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

Abstract. The purpose of the current paper is to investigate the existence of local and global continuous solutions of a nonlinear β−order modified Riemann-Liouville (RL) fractional differential equation with a continuous right-hand side function f(x, ω) on (0, T ] × R which has a singularity of order β at x = 0. The local existence of solutions to singular equations of such types cannot be established by a direct application of fixed point theorems alone or with 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 intervals can be extended to larger ones. By doing this and the proposed technique, the global existence theorem for both sublinear and linear equations is achieved.