Existence Of Multiple Periodic Solutions Research Articles

Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima

We study the stability of periodic solutions of the scalar delay differential equation where f(t) is a periodic forcing term and δ,p∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.

Multiple periodic solutions for autonomous conservative systems

We consider an autonomous conservative second order system defined by a potential which admits a connected set $\Gamma$ of critical points at level zero. We prove the existence of multiple periodic solutions of large period which are located near $\Gamma$.

Bourgin-Yang type theorem and its application to 𝑍₂-equivariant Hamiltonian systems

We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of Z 2 Z_2 –action. To that purpose we prove a new version of the Bourgin–Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.

Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity

An asymptotically linear Hamiltonian system with strong resonance at infinity is considered. The existence of multiple periodic solutions is proved via variational methods in an equivariant setting.

A sufficient condition for the existence of multiple periodic solutions of differential inclusions

It is known that under regularity conditions on Φ and a certain Nagumo-type boundary condition a solution of (1) exists (see e.g. [2]). In this note, following ideas of [4] and using results from [11], we prove the existence of two or more periodic trajectories if additional conditions are satisfied: Our method requires that certain complementary potential functions (comp. [9], [7]) in phase space can be detected. In our approach we apply the topological fixed point index theory to the Poincare (also called Poincare-Andronov) translation operator associated with (1). This well-known technique was developed by M. A. Krasnosel’skĭi in the single valued case, i.e. when we have unique solvability of the respective initial value problem (see [9]). For applications of the set-valued Poincare map we recommend [4] as a good survey. We shall proceed as follows: In section 2 we present the fixed point index theory for the so-called decomposable mappings (see our definition (2.3), comp. [4]) as it is developed in [1] and obtain some additional information necessary for our later work. In the third section we prove some abstract multiple fixed point results for compact decomposable mappings defined on cones of arbitrary ordered normed spaces. We choose this general setting since we believe that the results given there might be of independent interest. Our methods are somewhat connected with the theory given in [10] (for single valued maps), where applications to certain boundary value problems and Hammerstein integral

Multiple existence of periodic solutions for Liénard system

The existence of multiple periodic solutions of a nonlinear Lienard system is treated. The proof is based on the theory of topological degree and monotone operators.

Oscillatory behaviour of a catalyst pellet with narrow region of activity

An analysis of the dynamical behaviour of an isothermal continuous stirred tank reactor containing catalyst pellets with a Dirac delta distributed activity is presented. The case of bimolecular Langmuir—Hinshelwood kinetics in studied. The bifurcation diagram in the plane active point location vs parameter X is constructed. Existence of multiple periodic solutions is numerically established. The effects of active point location, Thiele modulus and Damkoehler number are discussed.

Multiple periodic solutions for a class of nonlinear nonautonomous wave equations

In this paper, by means of ourZp index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.

Existence of multiple periodic solutions for a semilinear evolution equation

In this paper, we consider the existence of multiple periodic solutions for the problem \[ d u d t + L u = g ( u ) + h , t > 0 , u ( 0 ) = u ( T ) , \frac {{du}}{{dt}} + Lu = g(u) + h,t > 0,u(0) = u(T), \] where L L is a uniformly strongly elliptic operator with domain D ( L ) = H 0 m ( Ω ) , g : R → R D(L) = H_0^m(\Omega ),g:R \to R is a continuous mapping, T > 0 T > 0 and h : ( 0 , T ) → H 0 m ( Ω ) h:(0,T) \to H_0^m(\Omega ) is a measurable function.