Abstract
Abstract This paper investigates the kernel identification problem (IP) in a Banach space for a hyperbolic integro-differential equation (IDE). The basic theorem on the existence and uniqueness of a bounded solution (EUBS) for the kernel IP is proved. This result provides a foundation for solving practical problems where the kernel function needs to be identified based on the given data. The paper extends this result to four types of kernel identification problems for hyperbolic IDEs. By proving results for several types of IPs, the paper demonstrates that the framework can handle a broad range of practical scenarios where IDEs may be with dependent coefficients. This is particularly relevant in real-world applications where the relationship between the terms in the integro-differential equation can be complex. The results contribute to the theory of IDEs in Banach spaces by addressing kernel identification, which is a challenging problem due to the unbounded nature of the operator and the integral term involving the kernel function.