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Existence Results for Nonlinear Boundary Value Problems of Fractional Integrodifferential Equations with Integral Boundary Conditions

Boundary Value Problems20092009:708576

  • Received: 9 December 2008
  • Accepted: 23 January 2009
  • Published:


This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.


  • Fractional Order
  • Fractional Derivative
  • Fractional Differential Equation
  • Caputo Fractional Derivative
  • Integral Boundary Condition

1. Introduction

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see [122] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.

Integrodifferential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero elastic phenomena, visco elasticity, visco elastic panel in super sonic gas flow, fluid dynamics, electrodynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors, and mathematical modeling of a hereditary phenomena. For details, see [2329] and the references therein.

Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [30]. For more details of nonlocal and integral boundary conditions, see [3137] and references therein.

In this paper, we consider the following boundary value problem for a nonlinear fractional integrodifferential equation with integral boundary conditions
where is the Caputo fractional derivative, for

and are real numbers. Here, is a Banach space and denotes the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by

2. Preliminaries

First of all, we recall some basic definitions [15, 18, 20].

Definition 2.1.

For a function the Caputo derivative of fractional order is defined as

where denotes the integer part of the real number

Definition 2.2.

The Riemann-Liouville fractional integral of order is defined as

provided the integral exists.

Definition 2.3.

The Riemann-Liouville fractional derivative of order for a function is defined by

provided the right hand side is pointwise defined on

In passing, we remark that the definition of Riemann-Liouville fractional derivative, which did certainly play an important role in the development of theory of fractional derivatives and integrals, could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. The same applies to the boundary value problems of fractional differential equations. It was Caputo definition of fractional derivative which solved this problem. In fact, the Caputo derivative becomes the conventional th derivative of the function as and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [20].

Lemma 2.4 (see [22]).

For the general solution of the fractional differential equation is given by

where ( ).

In view of Lemma 2.4, it follows that

for some ( ).

Now, we state a known result due to Krasnosel'skiĭ [38] which is needed to prove the existence of at least one solution of (1.1).

Theorem 2.5.

Let be a closed convex and nonempty subset of a Banach space Let be the operators such that (i) whenever , (ii) is compact and continuous, (iii) is a contraction mapping. Then there exists such that

Lemma 2.6.

For any the unique solution of the boundary value problem
is given by
where is the Green's function given by


Using (2.5), for some constants we have
In view of the relations and for we obtain
Applying the boundary conditions for (2.6), we find that
Thus, the unique solution of (2.6) is

where is given by (2.8). This completes the proof.

3. Main Results

Theorem 3.1.

Assume that is jointly continuous and maps bounded subsets of into relatively compact subsets of is continuous with and are continuous functions. Further, there exist positive constants such that

(A1) for all

(A2) with

Then the boundary value problem (1.1) has a unique solution provided


Define by
Setting (by the assumption on ) and Choosing
we show that where For we have
Now, for and for each we obtain

which depends only on the parameters involved in the problem. As therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

Theorem 3.2.

Assume that (A1)-(A2) hold with where and

Then the boundary value problem (1.1) has at least one solution on


Let us fix
and consider We define the operators and on as
For we find that
Thus, It follows from the assumption (A1), (A2) that is a contraction mapping for
Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator In view of (A1), we define and consequently we have

which is independent of So is relatively compact on Hence, By Arzela Ascoli Theorem, is compact on Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the boundary value problem (1.1) has at least one solution on

Example 3.3.

Consider the following boundary value problem:
Here, As therefore, (A1) and (A2) are satisfied with Further,

Thus, by Theorem 3.1, the boundary value problem (3.15) has a unique solution on



The authors are grateful to the anonymous referee for his/her valuable suggestions that led to the improvement of the original manuscript. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

