New results for boundary value problems of nonlinear fractional differential equations with nonseparated boundary conditions (Q763695)

The author study the nonlinear fractional differential equations \[ ^{c}D^{q}x(t)=f(t,x(t)), t\in [0,T], ~ T>0, ~~ 1<q\leq 2, \tag{1} \] \[ x(0)=\lambda _{1}x(T)+\mu _{1},x^{\prime }(0)=\lambda_{2}x^{\prime }(T)+\mu _{2}, \lambda _{1}\neq 1,\lambda _{2}\neq 1, \] where \(^{c}D^{q}\) is the Caputo fractional derivative of order \(q,\) and \(f:[0,T]\times \mathbb R \to \mathbb R\) and \(\lambda _{1},\lambda _{2},\mu _{1},\mu _{2}\in \mathbb R. \) It was proved using the Leray-Schauder criterion under certain assumption on the function \(f\) that the boundary value problem (1) has at least one solution. Furthermore, existence and uniqueness results were also obtained in a Banach space setting under certain assumptions on \(f.\) The results obtained generalized some known results in [\textit{R. P. Agarwal, M. Benchohra} and \textit{B. A. Slimani}, Mem. Differ. Equ. Math. Phys. 44, 1--21 (2008; Zbl 1178.26006)] and \textit{B. Ahmad} and \textit{J. J. Nieto} [Topol. Methods Nonlinear Anal. 35, No. 2, 295--304 (2010; Zbl 1245.34008)]. Examples are given to illustrate some consequences of the results obtained.