Stability And Bounded Oscillation In Critical For Nonlinear Functuinal Differential Equations

This Ph. D thesis is composed of four chapters.In the first chapter, we introduce the historical background of problems which will be investigated and the main works of this thesis.In Chaper 2, by relaxing the restriction in York-condition, we investigate asympototic stability for nonlinear delay differential equation x′(t)+λx(t)=F(t, x(T(t))) whereλ≥0 and global attractivity for nonlinear difference equation x(n+1)-λx(n)+r(n)h(x(g(n)))=0 where 0＜λ≤1. Our results not only include the known results, but also have more comprehensive applications.In Chaper 3, we mainly investigate in critical bounded oscillation for second order non-linear delay differential equation and for second order non-linear neutral delay differential equation The work of investigating oscillation in critical is more difficult than that in noncritical. In proofs, we use some techniques and a lot of calculations.In Chaper 4, oscillations for higher-order linear difference equation with continuous arguments where n≥1 integer and for higher-order linear non-autonomous systems of difference equations with continuous arguments where N≥1 integer are mainly investigated. Although the forms of difference equations with continuous arguments are simple, the work of investigating them is more difficult. So the references about them are less. Some new results are obtained.We do the research work in the above three respects and obtain many new results. These results are the most new at home and abroad in our known. Some results in this thesis have been published in Nonlinear Analysis, Journal of Mathematical Analysis and Applications, Journal of Computational and Applied Mathematics, etc.