A novel technique for the existence of solutions to nonlinear fractional differential equation having a singularity of the critical order

Abstract

This paper is devoted to a nonlinear singular Riemann-Liouville type fractional differential equation, the local existence of whose continuous solutions under the weakest condition remained as an open problem until now. The singularity of the equation arises from the discontinuity of the right-hand side function $f(x,\omega)$ at $x=0$, and its order of singularity is the same as the order of the fractional differential operator in the equation. The local existence of solutions to singular equations of such type cannot be established by a direct application of fixed point theorems only or other methods even though they are enough for the same purpose in the case of equations including a singularity of the order less than the order of the corresponding fractional R-L operator. For this reason, we here propose a novel technique with a result for continuous functions to establish a local existence theorem for the problem. Moreover, a continuation result is suggested for the local solutions to the problem and, thus, it will be revealed that in what circumstances their existing interval can be extended to larger ones. Thanks to this and the proposed technique, the global existence theorem for both sublinear and linear equations is achieved.