Some results for a four-point boundary value problems for coupled system involving Caputo derivative
Authors :
M. Houas, a,* M. Benbachira and Z. Dahmanib
Author Address :
aFaculty of Sciences and Technology Khemis Miliana University, Ain Defla, Algeria.
b LPAM, Faculty SEI, UMAB Mostaganem, Algeria.
*Corresponding author.
Abstract :
In this paper, we prove the existence and uniqueness of solutions for a system for fractional differential equations with four point boundary conditions. The results are obtained using Banach contraction principle and Krasnoselkii’s fixed point theorem
$left{
egin{tabular}
[c]{l}%
$D^{alpha}xleft( t ight) +fleft( t,yleft( t ight) ,D^{delta
}yleft( t ight) ight) =0,tin J,$
$D^{eta}yleft( t ight) +gleft( t,xleft( t ight) ,D^{sigma
}xleft( t ight) ight) =0,tin J,$
$xleft( 0 ight) =yleft( 0 ight) =0,xleft( 1 ight) -lambda
_{1}xleft( eta ight) =0,yleft( 1 ight) -lambda_{1}yleft(
eta ight) =0,$
$x^{primeprime}left( 0 ight) =y^{primeprime}left( 0 ight)
=0,x^{primeprime}left( 1 ight) -lambda_{2}x^{primeprime}left(
xi ight) =0,y^{primeprime}left( 1 ight) -lambda_{2}y^{^{prime
prime}}left( xi ight) =0,$%
end{tabular}
ight. $igskip
where $3<alpha,etaleq4,alpha-2<sigmaleqalpha-1,eta-2<deltaleq
eta-1,0<xi,eta<1,$ and $D^{alpha},D^{eta},D^{delta}$ and $D^{sigma},$
are the Caputo fractional derivatives, $J=left[ 0,1 ight] ,lambda
_{1},lambda_{2}$ are real constants with $lambda_{1}eta eq1,lambda_{2}%
xi eq1$ and $f,g$ continuous functions on $left[ 0,1 ight] imes%
%TCIMACRO{U{211d} }%
%BeginExpansion
mathbb{R}
%EndExpansion
^{2}.$
Keywords :
Caputo derivative; Boundary Value Problem; fixed point theorem.
DOI :
Article Info :
Received : February 12, 2014; Accepted : October 15, 2014.
