ON A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES

Transcription

1 J. Nonlinear Funct. Anal. 28 (28, Article ID 2 ON A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES SAÏD ABBAS, MOUFFAK BENCHOHRA 2,, JAMAL E. LAZREG 2, JUAN J. NIETO 3 Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 38, EN-Nasr, 2 Saïda, Algeria 2 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89, Sidi Bel-Abbès 22, Algeria 3 Departamento de Estatistica, Analisis Matematico e Optimizacion, Facultade de Matematicas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Abstract. In this paper, by using the technique that relies on the concept of the measure of noncompactness and the fixed point theory, we prove some existence results for some coupled systems of Hilfer and Hilfer-Hadamard fractional differential equations. Keywords. Coupled system; Fixed point; Fractional differential equation; Hilfer-Hadamard fractional derivative; Measure of noncompactness. 2 Mathematics Subject Classification. 26A33.. INTRODUCTION Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering, and other applied sciences [, 2]. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs of Abbas, Benchohra and N Guérékata [3, 4], Samko, Kilbas and Marichev [5], Kilbas, Srivastava and Trujillo [6] and Zhou [7], the papers by Abbas et al. [8, 9, ], and the references therein. Recently, considerable attention have been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative; see [,, 2, 3, 4, 5, 6, 7] and the references therein and a coupled system of Hadamard type sequential fractional differential equations was considered in [8]. In [9, 2, 2], the measure of noncompactness was applied to some classes of functional Riemann-Liouville or Caputo fractional differential equations in Banach spaces. Motivated Corresponding author. addresses: abbasmsaid@yahoo.fr (S. Abbas, benchohra@univ-sba.dz (M. Benchohra, lazregjamal@yahoo.fr (J.E. Lazreg, juanjose.nieto.roig@usc.es (J.J. Nieto. Received September 27, 27; Accepted March 4, 28. c 28 Journal of Nonlinear Functional Analysis

2 2 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO by the research going on in this direction, we discuss the existence of solutions for the following coupled system of Hilfer fractional differential equations (D α,β u(t = f (t,u(t,v(t (D α ; t I := [,T ], (. 2,β 2 v(t = f 2 (t,u(t,v(t with the following initial conditions (I γ u( = φ, (I γ 2 v( = φ 2, where T >, α i (,, β i [,], γ i = α i + β i α i β i, φ i E, f i : I E E E; i =,2, are given functions, E is a real (or complex Banach space with a norm, I γ i is the left-sided mixed Riemann-Liouville integral of order γ i, and D α i,β i is the generalized Riemann-Liouville derivative (Hilfer operator of order α i and type β i : i =,2. Next, we consider the following coupled system of Hilfer-Hadamard fractional differential equations ( H D α,β u(t = g (t,u(t,v(t ( H D α ; t [,T ], (.3 2,β 2 v(t = g 2 (t,u(t,v(t, with the following initial conditions ( H I γ u( = ψ ( H I γ 2 v( = ψ 2, where T >, α i (,, β i [,], γ i = α i + β i α i β i, ψ i E, g i : [,T ] E E E; i =,2 are given functions, H I γ i is the left-sided mixed Hadamard integral of order γ i, and H D α i,β i is the Hilfer-Hadamard fractional derivative of order α i and type β i ; i =,2. 2. PRELIMINARIES Let C be the Banach space of all continuous functions v from I into E with the supremum (uniform norm v := sup v(t. t I As usual, by AC(I we denote the space of absolutely continuous functions from I into E. We denote by AC (I the space defined by AC (I := {w : I E : d w(t AC(I}. dt By L (I, we denote the space of measurable functions v : I E which are Bochner integrable, with the norm T v = v(t dt. By C γ (I and Cγ (I, we denote the weighted spaces of continuous functions defined by C γ (I = {w : (,T ] E : t γ w(t C}, (.2 (.4 with the norm w Cγ := sup t γ w(t, t I

