Inverse Boundary Problem for a Fractional Order Integro-Differential Equations in a Locally Convex Space
Ключевые слова:
fractional order equation, locally convex space, equations with two unknown quantities, method of successive approximations, contraction mapping, existence, uniqueness, stabilityАннотация
This paper investigates an inverse boundary problem for a nonlinear fractional-order
integro-differential equation in a locally convex space. The considered model contains a nonlinear
Volterra-type integral structure together with a fractional Gerasimov–Caputo operator, which
reflects hereditary and memory effects arising in distributed parameter systems and nonlocal
dynamical processes.
Unlike the classical Banach-space framework, the analysis is developed in a locally convex setting
generated by a family of seminorms. This approach makes it possible to study a significantly
broader class of infinite-dimensional functional systems, including nonnormable spaces naturally
appearing in generalized function theory, operator analysis, and evolution equations.
The inverse problem is reduced to a coupled nonlinear system consisting of two integral equations:
one equation determines the main unknown function, while the second identifies the unknown
boundary value. Since each equation contains both unknown quantities, the resulting system
possesses a strongly nonlinear and nonlocal structure.
To establish solvability, the method of successive approximations is combined with a seminormbased contraction principle in locally convex spaces. Existence and uniqueness of the solution pair
are proved, and continuous dependence of the main function on the boundary data is established.
The obtained results extend several classical solvability results for fractional integro-differential
equations from Banach spaces to the wider class of locally convex spaces and provide a functionalanalytic framework for studying inverse problems with memory effects in nonnormable settings
Библиографические ссылки
O. Kh. Abdullaev, O. Sh. Salmanov, T. K. Yuldashev, ”Direct and inverse problems for a parabolichyperbolic equation involving Riemann–Liouville derivatives”, Trans. Natl. Acad. Sci. Azerb. Ser.
Phys.-Tech. Math. Sci. – Mathematics 43 (1), 21–33 (2023).
K. Diethelm, A. D. Freed, ”On the solution of nonlinear fractional order differential equations used in
the modeling of viscoplasticity”, Scientifice Computing in Chemical Engineering II-Computational
Fluid Dynamics, Reaction Engineering and Molecular Properties (F. Keil, W. Mackens, H. Voss,
and J. Werther, Eds), Springer–Verlag, Heidelberg, 1999. 217–224.
K. Diethelm, N. J. Ford, ”Analysis of fractional differential equations”, J. Math. Anal. Appl. 265,
–248 (2002).
D. Delbosco, L. Rodino, ”Existence and uniqueness for a nonlinear fractional differential equation”,
J. Math. Anal. Appl. 204, 609–625 (1996).
L. Gaul, P. Klein, S. Kempfle, ”Damping description involving fractional operators”, Mech. Systems
Signal Processing No. 5, 81–88 (1991).
W. G. Glockle, T. F. Nonnenmacher, ”A fractional calculus approach of selfsimilar protein dynamics”, Biophys. J. 68, 46–53 (1995).
R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
E. R. Kaufmann, E. Mboumi, ”Positive solutions of a boundary value problem for a nonlinear
fractional differential equation”, Electron. J. Qual. Theory Differ. Equ. 2007 (3), 1–11 (2007).
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204. Elsevier Science B.V., Amsterdam, 2006.
M. T. Kosmakova, T. K. Yuldashev, M. U. Sagdullaeva, ”Inverse problem for a nonlinear pseudiparabolic differential equation with final condition and Gerasimov–Caputo operator”, Journal of
Osh University ”Differential Equations” 1 (1), 1–14 (2026).
V. Lakshmikantham, A. S. Vatsala, ”Theory of fractional differential inequalities and applications”,
Commun. Appl. Anal. 11 (3-4), 395–402 (2007).
F. Mainardi, ”Some basic problems in continuum and statistical mechanics”, Fractals and Fractional
Calculus in Continuum Mechanics. (A. Carpinteri and F. Mainardi, Eds), 291–348, Springer-Verlag,
Wien, 1997.
M. K. Mamanov, G. A. Khasanov, Z. A. Madatova, ”Optimal control problem for an impulsive systems of functional-differential equations with a nonlinear function under the sign of a first-order
differential”, J. Osh University ”Differential Equations” 1 (1), 67–77 (2026).
N. K. Ochilova, T. K. Yuldashev, ”On a nonlocal boundary value problem for a degenerate parabolichyperbolic equation with fractional derivative”, Lobachevskii J. Math. 43 (1), 229–236 (2022).
https://doi.org/10.1134/S1995080222040175
I. Podlubny, ”Fractional differential equations”, Mathematics in Sciences and Engineering 198,
Academic Press, San Diego, 1999.
A. V. Pskhu, Partial differential equation of fractional order. Moscow, Nauka, 2000 (in Russian).
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, Yverdon, 1993.
T. K. Yuldashev, ”Determining of coefficients and the classical solvability of a nonlocal boundaryvalue problem for the Benney—Luke integro-differential equation with degenerate kernel”, J. Mathematical Sciences 254 (6), 793–807 (2021).
T. K. Yuldashev, O. Kh. Abdullaev, ”Inverse problems for the loaded parabolic-hyperbolic equation involves Riemann–Liouville operator”, Lobachevskii J. Math. 44 (3), 1080–1090 (2023).
https://doi.org/10.1134/S1995080223030034.
T. K. Yuldashev, B. J. Kadirkulov, ”On a boundary value problem for a mixed type fractional
differential equations with parameters”, Proceedings of the Institute of Mathematics and Mechanics,
National Academy of Sciences of Azerbaijan 47 (1), 112–123 (2021).
T. K. Yuldashev, B. J. Kadirkulov, R. A. Bandaliyev, ”On a mixed problem for Hilfer type fractional
differential equation with degeneration”, Lobachevskii J. Mathematics 43 (1), 263–274 (2022).
T. K. Yuldashev, B. J. Kadirkulov, Kh. R. Mamedov, ”An inverse problem for Hilfer type differential
equation of higher order”, Bulletin of the Karaganda University. Mathematics series 105 (1), 136–
(2022).
