An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces

Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 628916, 11 pages doi:10.1155/2009/628916 Research Article An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces Mouffak Benchohra,1Alberto Cabada,2and Djamila Seba3 1Laboratoire de Math´ ematiques, Universit´ e de Sidi Bel-Abb` es, BP 89, 22000 Sidi Bel-Abb` es, Algeria 2Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain 3D´ epartement de Math´ ematiques, Universit´ e de Boumerd` es, Avenue de l’Ind´ ependance, 35000 Boumerd` es, Algeria Correspondence should be addressed to Mouffak Benchohra, [email protected] Received 30 January 2009; Revised 23 March 2009; Accepted 15 May 2009 Recommended by Juan J. Nieto The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness. Copyright q2009 Mouffak Benchohra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer 1, Glockle and Nonnenmacher 2, Metzler et al.3, Podlubny 4, Gaul et al. 5, among others. Fractional differential equations are also often an object of mathematical investigations; see the papers of Agarwal et al.6, Ahmad and Nieto 7, Ahmad and Otero- Espinar 8, Belarbi et al.9, Belmekki et al 10, Benchohra et al.11–13, Chang and Nieto 14, Daftardar-Gejji and Bhalekar 15, Figueiredo Camargo et al. 16, and the monographs of Kilbas et al.17and Podlubny 4. Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain y0,y 0, and so forth. the same requirements of boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see 18,19. 2 Boundary Value Problems In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form cDrytft, y t,for each t∈J0,T, y0−y0T 0 gs, y sds, yTyTT 0 hs, y sds, 1.1 where cDr,1<r≤2 is the Caputo fractional derivative, f,g,andh:J×E→Eare given functions satisfying some assumptions that will be specified later, and Eis a Banach space with norm ·. Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics 20and cellular systems 21. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra 22, Benchohra et al.23,24,Infante25, Peciulyte et al.26, and the references therein. In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of M¨ onch type. This technique was mainly initiated in the monograph of Bana and Goebel 27and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni 28, Guo et al. 29, Lakshmikantham and Leela 30,M¨ onch 31, and Szufla 32. 2. Preliminaries In this section, we present some definitions and auxiliary results which will be needed in the sequel. Denote by CJ, Ethe Banach space of continuous functions J→E, with the usual supremum norm  y ∞sup yt,t∈J.2.1 Let L1J, Ebe the Banach space of measurable functions y:J→Ewhich are Bochner integrable, equipped with the norm  y L1T 0 ysds. 2.2 Boundary Value Problems 3 Let L∞J, Ebe the Banach space of measurable functions y:J→Ewhich are bounded, equipped with the norm  y L∞inf c>0:yt≤c, a.e. t∈J.2.3 Let AC1J, Ebe the space of functions y:J→E, whose first derivative is absolutely continuous. Moreover, for a given set Vof functions v:J→Elet us denote by Vt{vt,v∈V},t∈J, VJ{vt:v∈V},t∈J. 2.4 Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness. Definition 2.1 see 27.LetEbe a Banach space and ΩEthe bounded