3 and with the norm A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS 3 C γ (I = {w C : dw dt C γ }, w C γ := w + w Cγ. Also, by C := C γ C γ2 we denote the product weighted space with the norm (u,v C = u Cγ + v Cγ2. Let M X denote the class of all bounded subsets of a metric space X. Definition 2.. Let X be a complete metric space. A map µ : M X [, is called a measure of noncompactness on X if it satisfies the following properties for all B,B,B 2 M X. (a µ(b = if and only if B is precompact (Regularity, (b µ(b = µ(b (Invariance under closure, (c µ(b B 2 = max{µ(b, µ(b 2 } (Semi-additivity. Example 2.2. In every metric space X, the map φ : M X [, with φ(b = if B is relatively compact and φ(b = otherwise is a measure of noncompactness, the so-called discrete measure of noncompactness; see Example in [9]. Definition 2.3. [2] Let E be a Banach space and let Ω E be the family of bounded subsets of E. The Kuratowski measure of noncompactness is the map µ : Ω E [, defined by where M Ω E. Here, we list some important properties. µ(m = inf{ε > : M m j=m j,diam(m j ε} ( µ(m = M is compact (M is relatively compact. (2 µ(m = µ(m. (3 M M 2 µ(m µ(m 2. (4 µ(m + M 2 µ(m + µ(m 2. (5 µ(cm = c µ(m, c R. (6 µ(conv M = µ(m. Now, we give some results and properties of fractional calculus. Definition 2.4. [3, 5, 6] The left-sided mixed Riemann-Liouville integral of order r > of a function w L (I is defined by (Iw(t r = (t s r w(sds; f or a.e. t I, Γ(r where Γ( is the (Euler s Gamma function defined by Γ(ξ = t ξ e t dt; ξ >. For all r,r,r 2 > and each w C, we have I r w C, and (I r Ir 2 w(t = (Ir +r 2 w(t; f or a.e. t I.

4 4 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO Definition 2.5. [3, 5, 6] The Riemann-Liouville fractional derivative of order r (, ] of a function w L (I is defined by ( d (D r w(t = dt I r w (t = d Γ( r dt (t s r w(sds; f or a.e. t I. Let r (,], γ [, and w C γ (I. Then the following expression leads to the left inverse operator as follows (D r I r w(t = w(t; f or all t (,T ]. Moreover, if I r w C γ (I, then the following composition is proved in [5] (ID r r w(t = w(t (I r w( + t r ; f or all t (,T ]. Γ(r Definition 2.6. [3, 6, 5] The Caputo fractional derivative of order r (,] of a function w L (I is defined by ( ( c D r w(t = I r d dt w (t = Γ( r (t s r d w(sds; f or a.e. t I. ds In [], Hilfer studied applications of a generalized fractional operator having the Riemann-Liouville and the Caputo derivatives as specific cases (see also [4, 5]. Definition 2.7. (Hilfer derivative. Let α (,, β [,], w L (I, and I ( α( β w AC (I. The Hilfer fractional derivative of order α and type β of w is defined as ( (D α,β w(t = I β( α d w (t; f or a.e. t I. (2. dt I( α( β Here, we also list some properties. Let α (,, β [,], γ = α + β αβ, and w L (I.. The operator (D α,β w(t can be written as (D α,β w(t = ( Moreover, the parameter γ satisfies I β( α d w dt I γ ( (t = I β( α D γ (t; w f or a.e. t I. γ (,], γ α, γ > β, γ < β( α. 2. The generalization (2. for β = coincides with the Riemann-Liouville derivative and for β = with the Caputo derivative D α, = D α, and D α, = c D α. 3. If D β( α w exists and is in L (I, then (D α,β I α w(t = (I β( α D β( α w(t; f or a.e. t I.

5 A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS 5 Furthermore, if w C γ (I and I β( α w C γ (I, then 4. If D γ w exists and is in L (I, then (D α,β I α w(t = w(t; f or a.e. t I. (I α D α,β w(t = (I γ Dγ I γ ( + w(t = w(t t γ ; f or a.e. t I. Γ(γ Corollary 2.8. Let h C γ (I. Then the Cauchy problem (D α,β u(t = h(t; t I, (I γ u(t t= = φ, has the following unique solution u(t = φ Γ(γ tγ + (I α h(t. Lemma 2.9. [22] If Y is a bounded subset of a Banach space X, then for each ε >, there is a sequence {y k } k= Y such that µ(y 2µ({y k } k= + ε. Lemma 2.. [23] If {u k } k= L (I is uniformly integrable, then µ({u k } k= is measurable and for each t I, ({ } µ u k (sds 2 µ({u k (s} k=ds. k= In the sequel, we will make use of the following fixed point theorem. Theorem 2.. [24] Let F be a closed and convex subset of a real Banach space. Let G : F F be a continuous operator and Let G(F be bounded. If there exists a constant k [, such that for each bounded subset B F, µ(g(b kµ(b, then G has a fixed point in F. 3. A COUPLED SYSTEM OF HILFER FRACTIONAL DIFFERENTIAL EQUATIONS In this section, we are concerned with the existence of solutions of system (.-(.2. Definition 3.. By a solution of problem (.-(.2, we mean a coupled measurable functions (u, v C γ C γ2 those satisfy equations (. on I, and conditions (I γ u( + = φ, and (I γ 2 v( + = φ 2. The following hypotheses will be used in the sequel. (H The functions t f i (t,u,v; i =,2 are measurable on I for each (u,v C γ C γ2, and the functions (u,v f i (t,u,v are continuous on C γ C γ2 for a.e. t I, (H 2 There exist continuous functions p i : I (, ; i =,2 such that f i (t,u,v f i (t,w,z p i (t u w ; f or a.e. t I, and each u,v,w,z E, + u w + v z (H 3 For each bounded and measurable set B C γi ; i =,2 and for each t I, we have µ( f i (t,b,b p i (tµ(b,

