and only if, y is a solution of the initial value problem for the fractional differential equation D β y(t) = h(y(t)), t (, b], y() =. In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced in [8]. More precisely, B will be a linear space of all functions from (, ] to R endowed with a seminorm B satisfying the following axioms: (A) If y : (, b] R, b >, is continuous on J and y B, then for every t J the following conditions hold: (i) y t 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 : [, ) [, ) is continuous, M : [, ) [, ) is locally bounded and H, K, M are independent of y( ). (A-) For the function y( ) in (A), y t is a B-valued continuous function on [, b]. (A-2) The space B is complete. The next lemma is a consequence of the phase space axioms and is proved in [3].
542 M. A. DARWISH AND S. K. NTOUYAS Lemma 2.4. Let ϕ B and I = (γ, ] be such that ϕ t B for every t I. Assume that there exists a locally bounded function J ϕ : I [, ) such that ϕ t B J ϕ (t) ϕ B for every t I. If y : (, b] R is continuous on J and y = ϕ, then y t B (M b +J ϕ (max{γ, s }) ϕ B +K b sup{ y(θ) : θ [, max{, s}]}, s (γ, b], where, we denoted K b = sup t J K(t) and M b = sup t J M(t). Finally, we state the following generalization of Gronwall s lemma for singular kernels, whose proof can be found in [6], Lemma 7.., will be essential for our main results. Lemma 2.5. Let v : [, b] [, ) be a real function and w( ) is a nonnegative, locally integrable function on [, b] and there are constants a > and < β < such that v(s) v(t) w(t) + a (t s) ds, β then, there exists a constant K = K(β) such that for every t [, b]. v(t) w(t) + Ka w(s) (t s) β ds, 3. FDEs OF FRACTIONAL ORDER In this section, the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions of problem (.) (.2). Let us start by defining what we mean by a solution of problem (.) (.2). Definition 3.. A function y : (, b] R is said to be a solution of (.) (.2) if y = ϕ, y ρ(s,ys) B for every s J and y(t) = f(s, y ρ(s,ys)) (t s) β ds, t J. In what follows we assume that ρ : J B (, b] is continuous and ϕ B and f satisfies the following hypotheses: (H) 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 β p < + ;
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 543 (H3) The function t ϕ t is well defined and continuous from the set R(ρ ) = {ρ(s, ψ) : (s, ψ) J B, ρ(s, ψ) } into B. Moreover, there exists a continuous and bounded function J ϕ : R(ρ ) (, ) such that ϕ t B J ϕ (t) ϕ B for every t R(ρ ). Remark 3.2. The hypothesis (H3) is adapted from [3], where we refer for remarks concerning this hypothesis. Theorem 3.3. Assume that the hypotheses (H) (H3) hold. If ρ(t, ψ) t for every (t, ψ) J B, then the IVP (.) (.2) has at least one solution on (, b]. Proof. Let Y = {u C(J, R) : u() = ϕ() = } endowed with the uniform convergence topology and N : Y Y be the operator defined by Ny(t) = f(s, ȳ ρ(s,ȳs)) ds, t J, (t s) β where ȳ : (, b] R is such that ȳ = ϕ and ȳ = y on J. From axiom (A) and our assumption on ϕ, we infer that Ny( ) is well defined and continuous. Let ϕ : (, b] R be the extension of ϕ to (, b] such that ϕ(θ) = ϕ() = on J and J ϕ = sup{j ϕ : s R(ρ )}. We will prove that N( ) is completely continuous from B r ( ϕ J, Y ) into B r ( ϕ J, Y ). Step : N is continuous on B r ( ϕ J, Y ). This was proved in [3, p. 55, Step 3]. Step 2: The set N(B r ( ϕ J, Y ))(t) = {Ny(t) : y B r ( ϕ J, Y )} is relatively compact in R for every t J. The case t = is obvious. Let < ǫ < t b. If y B r ( ϕ J, Y ), from Lemma 2.4 follows that and so ȳ ρ(t,ȳt) B r = (M b + J ϕ ) ϕ B + K b r (Ny)(t) = t f(s, ȳ ρ(s,ȳs)) (t s) β ds p(s) + q(s) ȳ ρ(s,ȳs) B (t s) β ds = bβ p Γ(β + ) + bβ q Γ(β + ) r := l Step 3: N maps bounded sets into equicontinuous sets of Y.
544 M. A. DARWISH AND S. K. NTOUYAS Let t, t 2 [, b], t < t 2 and let B r as in Step 2. Let y B r. Then for each t [, b], we have (Ny)(t 2 ) (Ny)(t ) = [(t 2 s) β (t s) β ]f(s, ȳ ρ(s,ȳs)) ds + 2 f(s, ȳ ρ(s,ȳs)) t (t 2 s) ds β p + r q t [(t s) β (t 2 s) β ]ds + p + r q t2 ds t (t 2 s) β p + r q [(t 2 t ) β + t β Γ(β + ) tβ 2 ] + p + r q (t 2 t ) β Γ(β + ) 2( p + r q ) (t 2 t ) β. Γ(β + ) As t t 2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t < t 2 and t t 2 is obvious. As a consequence of Steps to 3, together with the Arzelá-Ascoli theorem, we can conclude that N is continuous and completely continuous. Step 4: (A priori bounds). with y λn(y) for λ (, ) and y U. We now show there exists an open set U Y Let y Y and y = λn(y) for some < λ <. Then for each t [, b] we have This implies by (H2) [ t f(s, ȳ ] ρ(s,ȳs)) y(t) = λ ds. (t s) β y(t) = t f(s, ȳ ρ(s,ȳs)) (t s) β ds bβ p Γ(β + ) + q p(s) + q(s)[(m b + J ϕ ) ϕ B + K b sup{ ȳ(s) : s [, t]}] ds (t s) β (M b + J ϕ ) ϕ B + K b sup{ ȳ(s) : s [, t]} (t s) β ds, since ρ(s, ȳ s ) s for every s J. Here J φ = sup{j φ (s) : s R(ρ )}. If µ(t) = (M b + J ϕ ) ϕ B +K b sup{ ȳ(s) : s [, max{, ρ(s, ȳ t )}] then we obtain µ(t) bβ p Γ(β + ) + q (t s) β µ(s)ds.
