Study Boundary Problem with Integral condition for Fractional Differential Equations

In last many years ago there was a great interest in studying the existence of positive solutions for fractional differential equations. Many authors have considered the existence of positive solutions of non-linear differential equations of non-integer order with integral boundary value conditions using fixed point theorems. G.wang etal(2012)in vest gated the following fractional differential equation ^cD^ W(t)+f(t,W(t))=0,0 is a positive number (0 < < 2),^CD^is the standard Caputo fractional derivative obtained his results by means of Guo-krosnosel'skii theorem in a cone also A.Cabada etat (2013) established the following non-linear fractional differential equation with integral boundary value conditions D^ W(t)+f(t,W(t))=0 ,00 , , D^is Riemann Liovuville standard fractional derivative and f is a continuous function the results was based on Guo-krasnosel'skii fixed point theorem in a cone . This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations: ^cD^ h(t)+k(t,h(t))=0 ,t(0,1) h(0)=h^' (0)=h^''' (0)=0 ,h(1)=_0^1h(n)dn where 3< 4 , is a positive number , 3 ,^CD^ denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem.