Existence of multiple positive solutions to the Caputo-type nonlinear fractional differential equation with integral boundary value conditions

. In this article, the existence criteria of at least one or at least three positive solutions to the Caputo-type nonlinear fractional differential equation with integral boundary value conditions has been established. The method applied in this study is formulated by the well-known Guo-Krasnoselskii’s fixed point theorem and Leggett-Williams fixed point theorem. First, the Green’s function for corresponding linear fractional differential equation of the main nonlinear fractional differential equation under same boundary value conditions has been constructed. Next, several essential properties of that Green’s function have been proved. Finally, in cone spaces some new existence and multiplicity results for the Caputo-type nonlinear fractional differential equation with integral boundary value conditions are obtained. To support the analytic proof appropriate illustrative examples has also been discussed.