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.
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 [1–22] 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 [23–29] 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 . For more details of nonlocal and integral boundary conditions, see [31–37] 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
For a function the Caputo derivative of fractional order is defined as
where denotes the integer part of the real number
The Riemann-Liouville fractional integral of order is defined as
provided the integral exists.
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 .
Lemma 2.4 (see ).
For the general solution of the fractional differential equation is given by
In view of Lemma 2.4, it follows that
for some ().
Now, we state a known result due to Krasnosel'skiĭ  which is needed to prove the existence of at least one solution of (1.1).
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
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
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
Then the boundary value problem (1.1) has a unique solution provided
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.
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
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.
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 Applications
Araya, D, Lizama, C: Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications. 69(11), 3692–3705 (2008). PubMed Abstract | Publisher Full Text
Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications. 311(2), 495–505 (2005). Publisher Full Text
Bonilla, B, Rivero, M, Rodríguez-Germá, L, Trujillo, JJ: Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation. 187(1), 79–88 (2007). Publisher Full Text
Chang, Y-K, Nieto, JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling. 49(3-4), 605–609 (2009). Publisher Full Text
Gafiychuk, V, Datsko, B, Meleshko, V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics. 220(1-2), 215–225 (2008). Publisher Full Text
Daftardar-Gejji, V: Positive solutions of a system of non-autonomous fractional differential equations. Journal of Mathematical Analysis and Applications. 302(1), 56–64 (2005). Publisher Full Text
Daftardar-Gejji, V, Bhalekar, S: Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications. 345(2), 754–765 (2008). Publisher Full Text
Ibrahim, RW, Darus, M: Subordination and superordination for univalent solutions for fractional differential equations. Journal of Mathematical Analysis and Applications. 345(2), 871–879 (2008). Publisher Full Text
Jafari, H, Seifi, S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation. 14(5), 2006–2012 (2009). Publisher Full Text
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies,p. xvi+523. Elsevier Science B.V., Amsterdam, The Netherlands (2006)
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. 13(4), 707–714 (2008). Publisher Full Text
Lazarević, MP: Finite time stability analysis of fractional control of robotic time-delay systems. Mechanics Research Communications. 33(2), 269–279 (2006). Publisher Full Text
Rida, SZ, El-Sherbiny, HM, Arafa, AAM: On the solution of the fractional nonlinear Schrödinger equation. Physics Letters A. 372(5), 553–558 (2008). Publisher Full Text
Varlamov, V: Differential and integral relations involving fractional derivatives of Airy functions and applications. Journal of Mathematical Analysis and Applications. 348(1), 101–115 (2008). Publisher Full Text
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. 179(6), 409–416 (2008). Publisher Full Text
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 Optimization
Luo, Z, Nieto, JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications. 70(6), 2248–2260 (2009). PubMed Abstract | Publisher Full Text
Nieto, JJ, Rodríguez-López, R: New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications. 328(2), 1343–1368 (2007). Publisher Full Text
Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Real World Applications. 9(4), 1727–1740 (2008). Publisher Full Text
Ahmad, B, Alsaedi, A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Analysis: Real World Applications. 10(1), 358–367 (2009). Publisher Full Text
Chang, Y-K, Nieto, JJ, Li, W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. Journal of Optimization Theory and Applications. 140(3), 431–442 (2009). Publisher Full Text
Yang, Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 68(1), 216–225 (2008). PubMed Abstract | Publisher Full Text