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An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces


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.

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. Fractionaldifferential 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. [1113], 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 , 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].

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


where is the Caputo fractional derivative, , , and are given functions satisfying some assumptions that will be specified later, and is 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 Mönch 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], Mönch [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 the Banach space of continuous functions , with the usual supremum norm


Let be the Banach space of measurable functions which are Bochner integrable, equipped with the norm


Let be the Banachspace of measurable functions which are bounded, equipped with the norm


Let be the space of functions , whose first derivative is absolutely continuous.

Moreover, for a given set of functions let us denote by


Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

Definition 2.1 (see [27]).

Let be a Banach space and the bounded subsets of . The Kuratowski measure of noncompactness is the map defined by



The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).

(a) is compact ( is relatively compact).






Here and denote the closure and the convex hull of the bounded set , 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 of order is defined by


where is the gamma function. When , we write where


for , and as .

Here is the delta function.

Definition 2.3 (see [17]).

For a function given on the interval , the Caputo fractional-order derivative of , of order is defined by


Here and denotes the integer part of .

Definition 2.4.

A map is said to be Carathéodory if

(i) is measurable for each

(ii) is continuous for almost each

For our purpose we will only need the following fixed point theorem and the important Lemma.

Theorem 2.5 (see [31, 33]).

Let be a bounded, closed and convex subset of a Banach space such that , and let be a continuous mapping of into itself. If the implication


holds for every subset of , then has a fixed point.

Lemma 2.6 (see [32]).

Let be a bounded, closed, and convex subset of the Banach space , G a continuous function on and a function satisfies the Carathéodory conditions, and there exists such that for each and each bounded set one has


If is an equicontinuous subset of , then


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 is said to be a solution of (1.1) if it satisfies (1.1).

Let be continuous functions and consider the linear boundary value problem


Lemma 3.2 (see [11]).

Let and let be continuous. A function is a solution of the fractional integral equation




if and only if is a solution of the fractional boundary value problem (3.1).

Remark 3.3.

It is clear that the function is continuous on , and hence is bounded. Let


For the forthcoming analysis, we introduce the following assumptions

(H1)The functions satisfy the Carathéodory conditions.

(H2)There exist , such that


(H3)For almost each and each bounded set we have


Theorem 3.4.

Assume that assumptions hold. If


then the boundary value problem (1.1) has at least one solution.


We transform the problem (1.1) into a fixed point problem by defining an operator as




and the function is given by (3.4). Clearly, the fixed points of the operator are solution of the problem (1.1). Let and consider the set


Clearly, the subset is closed, bounded, and convex. We will show that satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.

Step 1.

is continuous.

Let be a sequence such that in . Then, for each ,


Let be such that


By (H2) we have


Since and are Carathéodory functions, the Lebesgue dominated convergence theorem implies that


Step 2.

maps into itself.

For each , by and (3.8) we have for each


Step 3.

is bounded and equicontinuous.

By Step 2, it is obvious that is bounded.

For the equicontinuity of . Let , and . Then


As , the right-hand side of the above inequality tends to zero.

Now let be a subset of such that .

is bounded and equicontinuous, and therefore the function is continuous on . By (H3), Lemma 2.6, and the properties of the measure we have for each


This means that


By (3.8) it follows that , that is, for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 2.5 we conclude that has a fixed point which is a solution of the problem (1.1).

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:




Clearly, conditions (H1),(H2) hold with


From (3.4) the function is given by


From (4.4), we have


A simple computation gives


Condition (3.8) is satisfied with . Indeed


which is satisfied for each . Then by Theorem 3.4 the problem (4.1) has a solution on .


  1. Hilfer R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.

    MATH  Google Scholar 

  2. Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophysical Journal 1995, 68(1):46–53. 10.1016/S0006-3495(95)80157-8

    Article  Google Scholar 

  3. Metzler R, Schick W, Kilian H-G, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. The Journal of Chemical Physics 1995, 103(16):7180–7186. 10.1063/1.470346

    Article  Google Scholar 

  4. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.

    Google Scholar 

  5. Gaul L, Klein P, Kemple S: Damping description involving fractional operators. Mechanical Systems and Signal Processing 1991, 5(2):81–88. 10.1016/0888-3270(91)90016-X

    Article  Google Scholar 

  6. Agarwal RP, Benchohra M, Hamani S: Boundary value problems for differential inclusions with fractional order. Advanced Studies in Contemporary Mathematics 2008, 16(2):181–196.

