Anti-periodic Problems Research Articles

A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on L^{p}-conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a(t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS.

Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

Solvability of inclusions of Hammerstein type

A fairly general continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps is set forth. This result is then used to establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of $m$-accretive mutivalued differential equations, the anti-periodic problem for semilinear differential inclusions, abstract integral inclusions of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusions.

Green’s functions and spectral theory for the Hill’s equation

The aim of this paper is to show certain properties of the Green’s functions related to the Hill’s equation coupled with various two point boundary value conditions. We will obtain the expression of the Green’s function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Green’s function related to periodic ones.As a consequence we will prove suitable results in spectral theory and deduce some comparison results for the solutions of the Hill’s equation with various boundary value conditions.

Asymptotic Expansions of Eigenvalues of Periodic and Antiperiodic Boundary Problems for Singularly Perturbed Second Order Differential Equation with Turning Points

For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solved

Anti-periodic extremal problems for a class of nonlinear evolution inclusions in R N

The aim of this paper is to establish the existence of anti-periodic solutions to the following nonlinear anti-periodic problem: a.e. , , in where denotes the extremal point set of the multifunction , and is a nonlinear map from to . Sufficient conditions for the existence of extremal solutions are presented. Also, we prove that the extremal point set of this problem is compact in and dense in the solution set of nonlinear evolution problems with a convex valued perturbation which is multivalued. We apply our results on the control system with a priori feedback.

On an analog of periodic boundary value problems for the Poisson equation in the disk

We consider a periodic and an antiperiodic problem for the Poisson equation in the unit disk and prove their well-posedness. The possibility of separation of variables is justified. We construct the Green functions of these problems in closed form and obtain an integral representation of the solution. The problems are self-adjoint, and we construct all eigenvalues and eigenfunctions in closed form.

Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the formu′(t)=A(t)u(t)+f(t,u(t)),t∈R,u(t+T)=-u(t),t∈R, where(At)t∈R(possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach spaceX. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on(At)t∈R, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

Application of fixed point theorems for multivalued maps to anti-periodic problems of fractional differential inclusions

This paper deals with the existence and dimension of the solution set for an anti-periodic boundary value problem of fractional differential inclusions. Our results rely on Wegrzyk?s fixed point theorem and a result on the topological dimension of the set of fixed points for multivalued maps.

Anti-periodic problems for semilinear partial neutral evolution equations

Anti-periodic problems for semilinear partial neutral evolution equations

Anti-periodic solutions for nonlinear evolution equations

Abstract In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem { x ˙ + A ( t , x ) + B x = f ( t ) a.e. t ∈ R , x ( t + T ) = − x ( t ) , where A ( t , x ) is a nonlinear map and B is a bounded linear operator from R N to R N . Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W 1 , 2 ( I , R N ) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.

Anti-periodic solutions for differential inclusions in Banach spaces and their applications

We deal with anti-periodic problems for differential inclusions with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings. We then apply our results to evolution hemivariational inequalities and parabolic equations with nonmonotone discontinuities, which generalize and extend previously known theorems.

Anti-periodic solutions for semilinear evolution equations in Banach spaces

In this paper, we study the anti-periodic problem for semi-linear evolution equations in reflexive Banach spaces. Several existence results are obtained under suitable conditions.

Anti-periodic solutions for evolution equations associated with maximal monotone mappings

In this work, we study the anti-periodic problem for a nonlinear evolution inclusion where the nonlinear part is an odd maximal monotone mapping and the forcing term is an anti-periodic mapping. Several existence results are obtained under suitable conditions. An example is presented to illustrate the results.

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.

On the Floquet problem for second-order Marchaud differential systems

Solutions in a given set of the Floquet boundary value problem are investigated for second-order Marchaud systems. The methods used involve a fixed point index technique developed by ourselves earlier with a bound sets approach. Since the related bounding (Liapunov-like) functions are strictly localized on the boundaries of parameter sets of candidate solutions, some trajectories are allowed to escape from these sets. The main existence and localization theorem is illustrated by two examples for periodic and anti-periodic problems.

Periodic boundary value problems for nonlinear first order ordinary differential equations

We prove several new existence results for a nonlinear anti-periodic first order problem using a Leray-Schauder alternative. Two definitions of lower and upper solutions are presented and we show in this paper the validity of the lower and upper solution method. Also, we give a method to generate a sequence of approximate solutions converging to a solution of the anti-periodic problem. ∗First and second authors were supported in part by D.G.E.S.I.C. (Spain), project PB97 – 0552. †This is the preprint version of the paper published in Journal of Mathematical Inequalities and Applications, Vol. 6 (2003) 477–485. ‡Corresponding author