We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type , , . , , , , where, , , , the boundary parameters and is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results. 1. Introduction Fractional calculus goes back to the beginning of the theory of differential calculus and is developing since the 17th century through the pioneering work of Leibniz, Euler, Abel, Liouville, Riemann, Letnikov, Weyl, and many others. Fractional calculus is the generalization of ordinary integration and differentiation to an arbitrary order. For almost 300 years, it was seen as interesting but abstract mathematical concept. Nevertheless the applications of fractional calculus just emerged in the last few decades in various areas of physics, chemistry, engineering, biosciences, electrochemistry, and diffusion processes. For details, we refer the readers to [1–5]. The existence and uniqueness of solutions for fractional differential equations is well studied in [6–10] and references therein. It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of initial value problems for fractional differential equations. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quiet recently, and many aspects of the theory need to be further investigated. There are some recent development dealing with the existence and multiplicity of positive solutions to nonlinear boundary value problems for fractional differential equations, see, for example, [11–18] and the reference therein. However, few results can be found in the literature concerning the existence of positive solutions to nonlinear three-point boundary value problems for fractional differential equations. For example, Li and coauthors [19] obtained sufficient conditions for the existence and multiplicity results to the following three point fractional boundary value problem where is standard Riemann-Liouville fractional order derivative. Bai [20] studied the existence and uniqueness of positive solutions to the following three-point boundary value problem for fractional differential equations where , , is standard
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