A Cauchy type problem and inverse problems for equations with a Riemann–Liouville type regular integro-differential operator and a closed operator
https://doi.org/10.25587/2411-9326-2025-3-95-112
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.
About the Authors
V. E. Fedorov
Mathematical Analysis Department, Chelyabinsk State University
Russian Federation
Russian Federation
Vladimir E. Fedorov
129 Kashirin Brothers Street, Chelyabinsk 454001
A. V. Nagumanova
Mathematical Analysis Department, Chelyabinsk State University
Russian Federation
Russian Federation
Anna V. Nagumanova
129 Kashirin Brothers Street, Chelyabinsk 454001
A. O. Sagimbaeva
Mathematical Analysis Department, Chelyabinsk State University
Russian Federation
Russian Federation
Angelina O. Sagimbaeva
129 Kashirin Brothers Street, Chelyabinsk 454001
References
1. Kozhanov A. I., Composite Type Equations and Inverse Problems, Utrecht, VSP (1999).
2. Prilepko A. I., Orlovskii D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York; Basel (2000).
3. Tikhonov I. V. and Eidelman Yu. S., “An inverse problem for a differential equation in a Banach space and the distribution of zeros of an entire Mittag-Leffler function,” Differ. Equ., 38, No. 5, 669–677 (2002).
4. Abasheeva N. L., “Some inverse problems for parabolic equations with changing time direction,” J. Inverse Ill-Posed Probl., 12, No. 4, 337–348 (2004).
5. Fedorov V. E. and Urazaeva A. V., “An inverse problem for linear Sobolev type equations,” J. Inverse Ill-Posed Probl., 12, No. 4, 387–395 (2004).
6. Favini A. and Lorenzi A., Differential Equations, Inverse and Direct Problems, Chapman and Hall/CRC, New York (2006).
7. Falaleev M. V., “Abstract problem of prediction-control with degeneration in Banach spaces [in Russian],” Izv. Irkut. Gos. Univ., Ser. Mat., 3, No. 1, 126–132 (2010).
8. Pyatkov S. G. and Samkov M. L., “On some classes of coefficient inverse problems for parabolic systems of equations,” Sib. Adv. Math., 22, No. 4, 155–177 (2012).
9. Al Horani M. and Favini A., “Degenerate first-order inverse problems in Banach spaces,” Nonlinear Anal., 75, No. 1, 68–77 (2012).
10. Glushak A. V., “On an inverse problem for an abstract differential equation of fractional order,” Math. Notes, 87, No. 5, 654–662 (2010).
11. Orlovsky D. G., “Parameter determination in a differentia equation of fractional order with Riemann–Liouville fractional derivative in a Hilbert space,” J. Sib. Fed. Univ., Ser. Math. Phys., 8, No. 1, 55–63 (2015).
12. Fedorov V. E. and Nazhimov R. R., “Inverse problems for a class of degenerate evolution equations with Riemann–Liouville derivative,” Fract. Calc. Appl. Anal., 22, No. 2, 271–286 (2019).
13. Fedorov V. E., Nagumanova A. V., and Avilovich A. S., “A class of inverse problems for evolution equations with the Riemann–Liouville derivative in the sectorial case,” Math. Methods Appl. Sci., 44, No. 15, 11961–11969 (2021).
14. Fedorov V. E. and Ivanova N. D., “Identification problem for degenerate evolution equations of fractional order,” Fract. Calc. Appl. Anal., 20, No. 3, 706–721 (2017).
15. Orlovsky D. G., “Determination of the parameter of the differential equation of fractional order with the Caputo derivative in Hilbert space,” J. Phys. Conf. Ser., 1205, No. 1, article ID 012042 (2019).
16. Fedorov V. E. and Kostic M., “Identification problem for strongly degenerate evolution equations with the Gerasimov–Caputo derivative,” Differ. Equ., 56, No. 12, 1613–1627 (2020).
17. Fedorov V. E., Nagumanova A. V., and Kosti´c M., “A class of inverse problems for fractional order degenerate evolution equations,” J. Inverse Ill-Posed Probl., 29, No. 2, 173–184 (2021).
18. Ashurov R. R. and Fayziev Yu. E., “Inverse problem for finding the order of the fractional derivative in the wave equation,” Math. Notes, 110, No. 6, 842–852 (2021).
