Mathematical Researches

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In this paper we investigate a kind of boundary value problem involving a fractional differential equation.  We study the existence of positive solutions of the problem that fractional derivative is the Reimann-Liouville fractional derivative. At first the green function is computed then it is proved that the green function is positive. We present necessary and sufficient conditions for existence of positive solution by calculating supremum of integral of green function over the solution interval and by use of some expansions of contraction mapping that are presented recently. For this purpose, at first the existence and uniqueness of solution for the problem, by use of existence of lower solution for the problem and expansion of contraction mapping on ordered space, is proved. Then by use of another expansion of contraction mapping on ordered spaces, the existence and uniqueness of positive solution is proved. Also, an example is presented to illustrate the proven results. 

Keywords: Boundary value problem, Fractional differential equations, Reimman Liouville fractional derivatve, Fixed Point theorem

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