6 6 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO Set p i = sup p i (t, and fi t I = sup f i (t,, ; i =,2. t I Now, we shall prove the following theorem concerning the existence of solutions of system (.-(.2. Theorem 3.2. Assume that hypotheses (H (H 3 hold. If then coupled system (.-(.2 has at least one solution defined on I. Proof. Define the operators N : C γ C γ and N 2 : C γ2 C γ2 by l := 4p T α Γ( + α + 4p 2 T α2 <, (3. Γ( + α 2 (N u(t = φ Γ(γ tγ + (t s α f (s,u(s,v(s ds, (3.2 Γ(α and (N 2 v(t = φ 2 Γ(γ 2 tγ 2 + (t s α 2 f (s,u(s,v(s ds. (3.3 Consider the continuous operator N : C C defined by (N(u,v(t = ((N u(t,(n 2 v(t. (3.4 Clearly, the fixed points of the operator N are solutions of system (.-(.2. For any u C γ, and each t I, we have t γ (N u(t φ Γ(γ + t γ Γ(α φ Γ(γ + t γ Γ(α + t γ Γ(α φ Γ(γ + t γ Γ(α (t s α f (s,u(s,v(s ds (t s α f (s,, ds (t s α f (s,u(s,v(s f (s,, ds φ Γ(γ + ( f + p Γ(α Also, for any v C γ2, and each t I, we get Thus (t s α ( f (s,, + p (sds γ T t φ Γ(γ + ( f + p Γ( + α T γ+α (t s α ds t γ (N u(t φ 2 Γ(γ 2 + ( f 2 + p 2 Γ( + α 2 T γ+α N(u,v C φ Γ(γ + φ 2 Γ(γ 2 + ( f + p Γ( + α This proves that N transforms the ball. T γ2+α2 + ( f 2 + p 2 T γ 2+α 2 Γ( + α 2 B R := B(,R = {(u,v C : (u,v C R}. := R. (3.5 into itself. We next show that the operator N : B R B R satisfies all the assumptions of Theorem 2.. The proof will be given in several steps.

7 A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS 7 Step. N : B R B R is continuous. Let {(u n,v n } n N be a sequence such that (u n,v n (u,v in B R. Then, for each t I, we have t γ (N u n (t t γ (N u(t t γ Γ(α p T γ Γ(α (t s α f (s,u n (s,v n (s f (s,u(s,v(s ds Also, t γ 2(N 2 v n (t t γ 2(N 2 v(t p 2 T γ 2 (t s α u n (s u(s ds. (t sα 2 u n (s u(s ds. Hence t γ (N u n (t t γ (N u(t + t γ 2 (N 2 v n (t t γ 2 (N 2 v(t p T γ Γ(α (t s α u n (s u(s ds + p 2 T γ 2 (t s α 2 u n (s u(s ds. (3.6 Since u n u as n, we see that equation (3.6 implies N(u n,v n N(u,v C as n. Step 2. N(B R is bounded. Since N(B R B R and B R is bounded, we find that N(B R is bounded. Step 3. For each bounded subset D of B R, µ(n(d lµ(d. From Lemmas 2.9 and 2., for any D B R and any ε >, there exists a sequence {(u n,v n } n= D, such that, for all t I, ({ φ t µ((nd(t = µ Γ(γ tγ + (t s α f (s,u(s,v(s ds Γ(α + φ 2 Γ(γ 2 tγ 2 + ({ 2µ ( ( ( (t s α Γ(α (t s α 2 (t s α 2 f 2(s,u(s,v(s ds; (u,v D f (s,u n (s,v n (sds f 2 (s,u n (s,v n (sds } n= + ε (t s α µ ({ f (s,u n (s,v n (s} Γ(α n=ds (t s α 2 µ ({ f 2 (s,u n (s,v n (s} n=ds + ε (t s α Γ(α (t s α Γ(α (t s α p (s + p (sds + (t s α 2 p (sds + Γ(α ( p T α Γ( + α + p 2 T α2 Γ( + α 2 lµ(d + ε. (t s α 2 (t s α 2 µ(d + ε } p 2 (s µ ({(u n (s,v n (s} n=ds + ε p 2 (sds µ ({(u n,v n } n= + ε p 2 (sds µ(d + ε

