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.<br /> G.wang etal(2012)in vest gated the following fractional differential equation <br /> 〖^c〗D^α W(t)+f(t,W(t))=0,0 λ is a positive number (0 < λ < 2),〖^C〗D^αis 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<br /> D^α W(t)+f(t,W(t))=0 ,00 ,λ≠α ,〗 D^αis 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 . <br /> This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations:<br /> 〖^c〗D^β h(t)+k(t,h(t))=0 ,t∈(0,1)<br /> h(0)=h^' (0)=h^''' (0)=0 ,h(1)=δ∫_0^1▒h(n)dn<br /> where 3< β≤4 ,δ is a positive number , δ≠3 ,〖^C〗D^β denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem.