A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations

We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.     

Parameter estimation in nonlinear evolution equations

We develop a convergence theory for identifying parameters in a general class of nonau-

tonomous nonlinear evolution equations. An application of this theory to a nonlinear reaction diffusion equation is presented.     

覆盖广义粗糙集的模糊性

在研究覆盖广义粗糙集的基础上,利用两个距离函数Hamming和Euclidean距离函数,结合模糊集的最近寻常集,引入了覆盖广义粗糙集模糊度的概念,给出了一种模糊度计算方法,并证明了该模糊度的一些重要性质。这些结果在覆盖广义粗糙集的理论研究和应用都发挥着一定作用。     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

Existence of solutions of nonlinear integrodifferential equations of sobolev type with nonlocal condition in Banach spaces

In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem.     

Positive periodic solutions for nonlinear repulsive singular difference equations

The existence and multiplicity of positive solutions are established to the periodic boundary value problems for repulsive singular nonlinear difference equations. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.     

基于泛系粗糙集等价算子上的粒计算模型

标准粗糙集使用等价类作为粒来描述概念.本文弱化对等价关系的要求, 将更广泛的粒计算模型建立到泛系粗糙集上去.本文通过对全域的分割和覆盖来诱导出泛系粗糙集上的粒计算模型.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

Multiple nonnegative solutions of nonlinear integral equations on compact and semi-infinite intervals

Existence results are presented which guarantee the existence of multiple nonnegative continuous solutions of a nonlinear integral equation on both a compact interval and semi-infinite interval     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].     

一类常宽"等腰梯形"

本文由对角线等于底边长的等腰梯形构造了一类新的常宽“等腰梯形”, 而著名的常宽凸集圆盘与Reuleaux 三角形为退化的特例. 我们还证明了关于这类常宽“等腰梯形” 面积的Blaschke-Lebesgue定理.     

Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations

In this paper, we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. The associated Green’s function for the boundary value problems is first given. The proofs of our main results are based upon the Leggett–Williams fixed point theorem. Finally, we give an example to demonstrate our result.     

The method of lines applied to nonlinear nonlocal functional differential equations

This paper discusses nonlinear functional differential equation in a real reflexive Banach space with nonlocal history condition. By using the method of lines, the existence and uniqueness of a strong solution are established. Finally, some applications of the abstract results are presented.