Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Abstract

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: $\left({}^C{D}_{\mathrm{0}+}^{q}u\right)\left(t\right)=f\left(t,u\left(t\right)\right)$, $t\in \left(\mathrm{0,1}\right)$, $u\left(\mathrm{0}\right)=u^{\mathrm{\prime }\mathrm{\prime }}\left(\mathrm{0}\right)=\mathrm{0},\hspace{0.17em}\hspace{0.17em}\left({}^C{D}_{\mathrm{0}+}^{{\sigma }_{\mathrm{1}}}u\right)\left(\mathrm{1}\right)=\lambda \left(I_{\mathrm{0}+}^{{\sigma }_{\mathrm{2}}}u\right)\left(\mathrm{1}\right)$, where , , ${\sigma }_{\mathrm{2}}>\mathrm{0}$, and $\lambda \ne \mathrm{\Gamma }\left(\mathrm{2}+{\sigma }_{\mathrm{2}}\right)/\mathrm{\Gamma }\left(\mathrm{2}-{\sigma }_{\mathrm{1}}\right)$. The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.