Abstract
In this paper, we study existence and approximation of solutions to some three- point boundary value problems for fractional differential equations of the type c D q 0+u(t) + f(t,u(t)) = 0, t 2 (0,1),1 < q < 2 u 0 (0) = 0, u(1) = �u(�), where 0 < �, � 2 (0,1) and c D q is the fractional derivative in the sense of Caputo. For the existence of solution, we develop the method of upper and lower solutions and for the approximation of solutions, we develop the generalized quasilinearization technique (GQT). The GQT generates a monotone sequence of solutions of linear problems that converges monotonically and quadratically to solution of the original nonlinear problem.
Highlights
The study of fractional differential equations is of fundamental concern due to its important applications to real world problems
Many problems in applied sciences such as engineering and physics can be modeled by differential equations of fractional order [1, 2, 3]
It has been observed that the models with fractional differential equations provide more realistic and accurate results compared to the analogous models with integer order derivatives, see, [4, 5]
Summary
The study of fractional differential equations is of fundamental concern due to its important applications to real world problems. Existence theory for solutions to boundary value problems for fractional differential equations have attracted the attention of many researcher quite recently, see for example [6, 7, 8, 9, 10, 11, 12] and the references therein. The method of upper and lower solutions for the existence of solution is less developed and hardly few results can be found in the literature dealing with the upper and lower solutions method to boundary value problems for fractional differential equations [13, 14, 15, 16, 17]. Boundary value problems; Fractional differential equations; Three-point boundary conditions; Upper and lower solutions; Generalized quasilinearization.
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