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.     

On existence and multiplicity of positive solutions to singular multi-point boundary value problems

The existence and multiplicity of positive solutions are established for the multi-point boundary value problem

     

On existence of positive solutions of Sturm-Liouville boundary value problems for a nonlinear singular differential system

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for a nonlinear singular differential system. Some well-known results in the literature are generalized and improved. An example is presented to illustrate the application of our main result.     

On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

     

Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Existence of positive periodic solutions of periodic boundary value problem for second order ordinary differential equations

In this paper the existence of positive solutions are obtained for a class of second order differential equations. The proof is based on the fixed point index theory in cones.     

On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions

We prove the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions. Obtained results yield positive periodic solutions of the equation on the whole real axis, provided that the coefficients are periodic.     

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.     

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.     

Extremal solutions of second order nonlinear periodic boundary value problems

We study a periodic boundary value problem for a nonlinear ordinary differential equation of second order when the nonlinearity is given by a Carathéodory function. We generalize the monotone iterative method to cover the fully nonlinear case.     

Study of singular boundary value problems for second order impulsive differential equations

This paper studies the existence of extremal solutions for a class of singular boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions and the monotone iterative technique, criteria of the existence of extremal solutions are established.     

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.     

Twist periodic solutions of second order singular differential equations

We study the twist periodic solutions of second order singular differential equations. Such twist periodic solutions are stable in the sense of Lyapunov and present much interesting dynamical features around them. The proof is based on the third-order approximation method. The estimates of periodic solutions of Ermakov-Pinney equations and the estimates on rotation numbers of Hill equations play an important role in the analysis.