Analyzing inverse backward problem in nonlinear integro-differential equation with memory kernel

This paper focuses on the backward problem related to an integro-differential equation with a general convolutional derivative in time and nonlinear source terms. The existence, uniqueness, and regularity of the mild solution to the proposed problem are established under certain assumptions in a suitable space. The proposed problem is ill-posed in the sense of Hadamard. Moreover, the Fourier truncation method is used to construct a regularized solution. Finally, the convergence rate between the regularized solution and the exact solution is determined.