Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions

Zhang

Description

In this article, by employing a fixed point theorem in cones, we investigate the existence of a positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions. We also obtain some relations between the solution and Green’s function. MSC: 26A33, 34B15, 34B16, 34G20. In this article, by employing a fixed point theorem in cones, we investigate the existence of a positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions. We also obtain some relations between the solution and Green’s function. MSC: 26A33, 34B15, 34B16, 34G20.

Sujets

• Fractional calculus
• Coné

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

Zhang Boundary Value Problems 2012, 2012 :123 http://www.boundaryvalueproblems.com/content/2012/1/123 R E S E A R C H Open Access Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions Xingqiu Zhang * * Correspondence: [e-mail] Abstract School of Mathematics and In this article, by employing fi d point theorem in cones, we investigate the Statistics, Huazhong University of a xe Science and Technology, Wuhan, existence of a positive solution for a class of singular semipositone fractional Hubei 430074, P.R. China differential equations with integral boundary conditions. We also obtain some School of Mathematics, Liaocheng University, Liaocheng, Shandong relations between the solution and Green’s function. 252059, P.R. China MSC: 26A33; 34B15; 34B16; 34G20 Keywords: fractional differential equations; integral boundary value problem; positive solution; semipositone; cone 1 Introduction In this article, we consider the existence of a positive solution for the following singular semipositone fractional differential equations: ⎧ ⎨ D  α + u ( t ) + f ( t , u ( t )) = ,  < t < , s ) d s , () ⎩ u () = u  () = u  () = , u () = λ   η u ( where f ∈ C [(, ) × [, + ∞ ), (– ∞ , + ∞ )],  < α ≤ ,  < η ≤ ,  ≤ λ α η α < , D  α + is the stan- dard Riemann-Liouville derivative, f ( t , u ) may be singular at t =  and/or t = . Since the nonlinearity f ( t , x ) may change sign, the problem studied in this paper is called the semi- positone problem in the literature which arises naturally in chemical reactor theory. Up to now, much attention has been attached to the existence of positive solutions for semi- positone differential equations and the system of differential equations; see [ –] and references therein to name a few. Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat con- duction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics. Moreover, boundary value problems with integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, multi- point, and nonlocal boundary value problems as special cases, which have received much attention from many authors. For boundary value problems with integral boundary con- ditions and comments on their importance, we refer the reader to the papers by Gallardo [], Karakostas and Tsamatos [ ], Lomtatidze and Malaguti [ ], and the references therein. © 2012 Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.