Abstract
In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.
Highlights
The interest towards the study of functional operators with shift was stipulated by the development of solvability theory and Fredholm theory for some classes of linear operators, in particular, singular integral operators with Carleman and non-Carleman shift [1]-[3]
We provide the main idea and the scheme of proof of the conditions of invertibility
Using a known method of solving such equations, we can find densities ν1 and ν 2 of the cyclic equilibrium of the system
Summary
The interest towards the study of functional operators with shift was stipulated by the development of solvability theory and Fredholm theory for some classes of linear operators, in particular, singular integral operators with Carleman and non-Carleman shift [1]-[3]. Conditions of invertibility for functional operators with shift in weighted Lebesgue spaces were obtained [1]. Our study of functional operators with shift in Hölder spaces with weight has an additional motivation: on modeling systems with renewable resources, equations with shift arise [4] [5], and the theory of linear functional operators with shift is the adequate mathematical instrument for the investigation of such systems. These are to be used in the proof of invertibility conditions. At the end of the article, an application to modeling systems with renewable resources is specified
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