Periodic Boundary Value Conditions Research Articles

The spectrum for the p-Laplacian systems with rotating periodic boundary value conditions and applications

In this paper, we study the spectrum problem for the p-Laplacian system (ϕp(u′))′+λϕp(u)=θ with rotating periodic boundary value conditions u(T)=Qu(0),u′(T)=Qu′(0), where Q is an N×N orthogonal real matrix and θ represents the zero vector. We give the definitions of spectrum and eigenvalue for the above problem, prove the relationship between them, discuss the basic properties of the eigenvalues and obtain the variational characteristic of the first positive eigenvalue, which deduce the generalized Wirtinger inequality and Sobolev inequality. These results provide a theoretical basis for subsequent research. In order to use the variational method, we establish a Sobolev space corresponding to rotating periodic boundary value conditions and give some useful properties. As applications of the spectrum of the above problem, we prove the existence of solutions for the p-Laplacian system rotating periodic boundary value problem with growth of order p and the rotating periodic Liénard type system involving p-Laplacian. Our results extend and enrich the existing relative work.

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

The Existence Theorems of Fractional Differential Equation and Fractional Differential Inclusion with Affine Periodic Boundary Value Conditions

This paper is devoted to investigating the existence of solutions for the fractional differential equation and fractional differential inclusion of order α∈(2,3] with affine periodic boundary value conditions. Applying the Leray–Schauder fixed point theorem, the existence of the solutions for the fractional differential equation is established. Furthermore, for the fractional differential inclusion, we consider two cases: (i) the set-valued function has convex value and (ii) the set-valued function has nonconvex value. The main tools of our research are the Leray–Schauder alternative theorem, Covita and Nadler’s fixed point theorem and some set-valued analysis theories.

The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Affine Periodic Boundary Value Conditions

The solution to a sequential fractional differential equation with affine periodic boundary value conditions is investigated in this paper. The existence theorem of solution is established by means of the Leray–Schauder fixed point theorem and Krasnoselskii fixed point theorem. What is more, the uniqueness theorem of solution is demonstrated via Banach contraction mapping principle. In order to illustrate the main results, two examples are listed.

On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

Error analysis of a time fourth-order exponential wave integrator Fourier pseudo-spectral method for the nonlinear Dirac equation

In this paper, we propose a time fourth-order exponential wave integrator (EWI) Fourier pseudo-spectral method for solving the nonlinear Dirac equation with periodic boundary value conditions. The new numerical method is designed by using Fourier pseudo-spectral method for spatial derivatives combined with a new fourth-order EWI for temporal derivatives. We give rigourously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. In more details, the proposed method has the fourth-order temporal accuracy and spectral spatial accuracy, respectively, in general -norm. Extensive numerical experiments are reported to confirm the theoretical analysis and show the superiority of the new method.

Existence of solutions for subquadratic convex operator equations at resonance and applications to Hamiltonian systems

This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.

Fixed Point Theorem Based Solvability of 2-Dimensional Dissipative Cubic Nonlinear Klein-Gordon Equation

The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem (LS). First, the GLK method is used to construct some uniform priori estimates of approximate solution to the corresponding equation of 2DDCNLKGE. Finally, the LS fixed point theorem is applied to obtain the efficient and straightforward existence and uniqueness criteria of time periodic solution to the 2DDCNLKGE.

On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space

We consider the periodic boundary value problem (PBVP) for a semilinear fractional-order delayed functional differential inclusion in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove the main existence result (Theorem ). We present an example dealing with existence of a trajectory for a time-fractional diffusion type feedback control system with a delay satisfying periodic boundary value condition.

Existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance

We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.

Global Bifurcation for Fourth-Order Differential Equations with Periodic Boundary-Value Conditions

We establish the global structure of positive solutions of fourth-order periodic boundary-value problems u⁗(t) + Mu(t) = λf (t, u(t)), t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, 4(2πnM4/T)4) and u(4)(t) − Mu(t) + λg(t, u(t)) = 0, t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, (2πM4/T)4);here g, f ∈ C([0,T] × [0, ∞), [0, ∞)), M is constant, and λ> 0 is a real parameter. The main results are based on a global bifurcation theorem.

Index Theory for Second Order Linear Hamiltonian Systems with L1 Coefficient Matrix Satisfying Generalized Periodic Boundary Value Conditions

In this paper, we will first establish an index theory for second order linear Hamiltonian systems with L1 coefficient matrix satisfying generalized periodic boundary value conditions. And then we will investigate nontrivial solutions for asymptotically linear second-order Hamiltonian systems and obtain some new results.

Existence results for systems of conformable fractional differential equations

In this article, we study the existence of solutions to systems of conformable fractional differential equations with periodic boundary value or initial value conditions. where the right member of the system is $L^{1}_{\alpha }$-caratheodory function. We employ the method of solution-tube and Schauder’s fixed-point theorem.

Periodic boundary value problems for two classes of nonlinear fractional differential equations

By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper.

Periodic impulsive fractional differential equations

AbstractThis paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.

Boundary value problems for semilinear differential inclusions of fractional order in a Banach space

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an -set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented.

Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation

We study the fractional complex Ginzburg-Landau equation with periodic initial boundary value condition in three spatial dimensions. The problem is discretized fully by Fourier Galerkin spectral method. The dynamical behavior of the resulting discrete system is examined. The existence of a global attractor is established, and the corresponding convergence is proved through the error estimates of the discrete solution. Numerical stability and convergence of the discrete scheme are proved.

Local well-posedness for Boussinesq approximation with shear dependent viscosities in 3D

In this paper, we prove the existence and uniqueness of regular weak solutions for a nonlinear Boussinesq system in a small time interval. The viscosity is assumed to be with a p-structure depending on the shear rate. We mainly work on 3D case with periodic boundary and initial value conditions. The W2,p-type regularity for velocity and H2-type regularity for temperature were obtained for 75<p≤2.

Systems of Hammerstein integral inclusions in Banach spaces with mixed monotone conditions

In this paper, we establish the existence of solutions to systems of Hammerstein integral inclusions under mixed monotonicity type conditions. Existence of solutions to systems of differential inclusions with initial value condition or periodic boundary value condition are also obtained. Our results rely on fixed point theorems for multivalued weak contractions on complete metric spaces endowed with a graph.

Existence of solutions for sub-linear or super-linear operator equations

We investigate solutions to superlinear or sublinear operator equations and obtain some abstract existence results by minimax methods. These results apply to superlinear or sublinear Hamiltonian systems satisfying several boundary value conditions including Sturm-Liouville boundary value conditions and generalized periodic boundary value conditions, and yield some new theorems concerning existence of solutions or nontrivial solutions. In particular, some famous results about periodic solutions to superlinear or sublinear Hamiltonian systems by Rabinowitz or Benci-Rabinowitz are special cases of the theorems.