subsets of E.The Kuratowski measure of noncompactness is the map α:ΩE→0,∞defined by αBinf >0:B⊆n i1Biand diam Bi≤; here B∈ΩE.2.5 Properties The Kuratowski measure of noncompactness satisfies some properties for more details see 27. aαB0⇔Bis compact Bis relatively compact. bαBαB. cA⊆B⇒αA≤αB. dαAB≤αAαB. eαcB|c|αB;c∈R. fαcoBαB. Here Band coB denote the closure and the convex hull of the bounded set B, respectively. For completeness we recall the definition of Caputo derivative of fractional order. Definition 2.2 see 17. The fractional order integral of the function h∈L1a, b of order r∈R; is defined by Ir aht1 Γrt a hs t−s1−rdt, 2.6 4 Boundary Value Problems where Γis the gamma function. When a0, we write Irhth∗ϕrt,where ϕrttr−1 Γrfor t>0,2.7 ϕrt0fort≤0, and ϕr→δtas r→0. Here δis the delta function. Definition 2.3 see 17. For a function hgiven on the interval a, b, the Caputo fractionalorder derivative of h,oforderr>0,is defined by cDr aht1 Γn−rt a hnsds t−s1−nr.2.8 Here nr1andrdenotes the integer part of r. Definition 2.4. A map f:J×E→Eis said to be Carath´ eodory if it→ ft, uis measurable for each u∈E; iiu→ ft, uis continuous for almost each t∈J. For our purpose we will only need the following fixed point theorem and the important Lemma. Theorem 2.5 see 31,33.Let Dbe a bounded, closed and convex subset of a Banach space such that 0∈D, and let Nbe a continuous mapping of Dinto itself. If the implication VcoN Vor VNV∪{0}⇒αV02.9 holds for every subset Vof D,thenNhas a fixed point. Lemma 2.6 see 32.Let Dbe a bounded, closed, and convex subset of the Banach space CJ, E, G a continuous function on J×J, and a function f:J×E→Esatisfies the Carath´ eodory conditions, and there exists p∈L1J,Rsuch that for each t∈Jand each bounded set B⊂Eone has lim k→0αfJt,k ×B≤ptαB;where Jt,k t−k,t∩J. 2.10 If Vis an equicontinuous subset of D,then αJ Gs, tfs, y sds :y∈V≤J Gt, spsαVs ds. 2.11 Boundary Value Problems 5 3. Existence of Solutions Let us start by defining what we mean by a solution of the problem 1.1. Definition 3.1. A function y∈AC1J, Eis said to be a solution of 1.1if it satisfies 1.1. Let σ, ρ1,ρ 2:J→Ebe continuous functions and consider the linear boundary value problem cDrytσt,t∈J, y0−y0T 0 ρ1sds, yTyTT 0 ρ2sds. 3.1 Lemma 3.2 see 11.Let 1<r≤2and let σ, ρ1,ρ 2:J→Ebe continuous. A function yis a solution of the fractional integral equation ytPtT 0 Gt, sσsds 3.2 with PtT1−t T2T 0 ρ1sds t1 T2T 0 ρ2sds, 3.3 Gt, s⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ t−sr−1 Γr−1t T−sr−1 T2Γr−1t T−sr−2 T2Γr−1,0≤s≤t, −1t T−sr−1 T2Γr−1t T−sr−2 T2Γr−1,t≤s≤T, 3.4 if and only if yis a solution of the fractional boundary value problem 3.1. Remark 3.3. It is clear that the function t→ T 0|Gt, s|ds is continuous on J, and hence is bounded. Let  G:sup T 0 |Gt, s|ds, t ∈J.3.5 6 Boundary Value Problems For the forthcoming analysis, we introduce the following assumptions H1The functions f,g,h :J×E→Esatisfy the Carath´ eodory conditions. H2There exist pf,p g,p h∈L∞J,R, such that ft, y≤pftyfor a.e. t∈Jand each y∈E, gt, y≤pgty,for a.e. t∈Jand each y∈E, ht, y≤phty,for a.e. t∈Jand each y∈E. 3.6 H3For almost each t∈Jand each bounded set B⊂Ewe have lim k→0αfJt,k ×B≤pftαB, lim k→0αgJt,k ×B≤pgtαB, lim k→0αhJt,k ×B ≤phtαB. 