Existence Theorem For Periodic Solutions Research Articles

About one strengthening of the Massera’s existence theorem of periodic solutions of linear differential periodic systems

According to Massera’s theorem, an ordinary differential linear nonhomogeneous periodic system has a periodic solution with a period coinciding with that of the system if and only if this system has a bounded solution. We introduce the class L of vector functions called growing slower than a linear function. This class contains the class B of bounded vector functions in as its own subclass. It has been proved that Massera’s above-mentioned theorem will remain true if in its formulation a bounded solution is replaced by a slower growing solution than a linear function. It is shown that the set B in the metric space (L, distc ), where distc is the uniform convergence metric vector functions on intervals, has Baer’s first category, i. e. almost everything in the sense of the category of space vector functions (L, distc ) are not bounded. This fact shows the significance of the obtained strengthening of Massera’s theorem.

Existence of Periodic Solutions for Second-Order Ordinary p-Laplacian Systems

In this paper, we study the variational principle and the existence of periodic solutions for a new class of second-order ordinary p-Laplacian systems. The variational principle is given by making use of two methods. We obtain three existence theorems of periodic solutions to this problem on various sufficient conditions on the potential function F(t,x) or nonlinearity ∇F(t,x). Four examples are presented to illustrate the feasibility and effectiveness of our results.

Kolebatel'nye resheniya obyknovennogo differentsial'nogo uravneniya vtorogo poryadka s trekhpozitsionnym gisterezisnym rele bez vykhoda v zony nasyshcheniya

A second-order ordinary differential equation with an essential nonlinearity and an external perturbation in the form of a continuous periodic function is studied. Nonlinearity is given by a symmetrical relay characteristic with hysteresis, dead zone, and saturation zones. The traversal of the characteristic without entering the saturation zones is considered for some finite time and time commensurate with the period of the perturbation function. Conditions for the existence of an oscillating solution with a closed phase trajectory and four switching points during one traversal of the characteristic are obtained. Existence theorems for periodic solutions, including solutions with a symmetric trajectory, are proved. A numerical example is given.

On the variational principle and applications for a class of damped vibration systems with a small forcing term

<abstract> <p>This paper is dedicated to studying the existence of periodic solutions to a new class of forced damped vibration systems by the variational method. The advantage of this kind of system is that the coefficient of its second order term is a symmetric $N \times N$ matrix valued function rather than the identity matrix previously studied. The variational principle of this problem is obtained by using two methods: the direct method of the calculus of variations and the semi-inverse method. New existence conditions of periodic solutions are created through several auxiliary functions so that two existence theorems of periodic solutions of the forced damped vibration systems are obtained by using the least action principle and the saddle point theorem in the critical point theory. Our results improve and extend many previously known results.</p> </abstract>

Further study on periodic solutions of elliptic equations with a fractional Laplacian

We obtain some existence theorems for periodic solutions to several linear and nonlinear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.

On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

For eachx0∈[0,2π)andk∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problemu′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x)in(0,2π)withu(0)-u(2π)=u′(0)-u′(2π)=0wheng:(0,2π)×R→Ris a Caratheodory function which grows linearly inuasu→∞, andh∈L1(0,2π)may satisfy a generalized Landesman-Lazer condition(1+sign(β))∫02πh(x)v(x)dx<∫v(x)>0gβ+(x)vx1-βdx+∫v(x)<0gβ-(x)vx1-βdxfor allv∈N(L)\{0}. HereN(L)denotes the subspace ofL1(0,2π)spanned bysin⁡kxandcos⁡kx,-1<β≤0,gβ+(x)=lim infu→∞(gx,uu/u1-β), andgβ-(x)=lim infu→-∞(gx,uu/u1-β).

Existence of periodic solutions of sublinear Hamiltonian systems

The variational method is used to obtain some existence theorems of periodic solutions of sublinear systems with or not with impacts under suitable growth conditions. Compared with normal systems, impact systems need additional conditions to ensure the existence of periodic bouncing solutions.

Existence theorems of periodic solutions for second-order difference equations containing both advance and retardation

Using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations containing both advance and retardation is established. The proof is based on the Saddle Point Theorem in combination with variational technique.

About periodicity of impulsive evolution equations through fixed point theory

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existence theorems for periodic solutions. Moreover, the asymptotic stability of periodic solutions is obtained under suitable conditions. As one will see, the concept of an impulsive propagator is introduced for the first time in the paper.

