Continuous Solutions of a Class of Difference-Functional Equations

Тетяна Олександрівна Єрьоміна


The main object of research in this article is a study of the continuous solutions set structure of difference-functional equations of the form \[x\left ( qt \right )=a\left ( t \right )x\left ( t \right )+b\left ( t \right )x\left ( t+1 \right )+f\left ( t \right ),\] where \[a\left ( t \right ),b\left ( t \right ),f\left ( t \right )\]
– some real functions and q – real constant. Using methods of the theory of differential and difference equations a question of continuous solutions existence of linear difference-functional equations with constant coefficients was investigated and a building method for them was proposed. In particular for homogeneous equations, a continuous bounded solutions family that depends on an arbitrary continuous 1-periodic function with \[0< q< 1,a> 1\] and \[0< a< 1,q> 1,\] – positive was built. Furthermore, if \[0< q< 1,a> 1,\] then continuous solutions existence for heterogeneous equations with an arbitrary real value is proved and continuous bounded solutions family of a heterogeneous equation with an arbitrary nonnegative are built.


Difference equations; Functional equations; Difference-functional equations


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