Necessary and Sufficient Conditions for the Potentiality of Nonlinear Differential-Difference Operator in Partial Derivative

The purpose of the present paper is to investigate the potentiality of the differential difference operator with deviant arguments and to construct the functional, if the given operatoris a potential on a given set relatively to the some special bilinear form, i.e. the problem ofexistence of solutions of inverse problems of the calculus of variations for partial differentialdifference operators is investigated. Let ..,.. be normed linear spaces over the field of real numbers R. Take any operator .. : ..(..) > ..(..), where ..(..) . .., ..(..) . .. . A limit if it exists, is called the G.ateaux differential of .. at the point ... The operator ....(.., ·): .. > .. is called the G.ateaux derivative of .. at .. and will be denoted by ..... . Its domain of definition ..(..... ) consists of elements . . .. such that (.. + ...) . ..(..) for all .. sufficiently small. We obtain necessary and sufficient conditions for the adjusted partial differential difference op..,.. erator of ..class. The nonlinear differential operator of the second order and the nonlinear ..,.. differential operator of the second order with deviant arguments is consider as an example. Using the obtained conditions of potentiality corresponding functionals are constructed.

bilinear form, differential operator, differential-difference, functional