19. Kostin A. B. and Piskarev S. I., “Inverse source problem for the abstract fractional differential equation,” J. Inverse Ill-Posed Probl., 29, No. 2, 267–281 (2021).
20. Fedorov V. E., Ivanova N. D., Borel L. V., and Avilovich A. S., “Nonlinear inverse problems for fractional differential equations with sectorial operators,” Lobachevskii J. Math., 43, No. 11, 3125–3141 (2022).
21. Fedorov V. E. and Nagumanova A. V., “Inverse linear problems for a certain class of degenerate fractional evolution equations,” J. Math. Sci., 260, No. 3, 371–386 (2022).
22. Fedorov V. E., Plekhanova M. V., Ivanova N. D., Shuklina A. F., and Filin N. V., “Nonlinear inverse problems for some equations with fractional derivatives,” Chelyab. Phys. Math. J., 8, No. 2, 190–202 (2023).
23. Fedorov V. E., Plekhanova M. V., and Melekhina D. V., “Nonlinear inverse problems for equations with Dzhrbashyan–Nersesyan derivatives,” Fractal Fract., 7, No. 6, article ID 464 (2023).
24. Fedorov V. E., Plekhanova M. V., and Melekhina D. V., “On local unique solvability for a class of nonlinear identification problems,” Axioms, 12, No. 11, article ID 1013 (2023).
25. Fedorov V. E., Plekhanova M. V., and Sagimbaeva A. O., “Nonlinear inverse problems with a stationary unknown element for equations with Dzhrbashyan–Nersesyan derivatives,” Mat. Zamet. SVFU, 31, No. 3, 53–72 (2024).
26. Plekhanova M. V., Izhberdeeva E. M., Melekhina D. V., and Sagimbaeva A. O., “Global solvability of nonlinear inverse problems with Dzhrbashyan–Nersesyan derivatives and sectorial operators,” J. Math. Sci. (publ. 15 Feb. 2025) (2025). https://doi.org/10.1007/s10958-025-07570-1
27. Fedorov V. E., Godova A. D., and Kien B. T., “Integro-differential equations with bounded operators in Banach spaces,” Bull. Karaganda Univ. Math. Ser., No. 2, 93–107 (2024).
28. Fedorov V. E. and Godova A. D., “Integro-differential equations in Banach spaces and analytic resolving families of operators,” J. Math. Sci., 283, No. 2, 317–334 (2024).
29. Fedorov V. E. and Godova A. D., “Integro-differential equations of Gerasimov type with sectorial operators,” Proc. Steklov Inst. Math., 325, suppl. 1, S99–S113 (2024).
30. Fedorov V. E. and Godova A. D., “Linear inverse problems for integro-differential equations in Banach spaces with a bounded operator [in Russian],” Contemp. Math., Fund. Directions, 70, No. 4, 79–90 (2024)
31. Fedorov V. E. and Melekhina D. V., “Linear identifications problems for singular integrodifferential equations of Gerasimov type,” Mat. Zamet. SVFU, 32, No. 1, 46–64 (2025).
32. Caputo M., Fabrizio M., “A new definition of fractional derivative without singular kernel,” Progr. Fract. Differ. Appl., 1, No. 2, 73–85 (2015).
33. Atangana A. and Baleanu D., “New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model,” Thermal Sci., 20, 763–769 (2016).
34. Nagumanova A. V. and Fedorov V. E., “Direct and inverse problems for linear equations with Caputo–Fibrizio derivative and a bounded operator,” Chelyab. Phys. Math. J., 9, No. 3, 389–406 (2024).
35. Fedorov V. E. and Nagumanova A. V., “Direct and inverse problems for evolution equations with regular integrodifferential operators,” J. Math. Sci., 286, No. 2, 278–289 (2024).
36. LePage W. R., Complex Variables and the Laplace Transform for Engineers, Dover Publ., New York (1961).
37. Triebel H., Interpolation Theory, Functional Spaces, Differential Operators, VEB Deutsch. Verl. Wiss., Berlin (1978).
Review
For citations:
Fedorov V.E., Nagumanova A.V., Sagimbaeva A.O. A Cauchy type problem and inverse problems for equations with a Riemann–Liouville type regular integro-differential operator and a closed operator. Mathematical notes of NEFU. 2025;32(3):95-112. (In Russ.) https://doi.org/10.25587/2411-9326-2025-3-95-112
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