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 periodic solutions to second-order Hamiltonian systems with potential indefinite in sign

We have presented some new results for periodic solutions to a class of Hamiltonian systems with potential indefinite in sign. These results improve and correct some known results in the literature.     

Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.     

Generalized synchronization in linearly coupled time periodic systems

We consider the synchronization of a network of linearly coupled and not necessarily identical oscillators. We present an approach to the existence of the synchronization manifold which is based on some results developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that they are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski's principle. Several examples of application are presented.     

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations

We establish the existence, multiplicity and nonexistence of positive periodic solutions of systems of second order ordinary differential equations.     

Nagumo type existence results of Sturm-Liouville BVP for impulsive differential equations

In this paper, the method of upper and lower solutions and the Schauder degree theory are employed in the study of Sturm-Liouville boundary value problems for second order impulsive differential equations. We obtain the existence of at least three solutions to the problem under the assumption that the nonlinear term f satisfies a Nagumo condition with respect to the first order derivative.     

Existence of periodic solutions for a system of delay differential equations

In this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two-neuron network model of Hopfield type with time-delayed connections between the neurons. We first examine the local stability of the trivial solution, propose some sufficient conditions for the uniqueness of equilibria and then apply the Poincaré-Bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions. In addition, a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given. Numerical examples are provided to support the theoretical analysis.     

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.     

The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces

In this paper, the existence of solutions of infinite boundary value problems for first-order impulsive differential systems is obtained by means of the Schauder fixed point theorem in a Banach space.     

Nagumo and Landesman-Lazer type conditions for nonlinear second order systems

We study the existence of solutions for a nonlinear second order system of ordinary differential equations under various boundary conditions. Assuming suitable Nagumo type conditions we prove the existence of at least one solution applying the method of upper and lower solutions. Moreover, using topological degree methods we prove the existence of solutions under Landesman-Lazer type conditions.     

Multiple positive solutions of singularly perturbed differential systems with different orders

In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing.     

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.     

Periodic solutions of second order singular coupled systems

We establish the existence of periodic solutions for the second order non-autonomous singular coupled systems

     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

The existence of countably many positive solutions for singular multipoint boundary value problems

In this paper, we study the existence of countably many positive solutions for a singular multipoint boundary value problem. By using fixed-point index theory and the Leggett-Williams’ fixed-point theorem, sufficient conditions for the existence of countably many positive solutions are established.     

The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects

In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results.     

Boundedness of solutions for singular Hamiltonian differential systems with applications to deficiency indices

The paper gives boundedness estimation of solutions for singular Hamiltonian differential systems. As corollaries, limit-circle criteria are given and improve some previous results.     

Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem

We study the existence of positive solutions of the nonlinear boundary value problem

     

Existence and uniqueness of solutions of piecewise nonlinear systems

In this paper, we consider the existence and the uniqueness of solutions of piecewise nonlinear systems. We will present some necessary and sufficient conditions for the existence and the uniqueness of solutions of this class of systems.     

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$f$.