With Nonlinear Vibration Of The Existence Of Infinitely Many Solutions For The Problems

In this paper, we firstly study the existence of solutions for two-point boundary value problem with sublinear nonlinearity by using the minimax methods in critical point theory.Secondly, we study the existence of periodic solutions for the non-autonomous second-order system by using the minimax methods in critical point theory.Firstly, we consider the following two-point boundary value problem with sub-linear nonlinearity where f:[0, Ï€]×Râ†'R satisfies the following assumptions:(A)f(t,x) is measurable in t for all x∈R and continuous in x for almost every t∈[o,Ï€]..(B) f is sublinear,there exists g,h∈L1(0,Ï€; R+)å'ŒÎ±âˆˆ(0,1) such that(C)Then we can obtain the following theorems.Theorem1Suppose that f satisfies (A),(B),(C), then(i) There exists a sequence {un} of solutions of (P1) such that {un} is a critical point of φ and (ii) There exists a sequence {un*} of solutions of (P1) such that {un*} is a local minimum of φ andSecondly, we consider the following non-autonomous second-order system where T>0, F:[0, T]×RNâ†'R satisfies the following assumptions:(A\) F(t,x) is measurable in t for all x∈RN and continuous in x for almost every t∈[0,T]. Moreover, there exist a{x)∈C(R+,R+) and b(x)∈L1(R1,R+) such that for all x∈RN and a.e.t∈[0, T], where R+:=[0,+∞).(B1) There exists f,g∈L1(0,T; R+) with∫0T f(t)dt <12/T such that(C1) Then we can obtain the following theorems.Theorem2Suppose that f satisfies (A1),(B1),(C1), then there exists a sequence {un} of solutions of (P1) such that {un} is a critical point of φ and limnâ†'∞φ(un)=+∞.