Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales

Let T be a time scale such that 0, T ∈ T. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales where α ∈ Ctd（（O,T,[0,∞））,f∈ Ckd（[0,∞）×[0,∞））,β,γ ∈[0,∞）,ξi ∈（0,p（T）.b,ai∈ （0,∞） （for i = 1,..., m - 2） are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b 〉 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation （T = R） and difference equation （T = Z）.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions

     

Existence theorems for systems of nonlinear integro-differential equations

The existence of solutions for many systems of integro-differential equations discovered and generalized in the process of applying the Galerkin method for some initial-boundary value problems will be investigated in this paper. U. V. Le is currently supported by the Academy of Finland and the Emil Aaltonen Foundation.     

Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's

We give an example which says that Mizoguchi-Takahashi's fixed point theorem for set-valued mappings is a real generalization of Nadler's. We also give a very simple proof of Mizoguchi-Takahashi's theorem.     

Exact null controllability of semilinear integrodifferential systems in Hilbert spaces

Sufficient conditions for exact null controllability of the semilinear integrodifferential systems in Hilbert spaces are obtained. It is shown that under some natural conditions exact null controllability of the semilinear integrodifferential system is implied by the exact null controllability of the corresponding linear system with additive term. An application to partial integrodifferential equations is given.     

An asymptotic minimax theorem of order n

The asymptotic minimax theorem of LeCam and Hájek is refined by inclusion of terms of order n^−1/2. This renders more precise informations about the local properties of superefficient estimator-sequences.     

Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions

In this article, by using theory of linear evolution system and Schauder fixed point theorem, we establish a sufficient result of exact null controllability for a non-autonomous functional evolution system with nonlocal conditions. In particular, the compactness condition or Lipschitz condition for the function g in the nonlocal conditions appearing in various literatures is not required here. An example is also provided to show an application of the obtained result.     

Positive solutions of second-order differential systems with nonhomogeneous boundary conditions

This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.     

具有无限时滞的二阶微分方程解的存在性

§ 1.　IntroductionRecently ,thedifferentialequationswithdeviatingargumentswereusuallydiscussed(see[1 ],[4],[5 ]) .In [1 ],AGARWALRPandO’REGANDconsideredequationy″(t) =f(t,y(t) ,y(σ(t) ) ) ,　a.e .t∈ [0 ,1 ]y(t) =ψ(t) ,　　　　　　　　t∈ [-r ,0 ]y( 1 ) =a ,( )andtheydiscussedtheexistenceofatleastonesolutionforequation ( ) .Inthispaper ,weconsideramoregeneralequation-x″(t) =f(t ,xt) ,　t∈ [0 ,1 ]x(t) =ψ(t) ,　　　　t∈ ( -∞ ,0 ]x( 0 ) =x( 1 ) =0 ,( 1 .1 )andsomeexistencetheor…     

On solvability of a parabolic system arising in physical oceanography

As many equations in fluid mechanics, the thermocline equations, in standard notation, are composed of quasi-linear, first-order partial differential equations. The aim of this paper is to establish an existence, uniqueness result for the solution of such a problem.     

A generalized section theorem and a minimax inequality for a vector-valued mapping 1

A generalized Fan's section theorem has proposed by replacing convexity assumptions with merely topological properties. A generalized reformulation of Browder's fixed point theorem has derived. The Minimax Inequalities for vector-valued mapping in an ordered Banach space have established without the convexity and with convexity, respectively.     

一类具时滞的Rayleigh型泛函微分方程反周期解的存在性

应用Leray Schauder不动点定理,研究了一类具时滞的Rayleigh型泛函微分方程： x″（t）＋f（x′（t））＋g（x（t-τ（t）））=e（t）的反周期解问题,得到了反周期解存在的新的结果。     

Stabilization of coupled systems of quadratic integral equations of Chandrasekhar type

We present existence theorems for coupled system of quadratic integral equations of generalized Chandrasekhar type which has numerous application (cf. 2 , 3 , 4 and 13 ). Also, the asymptotic stability of solutions will be considered.     

The remainder term in asymptotic summation of linear difference systems

Quantitative estimates for the remainder terms in the asymptotic summation of linear difference systems are derived. For example, it is shown that any decay in excess of summability is passed on to the remainder.     

A non-oscillation theorem for differential matrix systems

We extend old results dealing with non-oscillations of Wintner (1951) and Opial (1957) from scalar to vector differential equations with symmetric coefficients using the Schauder fixed point theorem.     

二次周期系数微分方程的周期解

给出利用Schauder不动点定理求一类二次周期系数微分方程的周期解的一种方法,得到较好结果.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.