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Анотація

The article addresses the problem of finding fixed points for mappings in spaces of multidimensional sequences. The authors formulate and prove a fundamental theorem establishing the existence and uniqueness of a solution to the equation of the form x = h + Ax, where A is a linear operator acting in a Banach space endowed with a countable system of seminorms. These seminorms are consistent with the norm of the space and satisfy monotonicity and boundedness conditions.
By applying the method of successive approximations, the convergence of the corresponding iterative process is analyzed. The study demonstrates that the infinite series arising from the iterative scheme converges, providing an explicit representation of the solution. Furthermore, norm estimates for the solution are derived, showing that the operator under consideration is bounded. The uniqueness of the solution is rigorously established by proving that the associated homogeneous equation admits only the trivial solution under the stated assumptions.
The results presented in this paper extend the classical fixed-point theory into the framework of multidimensional sequence spaces, thereby contributing both to the theory of functional analysis and to applications. In particular, the developed approach is relevant for the study of discrete and continuous dynamical systems, the analysis of stochastic processes, and the design of models in optimal control theory, where identifying stationary states plays a central role.
Overall, the article provides a mathematically rigorous foundation for addressing solvability questions in spaces with complex structures. It enriches the theoretical toolbox available for researchers working on applied problems involving high-dimensional or structured state spaces, thereby opening perspectives for further studies in modern analysis and its applications.