Regularization of Nonlinear Volterra Integral Equations of the First Kind with Smooth Data
Karakeev T. Mustafayeva N.
December 2025Multidisciplinary Digital Publishing Institute (MDPI)
AppliedMath
2025#5Issue 4
The paper investigates the regularization of solutions to nonlinear Volterra integral equations of the first kind, under the assumption that a solution exists and belongs to the space of continuous functions. The kernel of the equation is a differentiable function and vanishes on the diagonal at an interior point of the integration interval. By applying an appropriate differential operator (with respect to x), the Volterra integral equation of the first kind is reduced to a Volterra integral equation of the third kind, equivalent with respect to solvability. The subdomain method is employed by partitioning the integration interval into two subintervals. Within the imposed constraints, a compatibility condition for the solutions is satisfied at the junction point of the partial subintervals. A Lavrentiev-type regularizing operator is constructed that preserves the Volterra structure of the equation. The uniform convergence of the regularized solution to the exact solution is proved, and conditions ensuring the uniqueness of the solution in Hölder space are established.
integral equation , regularization , small parameter , uniform convergence
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Department of Functional analysis, Institute of Mathematics and Computer Science, Kyrgyz National University Named After J. Balasagyn, Bishkek, 720033, Kyrgyzstan
Department of Information Systems, S. Seifullin Kazakh Agrotechnical Research University, Astana, 010000, Kazakhstan
Department of Functional analysis
Department of Information Systems
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