Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

Banach空间中半线性混合型发展方程的解

在不要求C0-半群为紧半群的前提下．利用函数e^-λt（其中λ〉0是常数）和Monch不动点定理,在更广泛的条件下,得到了Banach空间中一类半线性混合型发展方程初值问题的整体mild解和正mild解,本质上改进和推广了已有相关结果．     

The Pullback Asymptotic Behavior of the Solutions for 2D Nonautonomous G-Navier-Stokes Equations

The pullback asymptotic behavior of the solutions for 2D Nonautonomous G-Navier-Stokes equations is studied, and the existence of its $L^2$-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.     

Stability results for properly quasi convex vector optimization problems

In this paper, we discuss the stability of the sets of (weak) minimal points and (weak) efficient points of vector optimization problems. Assuming that the objective functions are (strictly) properly quasi convex, and the data ofthe approximate problems converges to the data of the original problems in the sense of Painlevé–Kuratowski, we establish the Painlevé–Kuratowski set convergence of the sets of (weak) minimal points and (weak) efficient points of the approximate problems to the corresponding ones of original problem. Our main results improve and extend the results of the recent papers.     

The initial value problems of first order impulsive differential equations in Banach spaces

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Banach空间中二阶常微分方程初值问题解的存在唯一性

用单调迭代方法给出了二阶常微分方程初值问题的解的存在唯一性的结果.     

二阶非线性积-微分方程的初值问题的上下解方法

建立了无限区间上二阶积－微分方程的比较定理,用上下解方法证明了无限区间上的Ｂａ－ｎａｃｈ空间二阶非线性积－微分方程的初值问题的解的存在性．     

An Isotonic Regression Problem for Infinite-Dimensional Parameters

We consider an infinite-dimensional isotonic regression problem which is an extension of the suitably revised classical isotonic regression problem. Given p-summable data, for p finite and at least one, there exists an optimal estimator to our problem. For p greater than one, this estimator is unique and is the limit in the p-norm of the sequence of unique estimators in canonical finite-dimensional truncations of our problem. However, for p equal to one, our problem, as well as the finite-dimensional truncations, admit multiple optimal estimators in general. In this case, the sequence of optimal estimator sets to the truncations converges to the optimal estimator set of the infinite problem in the sense of Kuratowski. Moreover, the selection of natural best optimal estimators to the truncations converges in the 1-norm to an optimal estimator of the infinite problem.     

Characterizing planarity using theta graphs

A theta graph is a homeomorph of K_2,3. In an embedded planar graph the local rotation at one degree-three vertex of a theta graph determines the local rotation at the other degree-three vertex. Using this observation, we give a characterization of planar graphs in terms of balance in an associated signed graph whose vertices are K_1,3 subgraphs and whose edges correspond to theta graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 17–20, 1998     

Shape sensitivity for the Laplace-Beltrami operator with singularities

This paper considers shape sensitivity analysis for the Laplace-Beltrami operator formulated on a two-dimensional manifold with a fracture. We characterize the shape gradient of a functional as a bounded measure on the manifold and decompose it into a “distributed gradient” supported on the manifold, plus a singular part that we derive as the limit of a “jump” through the crack and Dirac measures at the crack extremities. The important point is that we introduce a technique that is not dimension dependent, and makes no use of classical arguments such as the maximum principle or continuation uniqueness. The technique makes use of a family of envelopes surrounding the fracture which enable us to relax certain terms and to overcome the lack of regularity resulting from the presence of the fracture. We use the min-max differentiation in order to avoid taking the derivative of the state equation and to manage the crack's singularities. Therefore, we write the functional in a min-max formulation on a space which takes into account the hidden boundary regularity established by the tangential extractor method.