Study boundary problem with integral condition for fractional differential equations

Abstract EN

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〖^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 conditionsD^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.

This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations: 〖^c〗D^β h(t)+k(t, h(t))=0, t∈(0, 1)h(0)=h^' (0)=h^''' (0)=0, h(1)=δ∫_0^1▒h(n)dnwhere 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 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〖^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 conditionsD^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.

This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations: 〖^c〗D^β h(t)+k(t, h(t))=0, t∈(0, 1)h(0)=h^' (0)=h^''' (0)=0, h(1)=δ∫_0^1▒h(n)dnwhere 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.