Positive solutions of impulsive boundary value problems for first-order nonlinear functional differential equations

In this paper, we study the problem on the existence of positive solutions for a class of impulsive periodic boundary value problems of first-order nonlinear functional differential equations. By using the fixed point theorem in cones and some analysis techniques, we present some sufficient conditions which guarantee the existence of one and multiple positive solutions for the impulsive periodic boundary value problems. Our results generalize and improve some previous results. Moreover, our results show that positive solutions for the impulsive periodic boundary value problems may be yielded completely by some proper impulsive conditions (see Example 4.1 and Remark 4.2 in Sect. 4), and also implies that proper impulsive conditions are of great significance to simulate processes, optimal control, population model and so on.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

Solutions of periodic boundary value problems for second-order nonlinear differential equations via variational methods

By introducing a variational framework for a class of second order nonlinear differential equations with non-separated periodic boundary value conditions, some results on the existence of non-trivial, positive and negative solutions of the problems are obtained. Some results by Atici-Guseinov, Graef-Kong, etc. obtained by topological degree methods are extended. The resonant case of the problems where the nonlinearities are unbounded and satisfy Ahmad-Lazer-Paul type conditions is also considered.     

Positive solutions for third order semipositone boundary value problems

We obtain some sufficient conditions for the existence of positive solutions of a third order semipositone boundary value problem with a multi-point boundary condition. Applications of our results to some special problems are also discussed.     

Existence and Comparison Results for Difference φ-Laplacian Boundary Value Problems with Lower and Upper Solutions in Reverse Order

This paper is devoted to the study of the existence and comparison results for nonlinear difference φ-Laplacian problems with mixed, Dirichlet, Neumann, and periodic boundary value conditions. We deduce existence of extremal solutions of periodic and Neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order. We prove the optimality of some assumptions in φ.     

Positive solutions to a system of second-order nonlocal boundary value problems

In this paper we study the existence of positive solutions to a system of second-order nonlocal boundary value problems by using fixed point index theory in a cone. Our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems, and our boundary value conditions are expressed in terms of possibly nonlinear functions of Riemann–Stieltjes integrals, thus generalizing and unifying the boundary value conditions in the literature. Therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature.     

Existence and multiplicity of solutions to even order ordinary differential equations via variational methods

In this paper, we study the existence and multiplicity of solutions for two-point boundary value problems for even order ordinary differential equation. By using the critical point theory and the property of projective operator, we establish some conditions on the nonlinearity which guarantee that the nonlinear Hammerstein integral equation with quasi-positive definite kernel has at least one solution, two nontrivial solutions and finite pairs of solutions, respectively. As an application, we get the corresponding results for two-point boundary value problems for even order ordinary differential equation. Moreover, an example is presented to illustrate our main results.     

二阶离散周期边值问题的单个和多个正解

该文利用锥不动点理论获得了一类离散的周期边值问题的单个和多个正解存在的理论, 进而获得了带有周期边值条件的非线性差分方程在离散片段上的新结果.     

On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations

The existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations. The arguments are based only upon the positivity of the Green's functions and the Krasnosel'skii fixed point theorem. As an example, a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity.     

First order functional differential equations with periodic boundary conditions

In this article, the authors prove an existence theorem for periodic boundary value problems for first order functional differential equations (FDE) in Banach algebras under mixed generalized Lipschitz and Carathéodory conditions. The existence of extremal positive solutions is also proved under certain monotonicity conditions. An example illustrating the results is included.     

一类二阶三点边值问题单调正解的存在性

利用范数形式的锥上不动点定理,研究了一类二阶微分方程三点边值问题单调正解的存在性．分别给出了齐次和非齐次边界条件下的三点边值问题单调正解存在的充分条件,确定了解曲线的凹凸性,并且给出了一个应用实例．     

Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions

In this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments. We discuss our problem both for advanced or delayed arguments. We establish sufficient conditions under which such problems have positive solutions. To obtain the existence of multiple (at least three) positive solutions, we use a fixed point theorem due to Avery and Peterson. An example is also included to illustrate that corresponding assumptions are satisfied. The results are new.     

Existence of solutions to a discrete fourth order periodic boundary value problem

This paper is concerned with periodic boundary value problems for a fourth order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are given for the existence of at least one solution, two solutions, and nonexistence of solutions. Our conditions do not involve the primitive function of the nonlinear term. Examples are provided to illustrate the applicability of the results.     

Existence of Positive Solutions for a Nonlinear Second Order Periodic Boundary Value Problem

By using the first eigenvalue corresponding to the relevant linear operator and the topological degree theorem, sufficient conditions for the existence of positive solutions for a nonlinear second order periodic boundary value problem are given. Our results improve and generalize some preliminary works.     

Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations

In this article, we study a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. We obtain sufficient conditions for existence and uniqueness of positive solutions. We use the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem for uniqueness and existence results. As in application, we provide an example to illustrate our main results.     

Duffing型方程组的边界值问题的解的存在性

给出了带Dirichlet边界条件、Neumann边界条件和周期边界条件的Duffing型方程组的两点边界值问题的解的几个存在性定理。     

拟线性系统周期边值问题

The existence and uniqueness results about quasilinear periodic boundary value problems are established by using the global inverse function theorem and the result about the existence and uniquencess of periodic solutions for the periodic boundary value problem of nonhomogeneous linear periodic system.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Studies on anti‐periodic boundary value problems for two classes of special second order impulsive differential equations

We consider anti‐periodic boundary value problems for two classes of special second order impulsive differential equations. On the basis of several important impulsive differential inequalities, by using the monotone iterative technique coupled with lower and upper solutions, we obtain sufficient conditions to guarantee the existence and uniqueness of solutions for such problems. Further, we give two examples to illustrate our conclusions. Copyright © 2013 John Wiley & Sons, Ltd.     

The classification of self-adjoint boundary conditions: Separated, coupled, and mixed

There are three basic types of self-adjoint regular and singular boundary conditions: separated, coupled, and mixed. For even order problems with real coefficients, one regular endpoint and arbitrary deficiency index d, we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square-integrable solutions for some real value of the spectral parameter. In the separated case our construction yields non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available.