Existence of periodic solutions for a class of p-Laplacian equations

This paper is devoted to the existence of periodic solutions for the one-dimensional p-Laplacian equation − ( ϕ p ( u ′ ) ) ′ =f(t,u), where ϕ p (u)= | u | p − 2 u (1<p<+∞), f∈C([0,2π]×R,R). By using some asymptotic interaction of the ratios f ( t , u ) | u | p − 2 u and p ∫ 0 u f ( t , s ) d s | u | p with the Fučík spectrum of − ( ϕ p ( u ′ ) ) ′ related to periodic boundary condition, we establish a new existence theorem of periodic solutions for the one-dimensional p-Laplacian equation.

Existence theorems of periodic solutions for fourth-order nonlinear functional difference equations

By using of the critical point method, the existence of periodic solutions for fourth-order nonlinear functional difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear functional difference equations. Results obtained generalize and complement the existing one.

Periodic solutions for second order Hamiltonian systems

By using the least action principle and minimax methods in critical point theory, some existence theorems for periodic solutions of second order Hamiltonian systems are obtained.

Periodic solutions for sublinear systems via variational approach

The variational method is used to obtain some existence theorems for periodic solutions of second order systems with or without impulsive under weak sublinear growth conditions.

Periodic solutions of competition Lotka–Volterra dynamic system on time scales

In this paper, we rigorously establish an existence theorem of periodic solutions for the competition of Lotka–Volterra dynamic systems with a time delay and diffusion on time scales. It is shown that the existence of periodic solutions depend on the parameters of the model. It is also shown that a known result in the literature can can carry over quite easily to its discrete counterpart, and a much more accurate result can be obtained when studying the dynamic system on time scales. Moreover, one example is given to illustrate the result obtained.

Periodic Solution of Second-Order Hamiltonian Systems with a Change Sign Potential on Time Scales

This paper is concerned with the second-order Hamiltonian system on time scales&amp;#x1D54B;of the formuΔΔ(ρ(t))+μb(t)|u(t)|μ−2u(t)+∇¯H(t,u(t))=0, Δ-a.e.t∈[0,T]&amp;#x1D54B; ,u(0)−u(T)=uΔ(ρ(0))−uΔ(ρ(T))=0,where0,T∈&amp;#x1D54B;. By using the minimax methods in critical theory, an existence theorem of periodic solution for the above system is established. As an application, an example is given to illustrate the result. This is probably the first time the existence of periodic solutions for second-order Hamiltonian system on time scales has been studied by critical theory.

Periodic Solutions for a System of Difference Equations

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.

Existence of periodic solutions of nonlinear evolution inclusions in Banach spaces

In this paper, we consider nonlinear evolution problems, defined on an evolution triple of spaces, driven by a monotone operator, and with a perturbation term which is multivalued. We prove existence theorems of periodic solutions for the cases of a convex and of a nonconvex valued perturbation term which is defined on all of I × H with values in V ∗ (here V ⊂ H ⊂ V ∗ is the evolution triple). Also, we prove the existence of extremal periodic solutions and a strong relaxation theorem. Some examples of nonlinear parabolic problems are presented.

Boundedness, periodic solutions and global stability for cellular neural networks with variable coefficients and infinite delays

In this paper, we consider the cellular neural networks with variable coefficients and infinite distributed delays. By introducing the phase space C g ( R - ) and applying Lyapunov functional method and Young inequality technique, we first establish a series of criteria on the boundedness, globally asymptotic stability and globally exponential stability. Furthermore, by applying these results and combining the existence theorems of periodic solutions for general functional differential equations with infinite delays, we establish the existence of periodic solutions and its globally asymptotic stability and globally exponential stability for the periodic cellular neural networks with infinite distributed delays. At last, as a special case, we apply these results to the autonomous cellular neural networks with infinite distributed delays and the existence, uniqueness and global stability of equilibrium point are established.

Existence theorems of periodic solutions for a class of damped vibration problems

In the present paper, we research the following damped vibration problem (1.1) u ¨ ( t ) + g ( t ) u ˙ ( t ) = ∇ F ( t , u ( t ) ) a.e . t ∈ [ 0 , T ] u ( 0 ) - u ( T ) = u ˙ ( 0 ) - u ˙ ( T ) = 0 , where T > 0 , g ∈ L ∞ ( 0 , T ; R ) , G ( t ) = ∫ 0 t g ( s ) ds , G ( T ) = 0 and F : [ 0 , T ] × R N → R . The variational principle is given, and three existence theorems for periodic solutions are obtained.

Periodic solutions of some resonant Hamiltonian systems

Based on the Galerkin approximation and the Morse inequality, an existence theorem of periodic solutions of the Hamiltonian system J z ′ = H ′ ( t , z ) is given, where H ′ ( t , z ) is asymptotically linear and resonant at infinity. Existence of periodic solutions of the resonant asymmetric p-Laplacian equation is also discussed.