Geodesic
Nonlinear inverse problems for some equations with fractional derivatives
Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 2, pp. 190-202

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The solvability of nonlinear inverse problems with a time-dependent unknown element for evolution equations in Banach spaces with Gerasimov — Caputo derivatives is investigated. A theorem is obtained on the existence of a unique smooth solution of a nonlinear problem for an equation solved with respect to the highest fractional derivative with a bounded operator in the linear part. It is used in the study of degenerate evolution equations under the condition of $p$-boundedness of a pair of operators in the linear part of the equation — at the highest derivative and at the desired function. In the case of the action of a nonlinear operator into a subspace without degeneration, the existence of a unique smooth solution is proved; and for the independent of the nonlinear operator from elements of the degeneration subspace, the existence of a unique generalized solution is shown. The abstract results obtained for degenerate equations are used in the study of an inverse problem for a modified system of Sobolev equations with unknown coefficients at lower order fractional derivatives in time.
Keywords: Gerasimov — Caputo fractional derivative, inverse problem, degenerate evolution equation, Sobolev system of equations. 
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     author = {V. E. Fedorov and M. V. Plekhanova and N. D. Ivanova and A. F. Shuklina and N. V. Filin},
     title = {Nonlinear inverse problems for some equations with fractional derivatives},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {190--202},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_2_a2/}
}
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V. E. Fedorov; M. V. Plekhanova; N. D. Ivanova; A. F. Shuklina; N. V. Filin. Nonlinear inverse problems for some equations with fractional derivatives. Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 2, pp. 190-202. http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_2_a2/