Study Of The Existence Of Nonoscillatory Solutions Of Neutral Difference Equations With Variable Coefficients

Difference equations are considered as the discretization and digital solutions ofdifferential equations and delay differential equations，which have already become animportant branch in mathematical research， especially in dynamical systems。 Thenonoscillatory solution of difference equations is an important component of the theory ofdifference equations，and possesses important theoretical and practical significance。With therapid development of computer science，the study of difference equations has already becomea very active research field。In this paper，we study mainly the existence of nonoscillatory solutions for a class of firstorder nonlinear difference equations with a forced term and the existence of nonoscillatorysolutions for a higher-order neutral delay difference equations with variable coefficients。Some criteria conditions for the existence of nonoscillatory solutions are given，whichgeneralize some existed results。Firstly，the author reviews the development、research background and the researchresults of the difference equations obtained by the predecessors，and introduces the main workof this thesis。Secondly，the problem of the existence of nonoscillatory solutions for a class offirst order nonlinear difference equations with a forced term is presented，the involvedfunction with two variables is generalized to the function with three variables，and at the sametime， the sufficient conditions are modified to assure the existence of nonoscillatorysolutions。By adjusting the ideas and ways of proof and using the Banach contractionprinciple，we obtain a sufficient and necessary condition for the existence of nonoscillatorysolutions for this equation。Some examples are given to illustrate the correctness and validityof the theorem。Thirdly，Another kind of higher-order neutral delay difference equations withvariable coefficients is handled，the author not only gets rid of some quite stronger tentativeconditions in the prevenient literatures，but also extends the range of the parameter，then byusing similarly Banach contraction principle，the sufficient conditions are obtained。For thesituation of c1，one counter-example is presented to illustrate that the solution may be oscillate。Finally，the author summarizes the main work of the thesis and looks ahead for whatand how to do in the near future。...