2021/03/19 by Müfit Şan, Şan, Müfit
Mathematics · #26A33 #34A08 #34A12 #37C25 #74G20 #Advanced Differential Equations and Dynamical Systems #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #General Mathematics (math.GM) #Nonlinear Differential Equations Analysis #math.GM #msc:26A33 #msc:34A08 #msc:34A12 #msc:37C25 #msc:74G20
paper · pdf · doi:10.48550/arxiv.2103.11797
Due to many errors made in the paper
openalex publication_date 2021/03/19 · arxiv created 2021/11/28 · arxiv updated 2021/11/30 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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,ω) 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.