Periodic Boundary Value Conditions Research Articles

Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions

This work is devoted to the study of first order linear problems with involution and periodic boundary value conditions. We first prove a correspondence between a large set of such problems with different involutions to later focus our attention to the case of the reflection. We study then different cases for which a Greenʼs function can be obtained explicitly and derive several results in order to obtain information about its sign. Once the sign is known, maximum and anti-maximum principles follow. We end this work with more general existence and uniqueness of solution results.

Positive solutions for vector differential equations

In this paper, we are concerned with the existence and multiplicity of positive periodic solutions for first-order vector differential equations. By using the Leray-Schauder alternative theorem and the Kransnosel’skii fixed point theorem, we show that the differential equations under the periodic boundary value conditions have at least two positive periodic solutions.

The Fixed Point Problem for Systems of Coordinate-Wise Uniformly Monotone Operators and Applications

We study the fixed point problem for a system of multivariate operators that are coordinate-wise uniformly monotone, in the setting of quasi-ordered sets. We show that this problem is equivalent to the fixed point problem for a mixed monotone operator that can be explicitly constructed. As a consequence, we obtain a criterion for the existence and uniqueness of solution to the considered problem, together with an approximating iterative scheme, in the setting of partially ordered metric spaces. As an application, we investigate a new abstract multidimensional fixed point problem. To validate our results, we also provide an application to a first-order differential system with periodic boundary value conditions.

Eigenvalue criteria for existence of positive solutions of impulsive differential equations with non-separated boundary conditions

In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.

Existence Results for a Class of Fractional Differential Equations with Periodic Boundary Value Conditions and with Delay

We discuss the existence and uniqueness of solution for two types of fractional order ordinary and delay differential equations. Fixed point theorems are the main tool used here to establish the existence and uniqueness results. First we use Banach contraction principle to prove the uniqueness of solution and then Krasnoselskii's fixed point theorem to show the existence of the solution under certain conditions in a Banach space.

Comparison results for first order linear operators with reflection and periodic boundary value conditions

This work is devoted to the study of the first order operator x′(t)+mx(−t) coupled with periodic boundary value conditions. We describe the eigenvalues of the operator and obtain the expression of its related Green’s function in the non resonant case. We also obtain the range of the values of the real parameter m for which the integral kernel, which provides the unique solution, has constant sign. In this way, we automatically establish maximum and anti-maximum principles for the equation. Some applications to the existence of nonlinear periodic boundary value problems are shown.

Global attractor for a nonlinear model with periodic boundary value condition

In this paper, we consider the existence of a global attractor for a nonlinear model with periodic boundary value condition. Based on the iteration technique for regularity estimates and the classical existence theorem of global attractors, we prove that the equation possesses a global attractor on some affine space of H^k ( 0\leq k<+\infty ).

Existence of positive solutions for nth-order periodic difference equations

This paper is devoted to the study of difference equations coupled with periodic boundary value conditions. We deduce the existence of at least one positive solution provided that the nonlinear part of the equation satisfies some monotonicity assumptions and the existence of a positive upper solution. The result is obtained from a new fixed point theorem based on the classical Krasnoselskii's cone expansion/contraction theorem and the constant sign properties of the related Green's function.

On a class of singular Sturm–Liouville periodic boundary value problems

Keeping in mind the singular model for the periodic oscillations of the axis of a satellite in the plane of the elliptic orbit around its center of mass, we give sufficient conditions for the solvability of a class of singular Sturm–Liouville equations with periodic boundary value conditions. To this end, under a suitable change of variables, we present a new existence result for problems defined in the real half-line.

Solutions of periodic boundary value problems for second-order nonlinear differential equations via variational methods

By introducing a variational framework for a class of second order nonlinear differential equations with non-separated periodic boundary value conditions, some results on the existence of non-trivial, positive and negative solutions of the problems are obtained. Some results by Atici–Guseinov, Graef–Kong, etc. obtained by topological degree methods are extended. The resonant case of the problems where the nonlinearities are unbounded and satisfy Ahmad–Lazer–Paul type conditions is also considered.

Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities

This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii's fixed point theorems in cones. An exhaustive study of the Green's function related to a linear fourth order problem is done.

Periodic boundary value problem for the first order functional differential equations with impulses

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.

Existence of Periodic Solution for a Nonlinear Fractional Differential Equation

We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.

Existence of Solutions for Second-Order Nonlinear Impulsive Differential Equations with Periodic Boundary Value Conditions

We are concerned with the nonlinear second-order impulsive periodic boundary value problem , , , , , , new criteria are established based on Schaefer's fixed-point theorem.

Existence and multiplicity of solutions to 2 mth-order ordinary differential equations

In this paper, the existence and multiplicity of solutions are obtained for the 2 mth-order ordinary differential equation two-point boundary value problems ( − 1 ) m u ( 2 m ) ( t ) + ∑ i = 1 m ( − 1 ) m − i a i u ( 2 ( m − i ) ) ( t ) = f ( t , u ( t ) ) for all t ∈ [ 0 , 1 ] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, a i ∈ R for all i = 1 , 2 , … , m . Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u + ∑ i = 1 m a i T i u = T m f u , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2 mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

POSITIVE EQUILIBRIUM SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS

The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions: “subsolution ≰ supersolution”, the existence and stability/instability of equilibrium solutions are obtained.

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x ′ ( t ) + f ( t , x ( t ) ) = 0 , a.e. t ∈ [ 0 , ω ] under the periodic boundary value condition x ( 0 ) = x ( ω ) . Here ω is a positive constant, and f : [ 0 , ω ] × R n → R n is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.

Periodic boundary value problems for delay differential equations with impulses

Existence and approximation of solutions for an impulsive delay differential equation with periodic boundary value conditions are presented. We use comparison principles and monotone iterative techniques.

Existence and Comparison Results for Difference φ-Laplacian Boundary Value Problems with Lower and Upper Solutions in Reverse Order

This paper is devoted to the study of the existence and comparison results for nonlinear difference φ-Laplacian problems with mixed, Dirichlet, Neumann, and periodic boundary value conditions. We deduce existence of extremal solutions of periodic and Neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order. We prove the optimality of some assumptions in φ.

Differential inequalities for functional perturbations of first-order ordinary differential equations

The theory of differential inequalities plays a central role in the qualitative and quantitative study of differential equations. In this paper, we present several comparison results for a class of functional differential equations of first order with periodic boundary value conditions. The inequalities obtained are, generally speaking, of the following type: Pv ≤ 0 implies that v ≤ 0, where P is a functional differential operator subject to some boundary conditions, and v is an element of a prescribed space of functions. We first obtain several new results for the linear problem. Then, we consider a nonlinear differential equation as a functional perturbation of the original differential equation and give different comparison results. Our results improve and generalize previous estimates described in the literature.