Multiple Periodic Solutions Of Second-order Ordinary Differential Equations Across Resonance

The second-order ordinary differential equations are widely applied in various fields. The periodic solution problems of the differential equations become an important issue. In many fields such as in physics and mechanics, the resonance phenomenon is very common. So the resonant problems for the. differential equations have attracted much attention. In this paper we study the existence of multiple periodic solutions of second-order ordinary differential equations. New results of multiplicity of periodic solutions are obtained when the nonlinearity may cross multiple consecutive eigenvalues. The arguments are proceeded by a combination of variational and degree theoretic methods.Consider the following periodic boundary value problem of the second-order nonlinear ordinary differential equation where f∈C(R2,R), f{t+2π,x)=f{t,x) for all (t,x)€R2, T>0, f(t,O)=0. Clearly, the problem (1) has a trivial solution x=0. Our goal is to obtain mul-tiple nontrivial periodic solutions of problem (1). The linear periodic boundary value problem of (1) enjoys the eigenvalues{k2}keN.The eigenvalues{k2}kκN of linear periodic boundary value problem (2) are usually called resonant points of the problem (1).Various works in the literature are devoted to resonant problems for or-dinary or partial differential equations during the last several decades. A-mong these works, Fabry and Fonda, Omari and Zanolin established the solvability of the problem(1) under the following so-called double-resonance conditions, i.e., for some k€N and uniformly for a.e. t€[0,2?r] and some additional condi-tions imposed on f.Papageorgiou and Staicu’58^, Su and Zhao’57’studied the existence of multiple periodic solutions for the problem (1) when double resonance oc-curs, where they assumed double-resonance conditions and some generaliza-tion Landesman-Lazer conditions. Under the additional assumption (4) or f’{t,O)<0with/being C1, they both established the existence of at least two nontrivial solutions of the problem (1). Recently, Barletta and Papageorgioui59’ used the variational methods together with Morse theory to obtain six non-trivial solutions of the problem (1) when the nonlinearity is resonant both at infinity and at zero.In the above resonant problems, they all require the ratio stays asymptotically at infinity between two consecutive eigenvalues of linear prob-lem. This paper is devoted to the existence of multiple periodic solutions of problem (1) when the nonlinearity/cross multiple consecutive eigenvalues of the problem (2).In this paper, we do not require the ratio stays asymptotically at infinity between two consecutive eigenvalues of linear problem, so the ratio f(t,s)/s-may cross resonant points{k2} asymptotically. We show that if double-resonance conditions (3) with Landesman-Lazer conditions are replaced by the assumptions (4) and (5), the problem (1) still admits at least two nontrivial solutions. Furthermore, if (4) and (5) are replaced by some stronger condi-tions (7) and (8), then we may obtain more nontrivial solutions. Based on a combination of variational method and degree theory, we obtain our results as follows.Let H denote the Sobolev space equipped with the usual inner product and norm Define the functional I Obviously, It is well known that x is a weak solution of (1) if and only if x is a critical point of the functional I.Our main results are. as follows. Theorem1Assume that and the following conditions hold:(i) there exist η1, M, M1>0such that for a.e. t E [0,2tt] and some k G Z+;(iii) there exists s\>0such that Then the problem (1) admits at least one nontrivial nonnegative solution and one nontrivial nonpositive solution. Theorem2Assume that f<E C([0,2tt] x R,R) tuitfi/(t,x)=g{x)+e(t) and (6) holds. Suppose that the following conditions hold:(i) there exist t?2, C, Ci>0suc/i iftat(ii) i/iere exist p, q>0sue/i t/iai/or a.e. i€[0,27r] and some k€Z+.T/ien we/iaue i/ie following results:(i) i/fc is odd, then problem (1) possesses at least two nontrivial solutions; if k is even, then problem (1) possesses at least three nontrivial solutions, one of which is nonnegative, and another one is nonpositive;(ii) if all the critical points of the functional I is nondegenerate, then problem (1) admits at least four nontrivial solutions;(iii) if g■CX(R, R) and there exists j5>0such that for all s€R, then problem (1) admits at least four nontrivial solutions.