8 8 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO Since ε > is arbitrary, we ahve µ(n(b lµ(b. As a consequence of Steps to 3 together with Theorem 2., we can conclude that N has at least one fixed point in B R which is a solution of system (.- ( A COUPLED SYSTEM OF HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS Now, we are concerned with the existence of solutions of coupled system (.3-(.4. Set C := C([,T ], and denote the weighted space of continuous functions defined by with the norm C γ,ln ([,T ] = {w(t : (lnt γ w(t C}, w Cγ,ln := sup (lnt r w(t. t [,T ] Also, by C γ,γ 2,ln([,T ] := C γ,ln([,t ] C γ2,ln([,t ] we denote the product weighted space with the norm (u,v Cγ,γ 2,ln([,T ] = u Cγ,ln + v C γ2,ln. Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [6] for a more detailed analysis. Definition 4.. [6] (Hadamard fractional integral. The Hadamard fractional integral of order q > for a function g L ([,T ], is defined as ( H I q g(x = x ( ln x q g(s Γ(q s s ds, provided the integral exists. Example 4.2. Let < q < and g(x = lnx, x [,e]. Then and Set ( H I q g(x = Γ(2 + q (lnx+q ; f or a.e. x [,e]. δ = x d, q >, n = [q] +, dx AC n δ := {u : [,T ] E : δ n [u(x] AC(I}. Analogous to the Riemann-Liouville fractional derivative, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral in the following way: Definition 4.3. [6] (Hadamard fractional derivative. The Hadamard fractional derivative of order q > applied to the function w ACδ n is defined as In particular, if q (,], then ( H D q w(x = δ n ( H I n q w(x. ( H D q w(x = δ(h I q w(x.

9 A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS 9 Example 4.4. Let < q <. Let w(x = lnx, x [,e]. Then ( H D q w(x = Γ(2 q (lnx q ; f or a.e. x [,e]. It has been proved (see, e.g., Kilbas [[25], Theorem 4.8] that in the space L (I, the Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral, i.e., From Theorem 2.3 of [6], we have ( H D q (H I q w(x = w(x. ( H I q (H D q w(x = w(x (H I q w( (lnx q. Γ(q Analogous to the Hadamard fractional calculus, the Caputo-Hadamard fractional derivative is defined in the following way. Definition 4.5. (Caputo-Hadamard fractional derivative The Caputo-Hadamard fractional derivative of order q > applied to the function w ACδ n is defined as In particular, if q (,], then ( Hc D q w(x = (H I n q δ n w(x. ( Hc D q w(x = (H I q δw(x. From the Hadamard fractional integral, the Hilfer-Hadamard fractional derivative (introduced for the first time in [26] is defined in the following way. Definition 4.6. (Hilfer-Hadamard fractional derivative. Let α (,, β [,], γ = α +β αβ, w L (I, and H I ( α( β w AC (I. The Hilfer-Hadamard fractional derivative of order α and type β applied to the function w is defined as ( ( H D α,β w(t = H I β( α ( H D γ (t w ( (4. = H I β( α δ( H I γ w (t; f or a.e. t [,T ]. This new fractional derivative (4. may be viewed as interpolating the Hadamard fractional derivative and the Caputo-Hadamard fractional derivative. Indeed for β = this derivative reduces to the Hadamard fractional derivative and when β =, we recover the Caputo-Hadamard fractional derivative. H D α, = H D α, and H D α, = Hc D α. From Theorem 2 of [27], we concluded the following lemma. Lemma 4.7. Let g : [,T ] E E be such that g(,u( C γ,ln ([,T ] for any u C γ,ln ([,T ]. Then problem (.3 is equivalent to the following Volterra integral equation u(t = φ Γ(γ (lntγ + ( H I α g(,u( (t. Definition 4.8. By a solution of coupled system (.3-(.4, we mean a coupled measurable functions (u,v C γ,ln C γ2,ln those satisfy the conditions (.4 and the equations (.3 on [,T ].

