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 OteroEspinar [8], Belarbi et al. [9], Belmekki et al [10], Benchohra et al. [11–13], Chang and Nieto [14], DaftardarGejji 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 RiemannLiouville 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
Properties
The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).
(a) is compact ( is relatively compact).
(b).
(c).
(d)
(e)
(f).
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 fractionalorder 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.
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
with
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.
Proof.
We transform the problem (1.1) into a fixed point problem by defining an operator as
where
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 righthand 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 AscoliArzelà 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:
Set
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 .
Acknowledgments
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 MTM200761724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.
References

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

Glockle, WG, Nonnenmacher, TF: A fractional calculus approach to selfsimilar protein dynamics. Biophysical Journal. 68(1), 46–53 (1995). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Metzler, R, Schick, W, Kilian, HG, Nonnenmacher, TF: Relaxation in filled polymers: a fractional calculus approach. The Journal of Chemical Physics. 103(16), 7180–7186 (1995). Publisher Full Text

Podlubny, I: Fractional Differential Equations, Mathematics in Science and Engineering,p. xxiv+340. Academic Press, San Diego, Calif, USA (1999)

Gaul, L, Klein, P, Kemple, S: Damping description involving fractional operators. Mechanical Systems and Signal Processing. 5(2), 81–88 (1991). Publisher Full Text

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

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

Ahmad, B, OteroEspinar, V: Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions. Boundary Value Problems. 2009, (2009)

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

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

Benchohra, M, Graef, JR, Hamani, S: Existence results for boundary value problems with nonlinear fractional differential equations. Applicable Analysis. 87(7), 851–863 (2008). Publisher Full Text

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

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. 338(2), 1340–1350 (2008). Publisher Full Text

Chang, YK, Nieto, JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling. 49(34), 605–609 (2009). Publisher Full Text

DaftardarGejji, V, Bhalekar, S: Boundary value problems for multiterm fractional differential equations. Journal of Mathematical Analysis and Applications. 345(2), 754–765 (2008). Publisher Full Text

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

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies,p. xvi+523. Elsevier Science B.V., Amsterdam, The Netherlands (2006)

Heymans, N, Podlubny, I: Physical interpretation of initial conditions for fractional differential equations with RiemannLiouville fractional derivatives. Rheologica Acta. 45(5), 765–771 (2006). Publisher Full Text

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

Blayneh, KW: Analysis of agestructured hostparasitoid model. Far East Journal of Dynamical Systems. 4(2), 125–145 (2002)

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

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

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

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

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

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

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

Banaś, J, Sadarangani, K: On some measures of noncompactness in the space of continuous functions. Nonlinear Analysis: Theory, Methods & Applications. 68(2), 377–383 (2008). PubMed Abstract  Publisher Full Text

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

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

Mönch, H: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications. 4(5), 985–999 (1980). PubMed Abstract  Publisher Full Text

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

Agarwal, RP, Meehan, M, O'Regan, D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics,p. x+170. Cambridge University Press, Cambridge, UK (2001)