Inverse Boundary Problem for a Fractional Order Integro-Differential Equations in a Locally Convex Space
Keywords:
fractional order equation, locally convex space, equations with two unknown quantities, method of successive approximations, contraction mapping, existence, uniqueness, stabilityAbstract
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
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