Abstract:
The unique solvability of a Cauchy-type problem and linear inverse coefficient problems for an evolution equation in a Banach space with a first-order Riemann– Liouville integro-differential operator with a regular kernel is investigated. The operator at the unknown function in the equation is assumed to be closed. The conditions for the existence and uniqueness of a solution of the Cauchy type problem for a linear inhomogeneous equation are obtained. A criterion of correct solvability is found for the inverse problem with a stationary unknown coefficient and with an integral overdetermination condition in the Riemann–Stieltjes sense, which includes the condition of final overdetermination as a special case. The conditions for the solvability and stability of a solution of the inverse problem with a nonstationary unknown coefficient and an abstract overdetermination condition on the interval are found. The abstract results obtained are used in the study of linear inverse initial boundary value problems for equations with a firstorder Riemann–Liouville type regular integro-differential operator in a time variable, with polynomials with respect to a self-adjoint elliptic differential operator in spatial variables and with an unknown coefficient.
Keywords:Riemann–Liouville type regular integro-differential operator, linear evolution equation in a Banach space, Cauchy type problem, linear inverse coefficient problem, initial boundary value problem
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