Periodic solutions of second order non-autonomous singular dynamical systems

In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray-Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder's fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. Recent results in the literature are generalized and significantly improved.     

A rational spectral collocation method for solving a class of parameterized singular perturbation problems

A new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two-point boundary value problems with one boundary layer. By means of the sinh transformation, the original Chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem. The results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation. Numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.     

Doubly periodic solutions to a coupled telegraph system

While the nonlinearities are bounded below, we establish the existence and multiplicity results of doubly periodic solutions for a coupled nonlinear telegraph system with a parameter. The proof is based on a well known fixed theorem in a cone.     

Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals

Unbounded upper and lower solutions theories are established for the Sturm–Liouville boundary value problem of a second order ordinary differential equation on infinite intervals. By using such techniques and the Schäuder fixed point theorem, the existence of solutions as well as the positive ones is obtained. Nagumo conditions play an important role in the nonlinear term involved in the first-order derivatives.     

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.     

The existence of positive solutions for second-order multi-point BVPs with the first derivative

By using the fixed-point index theory, several theorems of existence of positive solutions for second-order m$m$-point boundary value problem with the first derivative are obtained, where the nonlinear term may be noncontinuous. An example illustrating the main results is given.     

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q$p:q$ resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θ is rational.     

Positive solutions for third-order variable-coefficient nonlinear equation with weak and strong singularities

By application of Green's function and a fixed-point theorem, i.e. Leray–Schauder alternative principle, we establish some new existence results of positive periodic solutions for nonlinear third-order singular equation with variable-coefficient, these results can be applied to study the case of a strong singularity as well as the case of a weak singularity.     

Multiple positive solutions for nonlinear periodic problems

We consider a nonlinear periodic equation driven by the scalar p-Laplacian and with a Caratheodory asymptotically (p−1)-linear nonlinearity. Using variational methods coupled with suitable truncation techniques, we show that the problem has at least two positive solutions. For the semilinear case (p=2), using Morse theory we show that the problem has three distinct positive solutions.     

Multiplicity of sign-changing solutions for some four-point boundary value problem

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f$f$.     

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we present existence results for solutions of nonlinear second-order boundary value problems with impulses. Our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution. Moreover we allow functional dependence on the solution. Our existence results follow from the existence of a pair of well ordered lower and upper solutions. We generalize earlier results of Cabada and Tomec?ek, allowing more general compatible boundary conditions, impulses and φ-Laplacian equations.     

Triple positive solutions for boundary value problems on infinite intervals

This paper deals with the existence of triple positive solutions for Sturm–Liouville boundary value problems of second-order nonlinear differential equation on a half line. By using a fixed point theorem in a cone due to Avery–Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term.     

Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations

In this paper, we study a linear Sturm-Liouville impulsive problem and a nonlinear Sturm-Liouville impulsive problem. By applying a new approach via critical point theory and variational methods, the existence results of positive solutions are obtained.     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations

By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Received: 10 May 2004     

Triple positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism

In this paper, we investigate the existence of triple positive solutions for nonlinear differential equations boundary value problems with increasing homeomorphism and positive homomorphism operator. By using fixed point theorems in cones, we establish results on the existence of three positive solutions with suitable growth conditions imposed on the nonlinear term. As applications, two examples are given to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.