Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition
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Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition Rengui Li* *
Correspondence: [email protected] Department of Mathematics, Jining University, Jining, Shandong 273155, P.R. China
Abstract In this paper, we investigate a boundary value problem for singular fractional differential equations with a fractional derivative condition. The existence and uniqueness of solutions are obtained by means of the fixed point theorem. Some examples are presented to illustrate our main results. Keywords: boundary value problem; singular fractional differential equation; Caputo fractional derivative; fixed point theory
1 Introduction Differential equations of fractional order have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics; see [, ]. On the other hand, fractional differential equations also serve as an excellent tool for the description of memory and hereditary properties of various materials and processes. With these advantages, the model of fractional order become more and more practical and realistic than the classical of integer order, such effects in the latter are not taken into account. As a result, the subject of fractional differential equations is gaining much attention and importance. Recently, much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problem of fractional differential equations with nonlocal boundary conditions by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, the upper and lower solution method, etc.); see [–]. In [], Agarwal et al. investigated the existence of solutions for the singular fractional boundary value problems ⎧ ⎨Dα u(t) + f (t, u(t), Dμ u(t)) = ,
< t < ,
⎩u() = u() = , where < α < , < μ ≤ α – are real numbers, Dα is the standard Riemann-Liouville fractional derivative, f satisfies the Caratheodory conditions on [, ] × (, ∞) × R, f is positive, and f (t, x, y) is singular at x = . © 2014 Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.
Li Advances in Difference Equations 2014, 2014:292 http://www.advancesindifferenceequations.com/content/2014/1/292
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In [], Yan et al. studied the existence and uniqueness of solutions for a class of fractional differential equations with integral boundary conditions ⎧ C α C β ⎪ ⎪ ⎨ D+ x(t) + f (t, x(t), D+ x(t)) = , t ∈ [, ], x() + x () = y(x), x(t) dt = m, ⎪ ⎪ ⎩ x () = x () = · · · = x(n–) () = , β
where C Dα+ , C D+ are the Caputo fractional derivatives, f : [, ] × R × R → R is a continuous function, y : [, ] → R is a continuous function, and m ∈ R, n
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