Resonance To Indefinite Problems:a Rotation Number Approach

In this paper,using a rotation number approach,we study the resonance to indefinite problems,concluding the following four problems1.Nonresonance of the indefinite problem associated to planar systems in the sense of rotation number2.The existence of periodic solutions associated to the planar systems with asym-metric nonresonance and half-side superlinear growth3.The existence of infinitely many subharmonic solutions associated to a sublinear equation with indefinite weight4.The existence and multiplicity of periodic solutions associated to conservative weakly coupled systems and nonconservative weakly coupled systemsThe first problem Nonresonance of the indefinite problem of planar systems in the sense of rotation number includes two sub-problems.One is nonresonance in the sense of rotation number,the angular velocity of the system z'=f(t,z)is allowed to be con-trolled by the angular velocity of two positive homogeneous systems z'=Li(t,z),i=1,2,and we describe the 2?-rotation numbers of the positive homogeneous systems by the rotation numbers of them,then through the discussion of the relationship between the 2?-rotation number of the planar system and the 2?-rotation numbers of the positive homogeneous systems,finally we realize the estimations of the 2?-rotation number of the planar system using the rotation numbers of the positive homogeneous systems,it is necessary not only to use new techniques to calculate the rotation numbers of positive homogeneous systems,but also to make it avoid resonance points in the sense of rotation number through the precise estimations of the rotation numbers of the positive homoge-neous systems,thus the existence of periodic solutions is obtained by using the Poincare-Bohl theorem.Another is the double resonance problem under the Landesman-Lazer conditions,we construct a generalized polar coordinate system by the orbits of positive homogeneous systems and obtain the existence of periodic solutions by the Poincare-Bohl fixed point theorem combining the rotation number approachThe second problem We consider the asymmetric nonresonance and nonreso-nance with half-side superlinear growth.We use the angular velocity of the odd symmet-ric positive homogeneous systems to control the angular velocity of the planar system.and characterize the rotation behavior of the planar system by the rotation numbers of the odd symmetric positive homogeneous systems.Because solutions may grow superlin-early in the right half plane of the phase plane,and are controlled by two odd symmetric positive homogeneous systems in the left half plane,then the time that solutions stay in the left half plane is intermittent in 2?-period,so we estimate the angle difference of the dwell time of solution orbits in one side of the phase plane,and develop the system method of "tracking" the angle difference of each small interval on the given side under the condition of sign-changingThe third problem We consider an infinite numbers of subharmonic solutions of sublinear equations with indefinite weight.In this part' our difficulties come from two aspects.One is the difficulty of sign-changing' by the estimations of increments and the geometric analysis of phase plane we complete the proof of the spiral property:The other is that solutions spiral slowly under sublinear conditions,and the twist produced in 2? time is very weak,so we consider it in 2m?,m ? N time,then the problem is that solutions may run to the origin when m is very large,therefore,we need to describe the spiral property of solutions,based on the spiral property we find appropriate parameters to modify the system near the origin so that solutions of the new system can not run to the origin,then by the Poincare-Birkhoff twist theorem we get the multiplicity of periodic solutions of the new system,and return to the multiplicity of periodic solutions of the original system by the spiral property at lastThe forth problem We consider the existence and multiplicity of periodic solu-tions of weakly coupled systems,concluding conservative coupled systems and noncon-servative coupled systems.For conservative coupled systems,we discuss the superlinear-sublinear coupled systems.Firstly,we overcome the difficulty in the proof of spiral prop-erty caused by the indefinite weight' that is,the difficulty of sign-changing.On the other hand,we overcome the difficulty of modifying systems.The method of modifying the mixed type coupled systems in the superlinear part is totally different from that in the sublinear part.The modification in the superlinear part is because of the lack of global existence of solution components,by the fast rotation of solution components' a priori estimate of solution components with a certain number of zeros is given,so that the sys-tem can be modified in the place where the polar diameter is sufficiently large to meet the global existence of solutions but it doesn't affect the discussion of the existence of pe-riodic solutions of the original system.Solutions spiral slowly in the sublinear part' and the twist produced in 2? time is very weak,so we consider it in 2m?,m ? N time,but solutions may run to the origin if m is very large.So based on the spiral property,we find appropriate parameters to modify the system near the origin so that solutions of the new system can not run to the origin.For the nonconservative coupled system,firstly,we prove a fixed point theorem for the coupling of twist conditions and Poincare-Bohl type condi-tions by the topological degree method,and then prove the existence of periodic solutions of the nonconservative weak coupled systems with several kinds of mixed type coupled conditions by this fixed point theorem,that is,the superlinear-oscillatory nonlinearities Coupling,the superlinear-sublinear coupling,or superlinear-nonresonant coupling.