Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval

摘要

We consider the existence of solutions for the following Hadamard-type fractional differential equations: $$ extstyleegin{cases} {}^{H}D^{lpha }u(t)+q(t)f(t,u(t), {}^{H}D^{eta _{1}}u(t),{}^{H}D^{ eta _{2}}u(t))=0,quad 1 t +infty , u(1)=0, {}^{H}D^{lpha -2}u(1)=int ^{+infty }_{1}g_{1}(s)u(s)rac{ds}{s}, {}^{H}D^{lpha -1}u(+infty )=int ^{+infty }_{1}g_{2}(s)u(s) rac{ds}{s}, end{cases} $$ where $2lpha leq 3$ , $0eta _{1}leq lpha -2eta _{2}leq lpha -1$ , $f:J imes mathbb{R}^{3}ightarrow mathbb{R}$ satisfies the q-Carathéodory condition, $q,g_{1},g_{2}:Jightarrow mathbb{R}^{+}$ are nonnegative, where $J=[1,+infty )$ . Nonlinear term f is dependent on the fractional derivative of lower order $eta _{1}$ , $eta _{2}$ , which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with ${}^{H}D^{eta _{2}}uin C(1,+infty )$ at the left endpoint $t=1$ (if $eta _{2}lpha -1$ ).