    MATH  MathSciNet  Google Scholar 

  7. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.

    Google Scholar 

  8. Ahmad B, Otero-Espinar V: Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. Boundary Value Problems 2009, 2009:-11.

    Google Scholar 

  9. Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Applicable Analysis 2006, 85(12):1459–1470. 10.1080/00036810601066350

    Article  MATH  MathSciNet  Google Scholar 

  10. Belmekki M, Nieto JJ, Rodriguez-Lopez RR: Existence of periodic solution for a nonlinear fractional differential equation. Boundary Value Problems. in press

  11. Benchohra M, Graef JR, Hamani S: Existence results for boundary value problems with non-linear fractional differential equations. Applicable Analysis 2008, 87(7):851–863. 10.1080/00036810802307579

    Article  MATH  MathSciNet  Google Scholar 

  12. Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order. Surveys in Mathematics and Its Applications 2008, 3: 1–12.

    MATH  MathSciNet  Google Scholar 

  13. Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008, 338(2):1340–1350. 10.1016/j.jmaa.2007.06.021

    Article  MATH  MathSciNet  Google Scholar 

  14. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3–4):605–609. 10.1016/j.mcm.2008.03.014

    Article  MATH  MathSciNet  Google Scholar 

  15. Daftardar-Gejji V, Bhalekar S: Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications 2008, 345(2):754–765. 10.1016/j.jmaa.2008.04.065

    Article  MATH  MathSciNet  Google Scholar 

  16. Figueiredo Camargo R, Chiacchio AO, Capelas de Oliveira E: Differentiation to fractional orders and the fractional telegraph equation. Journal of Mathematical Physics 2008, 49(3):-12.

    Article  MathSciNet  Google Scholar 

  17. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.

    Google Scholar 

  18. Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta 2006, 45(5):765–771. 10.1007/s00397-005-0043-5

    Article  Google Scholar 

  19. Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis for Theory and Applications 2002, 5(4):367–386.

    MATH  MathSciNet  Google Scholar 

  20. Blayneh KW: Analysis of age-structured host-parasitoid model. Far East Journal of Dynamical Systems 2002, 4(2):125–145.

    MATH  MathSciNet  Google Scholar 

  21. Adomian G, Adomian GE: Cellular systems and aging models. Computers & Mathematics with Applications 1985, 11(1–3):283–291.

    Article  MATH  MathSciNet  Google Scholar 

  22. Arara A, Benchohra M: Fuzzy solutions for boundary value problems with integral boundary conditions. Acta Mathematica Universitatis Comenianae 2006, 75(1):119–126.

    MATH  MathSciNet  Google Scholar 

  23. Benchohra M, Hamani S, Henderson J: Functional differential inclusions with integral boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations 2007, 2007(15):1–13.

    Article  MathSciNet  Google Scholar 

  24. Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. to appear in The Rocky Mountain Journal of Mathematics

  25. Infante G: Eigenvalues and positive solutions of ODEs involving integral boundary conditions. Discrete and Continuous Dynamical Systems 2005, 436–442.

    Google Scholar 

  26. Peciulyte S, Stikoniene O, Stikonas A: Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition. Mathematical Modelling and Analysis 2005, 10(4):377–392.

    MATH  MathSciNet  Google Scholar 

  27. Banaś J, Goebel K: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics. Volume 60. Marcel Dekker, New York, NY, USA; 1980:vi+97.

    Google Scholar 

  28. Banaś J, Sadarangani K: On some measures of noncompactness in the space of continuous functions. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(2):377–383. 10.1016/

    Article  MATH  MathSciNet  Google Scholar 

  29. Guo D, Lakshmikantham V, Liu X: Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications. Volume 373. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341.

    Book  Google Scholar 

  30. Lakshmikantham V, Leela S: Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications. Volume 2. Pergamon Press, Oxford, UK; 1981:x+258.

    Google Scholar 

  31. Mönch H: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 1980, 4(5):985–999. 10.1016/0362-546X(80)90010-3

    Article  MATH  Google Scholar 

  32. Szufla S: On the application of measure of noncompactness to existence theorems. Rendiconti del Seminario Matematico della Università di Padova 1986, 75: 1–14.

    MATH  MathSciNet  Google Scholar 

  33. Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.

    Book  Google Scholar 

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The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

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Correspondence to Mouffak Benchohra.

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Benchohra, M., Cabada, A. & Seba, D. An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces. Bound Value Probl 2009, 628916 (2009).

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