Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

• Published in: Central European journal of mathematics 2014-02, Vol.12 (2), p.308-321
• Main Authors: Bulatov, Mikhail V., Lima, Pedro M., Weinmüller, Ewa B.
• Format: Article
• Language: English
• Subjects: 45F15; 65R20; Algebra; Geometry; Integro-differential equations; Kernels; Lie Groups; Mathematical analysis; Mathematics; Mathematics and Statistics; Number Theory; Partial differential equations; Probability Theory and Stochastic Processes; Research Article; Topological Groups; Two dimensional analysis; Two-dimensional Volterra integral-algebraic equations; Volterra integral equations; Weakly singular
• ISSN: 1895-1074; 2391-5455; 1644-3616; 2391-5455
• DOI: 10.2478/s11533-013-0334-5

We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k ( t , x ) vanishes when t = x . Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.