Behavior Of The Solutions Of Differentia Equations

This paper study the qualitatic behavior of the solutions of differential equations . It mainly deal with the stability and boundedness of solutions to the neatral functional differential equatins with unbounded delay (p- NFDE ),the asymptotic behavior and oscillation properties for second order nonlinear elliptic partial differential equations. Four chapters are includedIn Chapter one. the author make a survey of the course of study and the known classical basic theorems on the stability and boundedness of solutions to the retarded functional differential equatins with bounded delay ( FDE) and the neatral functional differential equatins with bounded delay ( NFDE ), the asymptotic behavior and oscillation properties for second order ordnary differential equations and second order elliptic partial differential equations. In the meanwhich, we introduce the main results in this paper.Chapter two considers the stability of p-NFDE p-NFDE is a new class of FDE, Which is different from FDE with bounded delay and FDE with infinite delay. We give a new defination of "stability of D-operayor" which generalize the traditional defination of inequatity, and obtain some stability theorems for p-NFDE by using Liapunov-Razumikhin functional, which improveand generalize some known classical theorems of RFDE, none of which requires f(t,φ) to be bounded for φ bounded.Chapter three consider the boundedness of p-NFDE. We frist present a concept of "bounded of D-operater". By liapunov functional and Jensen's inequality techniques, the aurthor establishfour critreion theorems for U.B and U.U.B of />-NFDE Moreover, some example are give to illustrate the effect of the obtained results.Chapter four is the last chapter. It contains three aspects. (I) the study of asymptotic behacvior of solutions of second order nonhomgeneous elliptic partial differential equations,some general theorems for the any solution y(x)of the equations to be satisfied liminf y(x)\ = 0 are obtained. (2) the reasarch for second order nonliner elliptic partial|jt(-MOdifferential equations with "weakly integrally small" cofBcient, some new and simple oscillatory conditions are obtained by method of function sequences. (3) the discussion on second order nonliner elliptic partial differential equations with " integrally small "cofBcient, some sufficient conditions for all solutions of the equation te be oscillatory are given by auxiliary ruction and integral inequality techniques. Particuarly, there results generalize and improve greatly thr previous results for second order ordinary differential equations and elliptic partial differential equations of second order.