Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

A numerical solution to nonlinear multi‐point boundary value problems in the reproducing kernel space

In this paper, a new numerical algorithm is provided to solve nonlinear multi‐point boundary value problems in a very favorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel function is discussed in detail. The theorem proves that the approximate solution and its first‐ and second‐order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear multi‐point boundary value problems. Copyright © 2010 John Wiley & Sons, Ltd.     

带P-Laplacian算子的四点四阶奇异边值问题的对称正解

本文主要利用上下解方法和Schauder不动点定理,在更广泛的条件下研究了一类带PLaplacian算子的四点四阶奇异边值问题的对称正解的存在性.克服了对非线性微分算子[φp(u″)]″Fredholm抉择定理和极大值原理不能使用的困难,改进并推广了最近的一些已知结果.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

非紧集上不连续函数的Ky Fan不等式及其等价形式和应用

证明了非紧集上不具有任何连续性的函数弱Ky Fan点的存在性,给出了在函数只具非常弱的连续性和凸性条件下非紧集上Ky Fan不等式解的存在性,并给出它的两种等价形式.作为应用:(1)得到Ky Fan截口定理和Fan-Browder不动点定理的推广;(2)应用于博弈理论,得到几个新的Nash平衡存在性定理.     

集值映射的对称向量拟均衡问题

研究向量集值映射的拟均衡问题的有效解,利用数值化方法与不动点定理,得到解的存在性定理.作为应用,得到广义向量鞍点,向量变分不等式与向量互补问题的存在性定理.     

关于良定问题

本文应用有限理性模型M,对非线性问题的良定性进行了统一的研究,对最优化、多目标最优化、非合作博弈和广义博弈得到了一些新的良定性结果.     

On cyclic Meir-Keeler contractions in metric spaces

Cyclic Meir-Keeler contractions are considered under the recently introduced WUC and HW properties on pairs of subsets of metric spaces. We show that, in contrast with previous results in the theory, best proximity point theorems under these properties do not directly extend from cyclic contractions to cyclic Meir-Keeler contractions. We obtain, however, a positive result for cyclic Meir-Keeler contractions under additional properties which is shown to be an extension of already existing results for cyclic contractions. Moreover, we give examples supporting the necessity of our additional conditions.     

Geodesic Ptolemy spaces and fixed points

In this paper we study the regularity of geodesic Ptolemy spaces and apply our findings to metric fixed point theory. It is an open question whether such spaces with a continuous midpoint map are CAT(0) spaces. We prove that if a certain uniform continuity is imposed on such a midpoint map then these spaces, if complete, are reflexive (that is, the intersection of decreasing families of bounded closed and convex subsets is nonempty) and that bounded sequences have unique asymptotic centers. These properties will then be applied to yield a series of fixed point results specific to CAT(0) spaces.     

Multiple limit cycle bifurcations of the FitzHugh–Nagumo neuronal model

In this paper, we complete the global qualitative analysis of the well-known FitzHugh–Nagumo neuronal model. In particular, studying global limit cycle bifurcations and applying the Wintner–Perko termination principle for multiple limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.