3.7 Theorem 3.4. Assume that assumptions H1–H3hold. If TT1 T2 pg L∞ ph L∞ G pf L∞<1,3.8 then the boundary value problem 1.1has at least one solution. Proof. We transform the problem 1.1into a fixed point problem by defining an operator N:CJ, E→CJ, Eas NytPytT 0 Gt, sfs, y sds, 3.9 where PytT1−t T2T 0 gs, y sds t1 T2T 0 hs, y sds, 3.10 and the function Gt, sis given by 3.4. Clearly, the fixed points of the operator Nare solution of the problem 1.1.LetR>0 and consider the set DRy∈CJ, E: y ∞≤R.3.11 Clearly, the subset DRis closed, bounded, and convex. We will show that Nsatisfies the assumptions of Theorem 2.5. The proof will be given in three steps. Boundary Value Problems 7 Step 1. Nis continuous. Let {yn}be a sequence such that yn→yin CJ, E. Then, for each t∈J, Nynt−Nyt≤T1 T2T 0 gs, yns−gs, y s ds T1 T2T 0 hs, yns−hs, y s ds T 0 |Gt, s| fs, yns−fs, y s ds. 3.12 Let ρ>0 be such that  yn ∞≤ρ,  y ∞≤ρ. 3.13 By H2we have  gs, yns−gs, y s ≤2ρpgs:σ1s;σ1∈L1J, R,  hs, yns−hs, y s ≤2ρphs:σ2s;σ2∈L1J, R, |G·,s | fs, yns−fs, y s ≤2ρ|G·,s |pfs:σ3s;σ3∈L1J, R. 3.14 Since f,g, and hare Carath´ eodory functions, the Lebesgue dominated convergence theorem implies that  Nyn−Ny ∞−→ 0asn−→ ∞ .3.15 Step 2. Nmaps DRinto itself. For each y∈DR,byH2and 3.8we have for each t∈J Nyt≤T1 T2T 0 gs, y s ds T1 T2T 0 hs, y s ds T 0 |Gt, s| fs, y s ds ≤RTT1 T2 pg L∞TT1 T2 ph L∞ G pf L∞ <R. 3.16 Step 3. NDRis bounded and equicontinuous. By Step 2, it is obvious that NDR⊂CJ, Eis bounded. 8 Boundary Value Problems For the equicontinuity of NDR.Lett1,t 2∈J,t1<t 2and y∈DR. Then Nyt2−Nyt1    −t2−t1 T2T 0 gs, y sds t2−t1 T2T 0 hs, y sds T 0 Gt2,s −Gt1,s  fs, y sds     ≤t2−t1 T2TR pg L∞ ph L∞ R pf L∞T 0 |Gt2,s −Gt1,s |ds. 3.17 As t1→t2, the right-hand side of the above inequality tends to zero. Now let Vbe a subset of DRsuch that V⊂coNV∪{0}. Vis bounded and equicontinuous, and therefore the function v→vtαVt is continuous on J.ByH3,Lemma 2.6, and the properties of the measure αwe have for each t∈J vt≤αNV t∪{0} ≤αNV t ≤T 0T1−t T2pgsαVs ds T 0t1 T2phsαVs ds T 0 |Gt, s|pfsαVs ds ≤TT1 T2 pg L∞vsTT1 T2 ph L∞vs G pf L∞vs ≤v∞TT1 T2 pg L∞ ph L∞ G pf L∞. 3.18 This means that v∞1−TT1 T2 pg L∞ ph L∞ G pf L∞≤0.3.19 By 3.8it follows that v∞0, that is, vt0 for each t∈J, and then Vtis relatively compact in E. In view of the Ascoli-Arzel` a theorem, Vis relatively compact in DR. Applying now Theorem 2.5 we conclude that Nhas a fixed point which is a solution of the problem 1.1. Boundary Value Problems 9 4. An Example In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional boundary value problem: cDryt2 19 etyt,t∈J:0,1,1<r≤2, y0−y01 0 1 5e5sysds, y1y11 0 1 3e3sysds. 4.1 Set ft, x2 19 etx, t, x∈J×0,∞, gt, x1 5e5tx, t, x∈0,1×0,∞, ht, x1 3e3tx, t, x∈0,1×0,∞. 4.2 Clearly, conditions H1,H2hold with pft2 19 et,p gt1 5e5t,p ht1 3e3t.4.3 From 3.4the function Gis given by Gt, s⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ t−sr−1 Γr−1t 1−sr−1 3Γr−1t 1−sr−2 3Γr−1,0≤s≤t, −1t 1−sr−1 3Γr−1t 1−sr−2 3Γr−1,t≤s≤1. 4.4 From 4.4, we have 1 0 Gt, sds t 0 Gt, sds 1 t Gt, sds tr Γr11t 1−tr 3Γr1−1t 3Γr1 1t 1−tr−1 3Γr−1t 3Γr −1t 1−tr 3Γr1−1t 1−tr−1 3Γr. 4.5