Authors’ Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain


  1. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprintGoogle Scholar
  2. Ahmad B, Sivasundaram S: Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. to appear in Dynamic Systems and ApplicationsGoogle Scholar
  3. Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3692–3705. 10.1016/ ArticleGoogle Scholar
  4. Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495–505. 10.1016/j.jmaa.2005.02.052MATHMathSciNetView ArticleGoogle Scholar
  5. Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. preprintGoogle Scholar
  6. Benchohra M, Hamani S, Nieto JJ, Slimani BA: Existence results for differential inclusions with fractional order and impulses. preprintGoogle Scholar
  7. Bonilla B, Rivero M, Rodríguez-Germá L, Trujillo JJ: Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation 2007, 187(1):79–88. 10.1016/j.amc.2006.08.105MATHMathSciNetView ArticleGoogle Scholar
  8. 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.2008.03.014MATHMathSciNetView ArticleGoogle Scholar
  9. Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics 2008, 220(1–2):215–225. 10.1016/ ArticleGoogle Scholar
  10. Daftardar-Gejji V: Positive solutions of a system of non-autonomous fractional differential equations. Journal of Mathematical Analysis and Applications 2005, 302(1):56–64. 10.1016/j.jmaa.2004.08.007MATHMathSciNetView ArticleGoogle Scholar
  11. 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.2008.04.065MATHMathSciNetView ArticleGoogle Scholar
  12. El-Shahed M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstract and Applied Analysis 2007, 2007:-8.Google Scholar
  13. Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations. Journal of Mathematical Analysis and Applications 2008, 345(2):871–879. 10.1016/j.jmaa.2008.05.017MATHMathSciNetView ArticleGoogle Scholar
  14. Jafari H, Seifi S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 2009, 14(5):2006–2012. 10.1016/j.cnsns.2008.05.008MATHMathSciNetView ArticleGoogle Scholar
  15. 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
  16. Ladaci S, Loiseau JJ, Charef A: Fractional order adaptive high-gain controllers for a class of linear systems. Communications in Nonlinear Science and Numerical Simulation 2008, 13(4):707–714. 10.1016/j.cnsns.2006.06.009MATHMathSciNetView ArticleGoogle Scholar
  17. Lazarević MP: Finite time stability analysis of fractional control of robotic time-delay systems. Mechanics Research Communications 2006, 33(2):269–279. 10.1016/j.mechrescom.2005.08.010MATHMathSciNetView ArticleGoogle Scholar
  18. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
  19. Rida SZ, El-Sherbiny HM, Arafa AAM: On the solution of the fractional nonlinear Schrödinger equation. Physics Letters A 2008, 372(5):553–558. 10.1016/j.physleta.2007.06.071MATHMathSciNetView ArticleGoogle Scholar
  20. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
  21. Varlamov V: Differential and integral relations involving fractional derivatives of Airy functions and applications. Journal of Mathematical Analysis and Applications 2008, 348(1):101–115. 10.1016/j.jmaa.2008.06.052MATHMathSciNetView ArticleGoogle Scholar
  22. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, 2006(36):1–12.Google Scholar
  23. Ahmad B, Sivasundaram S: Some existence results for fractional integrodifferential equations with nonlinear conditions. Communications in Applied Analysis 2008, 12: 107–112.MATHMathSciNetGoogle Scholar
  24. Ahmad B, Alghamdi BS: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Computer Physics Communications 2008, 179(6):409–416. 10.1016/j.cpc.2008.04.008MATHMathSciNetView ArticleGoogle Scholar
  25. Ahmad B: On the existence of -periodic solutions for Duffing type integro-differential equations with -Laplacian. Lobachevskii Journal of Mathematics 2008, 29(1):1–4.MATHMathSciNetView ArticleGoogle Scholar
  26. Chang YK, Nieto JJ: Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. to appear in Numerical Functional Analysis and OptimizationGoogle Scholar
  27. Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(6):2248–2260. 10.1016/ ArticleGoogle Scholar
  28. Mesloub S: On a mixed nonlinear one point boundary value problem for an integrodifferential equation. Boundary Value Problems 2008, 2008:-8.Google Scholar
  29. Nieto JJ, Rodríguez-López R: New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications 2007, 328(2):1343–1368. 10.1016/j.jmaa.2006.06.029MATHMathSciNetView ArticleGoogle Scholar
  30. Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Real World Applications 2008, 9(4):1727–1740. 10.1016/j.nonrwa.2007.05.005MATHMathSciNetView ArticleGoogle Scholar
  31. Ahmad B, Alsaedi A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Analysis: Real World Applications 2009, 10(1):358–367. 10.1016/j.nonrwa.2007.09.004MATHMathSciNetView ArticleGoogle Scholar
  32. Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. Rocky Mountain Journal of Mathematics. In pressGoogle Scholar
  33. Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):364–371. 10.1016/ ArticleGoogle Scholar
  34. Chang Y-K, Nieto JJ, Li W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. Journal of Optimization Theory and Applications 2009, 140(3):431–442. 10.1007/s10957-008-9468-1MATHMathSciNetView ArticleGoogle Scholar
  35. Chang YK, Nieto JJ, Li WS: Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. to appear in Journal of Optimization Theory and ApplicationsGoogle Scholar
  36. Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Nonlinear Analysis: Theory, Methods & Applications. In pressGoogle Scholar
  37. Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(1):216–225. 10.1016/ ArticleGoogle Scholar
  38. Krasnosel'skiĭ MA: Two remarks on the method of successive approximations. Uspekhi Matematicheskikh Nauk 1955, 10(1(63)):123–127.Google Scholar


© B. Ahmad and J.J. Nieto 2009

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