Abstract:
Sufficient conditions for unique solvability in the classical and generalized sense of the inverse problem for a nonlinear equation in a Banach space resolved with respect to the highest fractional derivative of Dzhrbashyan– Nersesyan are obtained. The overdetermination condition of the inverse problem is given by the Stieltjes integral; the lower derivatives are present in the equation non-linearly. The operator by the unknown function in the linear part of the equation is assumed to be bounded or generating an analytical resolving family of the corresponding linear homogeneous equation. Using our previous results for the direct problem for a linear inhomogeneous equation we obtain the main results here by the method of contraction mappings. An example of an inverse problem for a partial differential equation for which the conditions of an abstract theorem are fulfilled is given.
Keywords:nonlinear inverse problem, generalized solution, classical solution, fractional derivative of Dzhrbashyan–Nersesyan, initial boundary value problem
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