10 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO Now we give (without proof similar existence results for system (.3-(.4. Let us introduce the following hypotheses: (H The functions t g i(t,u,v; i =,2 are measurable on [,T ] for each (u,v C γ,ln C γ2,ln, and the function (u,v g i (t,u,v is continuous on C γ,ln C γ2,ln for a.e. t [,T ], (H 2 There exist continuous functions q i : [,T ] (, such that g i (t,u,v g i (t,w,z q i (t u w ; f or a.e. t [,T ], and each u,v,w,z E, + u w + v z (H 3 For each bounded and measurable set B C γ,γ 2,ln and for each t [,T ], we have µ(g i (t,b,b q i (tµ(b; i =,2. Theorem 4.9. Assume that hypotheses (H (H 3 hold. If l := 4q (lnt α Γ( + α + 4q 2 (lnt α2 <, (4.2 Γ( + α 2 where q i = supq i (t; i =,2, then system (.3 -(.4 has at least one solution defined on [,T ]. t I Let E = l = be the Banach space with the norm { 5. AN EXAMPLE u = (u,u 2,...,u n,..., u E = n= u n. n= u n < Consider the following coupled system of Hilfer fractional differential equation of the form 2 (D, 2 u n (t = f n (t,u(t,v(t; (D 2, 2 v n (t = g n (t,u(t,v(t; ; t [,], (5. (I 4 u n (t t= = (,,...,,..., (I 4 v n (t t= = (,,...,,..., where f n (t,u,v = f n (,u,v =, t 4 u n (tsint 64( + t( + u E + v E ; t (,], } with g n (t,u,v = u n (tcost 64( + u E + v E ; t (,], f = ( f, f 2,..., f n,..., g = (g,g 2,...,g n,..., u = (u,u 2,...,u n,..., ( v = (v,v 2,...,v n,..., and c := e3 8 Γ. 2

11 A COUPLED SYSTEM OF HILFER AND HILFER-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS Setting α i = β i = 2 ; i =,2, one has γ i = 3 4 ; i =,2. The hypothesis (H 2 is satisfied with p (t = t 4 sint 64( + t ; t (,], p ( =, and p 2 (t = cost 64 ; t (,]. Hence, Theorem 3.2 implies that system (5. has at least one solution defined on [,]. REFERENCES [] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [2] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, 2. [3] S. Abbas, M. Benchohra, G. M. N Guérékata, Topics in Fractional Differential Equations, Springer, New York, 22. [4] S. Abbas, M. Benchohra, G. M. N Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 25. [5] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 993. [6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 26. [7] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 24. [8] S. Abbas, E. Alaidarous, M. Benchohra, J.J. Nieto, Existence and stability of solutions for Hadamard-Stieltjes fractional integral equations, Discret. Dyn. Nat. Soc. 25 (25, Article ID [9] S. Abbas, M. Benchohra, J. Henderson, J.E. Lazreg, Measure of noncompactness and impulsive Hadamard fractional implicit differential equations in Banach spaces, Math. Eng. Science Aerospace 8 (27, -9. [] S. Abbas, M. Benchohra, J.E. Lazreg, Y. Zhou, A Survey on Hadamard and Hilfer fractional differential equations: analysis and stability, Chaos, Solitons Fractals 2 (27, [] K. M. Furati, M. D. Kassim. Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations 23 (23, Article ID 235. [2] K. M. Furati, M. D. Kassim, and N. e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. 64 (22, [3] H. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput. 257 (25, [4] R. Kamocki, C. Obczńnski, On fractional Cauchy-type problems containing Hilfer s derivative, Electron. J. Qual. Theory Differ. Equ. 26 (26, Article ID 5. [5] Ž. Tomovski, R. Hilfer, H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct. 2 (2, [6] J.-R. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (25, [7] M. Yang, Q.-R. Wang, Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions, Math. Methods Appl. Sci. 4 (27, [8] S. Aljoudi, B. Ahmad, J. Nieto, A. Alsaedi, A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos, Solitons Fractals 9 (26, [9] J. M. Ayerbee Toledano, T. Dominguez Benavides, G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications, vol 99, Birkhäuser, Basel, Boston, Berlin, 997. [2] J. Bana`s, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 98. [2] M. Benchohra, J. Henderson, D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal. 2 (28,

12 2 S. ABBAS, M. BENCHOHRA, J.E. LAZREG, J.J. NIETO [22] D. Bothe, Multivalued perturbation of m-accretive differential inclusions, Isreal J. Math. 8 (998, [23] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (98, [24] L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 39 (25, [25] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2, [26] M. D. Qassim, K. M. Furati, N.-e. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abst. Appl. Anal. 22 (22, Article ID [27] M. D. Qassim, N.-e. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abst. Appl. Anal. 23 (23, Article ID 6529.