Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)1836510.22363/2312-9735-2018-26-2-103-118Research ArticleThe Solvability of the Inverse Problem for the Evolution Equation with a Superstable SemigroupTikhonovI V<p>Doctor of Physical and Mathematical Sciences, professor of Department of Mathematical Physics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University</p>ivtikh@mail.ruVu NguyenSon Tung<p>Postgraduate Student of Mathematical Analysis Department, Moscow State University of Education</p>vnsontung@mail.ruLomonosov Moscow State UniversityMoscow State University of Education1512201826210311821042018Copyright © 2018, Tikhonov I.V., Vu Nguyen S.T.2018<p>For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.</p>evolution equationinverse problemsuperstable semigroupoperator equationNeumann seriesexistence and uniqueness theorem of the solutionэволюционное уравнениеобратная задачасуперустойчивая полугруппаоператорное уравнениеряд Нейманатеорема существования и единственности решения[E. Hille, R. Phillips, Functional Analysis and Semigroups, IL, Moscow, 1962, in Russian.][N. Dunford, J. Schwartz, Linear Operators. P. 1. General Theory, IL, Moscow, 1962, in Russian.][S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967, in Russian.][A. 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