Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods

This paper uses critical point theory and variational methods to investigate the multiple solutions of boundary value problems for second order impulsive differential equations. The conditions for the existence of multiple solutions are established. An example is constructed to illustrate the proposed result.     

Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Fast homoclinic solutions for some second order non-autonomous systems

This paper is concerned with the existence of homoclinic solutions for the following second order non-autonomous system

(FHS)     

Homoclinic solutions of some second order non-autonomous systems

In this paper we investigate the existence of homoclinic solutions for the following second order non-autonomous system

where A is an antisymmetric constant matrix, is a symmetric and positive definite matrix for all , W(t,q)=a(t)V(q) such that is a continuous function and . Assuming that V(q) is subquadratic as q→+∞ and some technical assumptions on A and L, we establish two existence criteria to guarantee that (DS) has at least one nontrivial homoclinic solution by using a standard minimizing argument. Besides that, in some particular case, for the first time the uniqueness of homoclinic solutions of (DS) is also obtained. Recent results in the literature are generalized and significantly improved.     

Existence of multiple solutions for a perturbed discrete anisotropic equation

In this paper, we are concerned with the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. The approach is based on variational methods. We also provide two examples in order to illustrate the results.     

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.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

Existence of three weak solutions for a perturbed anisotropic discrete Dirichlet problem

In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results.     

Existence of multiple positive periodic solutions for nonlinear functional difference equations

In this paper, we investigate the existence of multiple positive periodic solutions to a class of functional difference equations. We answer the open problems proposed by Y. Raffoul in [Electron. J. Differential Equations 55 (2002) 1-8] and the conditions obtained improve some recent results established there.     

Multiple positive solutions of singular problems by variational methods

The purpose of this paper is to use an appropriate variational framework to obtain positive solutions of some singular boundary value problems.

     

Multiple positive solutions for a semilinear equation with prescribed singularity

Variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.     

Multiplicity of positive solutions to period boundary value problems for second order impulsive differential equations

This paper deals with the existence and multiplicity of positive solutions to second order period boundary value problems with impulse effects. The proof of our main results relies on a well-known fixed point theorem in cones. The paper extends some previous results and reports some new results about impulsive differential equations.     

Existence of positive solutions for some problems with nonlinear diffusion

In this paper we study the existence of positive solutions for problems of the type

where is the -Laplace operator and . If , such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases , and , respectively. Also, some systems of equations are considered.

     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.     

Multiplicity results for positive solutions of some semi-positone three-point boundary value problems

In this paper, some multiplicity results for positive solutions of some singular semi-positone three-point boundary value problem be obtained by using the fixed point index method.