Abstract
The aims and objectives of this manuscript are concerned with the investigation of some appropriate conditions to establish existence theory of solutions to a class of nonlinear four-point boundary value problem (BVP) corresponding to fractional order differential equations (FODEs) provided as
{c .. y( t) =. t, y( t), c .. - 1 y( t) , 1 <.. = 2, t. = [ 0, 1],
y(0) =.. y(..), y(1) =.. y(..),..,..,..,... (0, 1),
where cD is Caputo's fractional derivative of order q and F(JxRxR,R) may be nonlinear. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Further, we establish some adequate conditions for the Ulam-Hyers stability and generalized Ulam-Hyers stability for the solutions to the considered BVP of nonlinear FODEs. We include a proper problem to illustrate our established results.