Initial Value Problems For Second-order Nonlinear Functional Differential Equations

Initial value problems are generated from applied mathematics and physics,for example,epidemiology,nuclear physics,cyb ernetics.Most phenomenas of the life can be characterized by the functional differential equations,so the research of the functional differential equations has great theoretical significance and applicational value.In the seventies of the last century,there has been the basic theory of functional differential equations on the books.Then the functional differential equations has leaded to the in-terest of many foreign scholars and to promote the development of functional differential equations theory, which has got the existence of solutions for functional differential equa-tions under different conditions,for example,Peiqing Jiang,Peixuan Weng in China and Ravi P.Agarwal,L.H.Erbe abroad.They were mostly using the fixed point theorem and approximation technique.The thesis contains two chapters.In chapter one,we consider the blow-up phenomena of second-order functional differential equations.Most research work about the blow-up phenomena is about the partial differential equations,for example,the research of evo-lution equations.Evolution equation is in the physical,mechanical or other natural disci-plines to describe the state or process over time,for example,the BBM equation,Sob-olev Galpem equations.Some people has also studied the blow-up phenomena of or-dinary differential equations.Zhijun Zhang and Yunhui Wang studied this ordinary differential equation in the year of 1999, u″＝λf(u)(1＋u′2), onλ>0,f∈C1 (R),and shown that under appropriate conditions some solutions of the equation blow-up.Yu Cheng studied a class of more generally ordinary differential equation, u″=λf(u)g(u′), in the case ofλ>0,f∈C1(R), g∈C1(R),and got the necessary and sufficient conditions of blow-up.For the second-order nonlinear differential equation x″=f(t,x,x′), the asymptotic behavior of solutions such as boundedness,dissipativity,asymptotic stabil-ity of equilibria and periodic solutions has been extensively studied in the literature;see,for instance,etc.Desheng Li and Haiyang Huang studied the sufficient conditions of some solutions about the above equation occurring blow-up phenomena.Inspired by the work of Desheng Li and Haiyang Huang in this chapter we consider the blow-up phe-nomena of solutions of second-order functional differential equation Under appropriate conditions it is shown that some solutions of the equation blow up in finite time.In the second chapter,we will overcome the difficulties which lie in changeful sign and singularity using the fixed point theorem and approximation technique.Thus we can get the existence of positive solutions for initional value problems of the second-order sin-gular functional differential equation.In [3],R.P.Agarwal Ch.G.Philos and P.Ch.Tsamatos discussed the functional differential equation of the form where f(t,φ,y) is singular at t=0.In [20] Fengfei Jin considered singular initial value problem (?)0t 1/p(s)ds<+∞,(?)0+∞1/p(s)ds=+∞,f>0 and be singular at z=0. In the section three of this chapter we consider this equation in the case of that f has changeful sign and can being singular at y=0. In the section four we consider the existence of positive solutions to the equation () with...