Gonchar I. Bounded and summable solutions of a differece equation with jumps of an operator coefficient

The work is devoted to investigation of properties of solutions of difference equations with jumps of an operator coefficient in finite–dimensional and infinite–dimensional Banach space. For a difference equation with one and two jumps of an operator coefficient necessary and sufficient conditions for the existance of the unique bounded solution of such an equation for arbitrary bounded «input» sequence are obtained in finite–dimensional complex Banach space. Section results can be applied to the study of the difference equation with any finite amount of jumps of an operator coefficient. Sufficient conditions for the existance of the unique bounded solution and –power summable solution of linear difference equation with one jump of an operator coefficient are obtained in infinite–dimensional Banach space. Sufficient conditions for the existance of the unique bounded solution of weakly nonlinear difference equation are obtained. Also the application of the obtained results to the research of the bounded in the mean of order solution of a stochastic analog of a difference equation with one jump of an operator coefficient is shown.