An investigation of non-local and nonlinear boundary value problems for the fractional subdiffusion equation

The paper focuses on the analysis of boundary value problems that are both nonlocal and nonlinear for the fractional subdiffusion equation within the domain [0; 1] × (0; T]. The equation is defined using the Caputo fractional derivative of order 0 < α < 1 with respect to the time variable. An a priori estimate is established for the solution, ensuring its stability. The existence of the solution is analyzed through an auxiliary problem and the Greens function method. The solution to the integral equation is proven in the space C[0; 1] using the method of successive approximations. The main result demonstrates that, under certain conditions, the problem has a unique solution.