Abstract.

This paper investigates kernel identification problems (IPs) in a Banach space for parabolic integro-differential equations (IDEs). We prove a basic theorem on the existence and uniqueness of a bounded solution (EUBS), providing a theoretical foundation for identifying the kernel function from given data. The result is extended to four types of kernel identification problems for parabolic integro-differential equations (IDEs) with dependent coefficients. These results demonstrate the applicability of the proposed framework to a broad class of problems arising in practical applications. The study contributes to the theory of IDEs in Banach spaces by addressing kernel IPs involving an unbounded operator and an integral term containing the unknown kernel function.

Keywords:

convolution integral, integral equation (IE), parabolic IDE, kernel identification problem, existence and uniqueness, positive operators, Banach spaces.