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 545 Then from Lemma 2.5, there exists K = K(β) such that we have Then Set µ(t) bβ p Γ(β + ) + q = bβ p Γ(β + ) bβ p Γ(β + ) K(β) (t s) β bβ p Γ(β + ) ds { + q } K(β) (t s) β ds { + q } Γ(β + ) K(β)bβ. { µ bβ p + q } Γ(β + ) Γ(β + ) K(β)bβ := M. U = {y Y : y < M + }. N : U Y is continuous and completely continuous. From the choice of U, there is no y U such that y = λn(y), for λ (, ). As a consequence of the nonlinear alternative of Leray-Schauder type [], we deduce that N has a fixed point y in U. 4. NFDEs OF FRACTIONAL ORDER In this section we give existence results for the IVP (.3) (.4). Definition 4.. A function y : (, b] R is said to be a solution of (.3) (.4) if y = ϕ, y ρ(s,ys) B for every s J and y(t) = g(s, y ρ(s,ys)) + (t s) β f(s, y ρ(s,ys))ds, t J. Theorem 4.2. Assume (H) (H2) and the following condition: (H4) the function g is continuous and completely continuous, and for any bounded set Q in B C([, b], R), the set {t g(t, y t ) : y Q} is equicontinuous in C([, b], R), and there exist constants K b d <, d 2 such that g(t, u) d u B + d 2, t [, b], u B. If ρ(t, ψ) t for every (t, ψ) J B, then the IVP (.3) (.4) has at least one solution on (, b]. Proof. Consider the operator N : C((, b], R) C((, b], R) defined by, ϕ(t), if t (, ], N (y)(t) = ϕ() g(, ϕ) + g(t, y ρ(t,yt)) + (t s) β f(s, y ρ(s,ys))ds, if t [, b].
546 M. A. DARWISH AND S. K. NTOUYAS In analogy to Theorem 3.3, we consider the operator N : Y Y defined by, t, (N y)(t) = g(t, ȳ ρ(s,ȳs)) + (t s) β f(s, ȳ ρ(s,ȳs))ds, t [, b]. We shall show that the operator N is continuous and completely continuous. Using (H4) it suffices to show that the operator N 2 : Y Y defined by, N 2 (y)(t) = (t s) β f(s, ȳ ρ(s,ȳs)) ds, t [, b], is continuous and completely continuous. This was proved in Theorem 3.3. y U. and We now show there exists an open set U Y with y λn (y) for λ (, ) and Let y Y and y = λn (y) for some < λ <. Then [ y(t) = λ g(s, ȳ ρ(s,ȳs)) + ] (t s) β f(s, ȳ ρ(s,ȳs)) ds, t [, b], y(t) d ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]}) + d 2 + (t s) β [p(s) + q(s)((m b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]})ds d ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]}) + d 2 + bβ p Γ(β + ) + q for t (, b]. If µ(t) = sup{ y(s) : s [, t]} then (t s) β ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]})ds, µ(t) d (M b + J ϕ ) ϕ B + d K b µ(t) + d 2 + bβ p Γ(β + ) + bβ q Γ(β + ) + q K b (t s) β µ(s) ds d (M b + J ϕ ) ϕ B + d K b µ(t) + d 2 (t s) β (M b + J ϕ ) ϕ B + bβ p Γ(β + ) + bβ q Γ(β + ) bβ (M b + J ϕ ) ϕ B +K b q (t s) β µ(s) ds, t (, b],
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 547 or µ(t) K b d + K b q K b d Consequently µ K b d + K b q K b d [ d (M b + J ϕ ) ϕ B + d 2 + bβ p (t s) β µ(s) ds, t (, b]. [ d (M b + J ϕ ) ϕ B + d 2 + bβ p (t s) β µ(s) ds and by Lemma 2.5, there exists K = K(β) such that where µ Λ + Λ 2 K(β) Λ = K b d Set ] Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B ] Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B (t s) β Λ ds Λ + Λ 2 K(β)Λ b β := L, [ ] d (M b + J ϕ ) ϕ B + d 2 + bβ p Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B, Λ = K b K b d q. U = {y Y : y < L + }. From the choice of U there is no y U such that y = λn (y) for λ (, ). As a consequence of the nonlinear alternative of Leray-Schauder type [], we deduce that N has a fixed point y in U. Then N has a fixed point, which is a solution of the IVP (.3) (.4). Acknowledgement. This work was completed when the first author was visiting the Department of Mathematics, Massachusetts Institute of Technology, USA. It is a pleasure for him to express gratitude for the warm hospitality. Also, the authors would like to thank the referee for his corrections and valuable remarks. REFERENCES [] O.P. Agrawal, Analytical schemes for a new class of fractional differential equations, J. Phys. A 4 (2) (27), 5469 5477. [2] W. Aiello, H.I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal Applied Mathematics, 52(3) (992), 855 869.
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