Quadratic functional equations in a set of Lebesgue measure zero

Let R$\mathbb{R}$ be the set of real numbers, Y   a Banach space and f:R→Y$f:\mathbb{R}\to Y$. We prove the Hyers–Ulam stability theorem for the quadratic functional inequality

‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤?$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \le ?$

for all (x,y)∈Ω$(x,y)\in \Omega$, where Ω⊂R²$\Omega \subset {\mathbb{R}}^2$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.     

Hyperstability of a logarithmic-type functional equation on restricted domains

Let $X$ be a real normed vector space and $f:(0,\infty )\to X$. In this paper, we prove the hyperstability of the logarithmic functional equation

$f\left(x^y\right)?yf\left(x\right)=0$

on $\Gamma$ of Lebesgue measure zero. More precisely, we prove that if $f:(0,\infty )\to X$ satisfies

$6f\left(x^y\right)?yf\left(x\right)6\le ?(x,y)$

for all $(x,y)\in {\Gamma }^{+}?\{(x,y):y>\alpha \left(x\right)\}[\mathrm{resp.}{\Gamma }^{?}?\{(x,y):0<y<\alpha \left(x\right)\}]$ of Lebesgue measure zero, where $\alpha :(0,\infty )\to (0,\infty )$ is an arbitrary given function and $?:(0,\infty )\times (0,\infty )\to [0,\infty )$ satisfies the condition $?(x,y)∕y\to 0$ as $y\to \infty$ [resp. $y\to 0$], then $f$ satisfies the functional equation $f\left(x^y\right)=yf\left(x\right)$ for all $x>0$ and$y>0$.     

Infinitely many homoclinic solutions for a class of damped vibration problems

In this paper, we deal with the existence of infinitely many homoclinic solutions for the damped vibration problems where A is an antisymmetry N × N constant matrix, we establish some new existence results to guarantee that the above system has infinitely many homoclinic solutions under more relaxed assumptions on W(t,x), which satisfies a kind of new subquadratic condition by using fountain theorem. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.     

关于上半连续集值映象的连续选择问题

本文进一步研究了上半连续集值映象的连续选择问题．首先,用一反例指出Ａｕｂｉｎ Ｊ、 Ｐ［１］中一结果不能成立．另外,对上半连续映象的σ－可选择性给出了另一逼近形式的结果．     

On the existence of solutions for discrete elliptic boundary value problems

Using critical point theory and some monotonicity results we consider the existence of solutions to a boundary value problem connected with the discrete elliptic equation with a positive parameter.     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.     

The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

We consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.      

Upper bounds for the number of solutions of a Diophantine equation

We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the -norm of certain exponential sums.

     

Characterization and representation of the lower semicontinuous envelope of the elastica functional

We characterize the lower semicontinuous envelope of the functional , defined on boundaries of sets , where κ_∂E denotes the curvature of ∂E and p>1. Through a desingularization procedure, we find the domain of and its expression, by means of different representation formulas.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

On existence and stability of solutions for higher order semilinear Dirichlet problems

We provide existence and stability results for semilinear Dirichlet problems with nonlinearity satisfying general growth conditions. We consider the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We show applications for the fourth order semilinear Dirichlet problem.     

On computational proofs of the existence of solutions to nonlinear parabolic problems

This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C⁰ finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.     

Existence and uniqueness of positive solutions to a linear transport equation in a metric space

The linear functional equation ∂_tz = L(z) − rz is considered. The linear operator L acts on a linear metric space of real functions z depending on t and on a parameter ω belonging to a subset of ^m. The existence and uniqueness to a nonnegative solution of the initial value problem is shown. An application to a kinetic equation is performed.     

New existence criteria of the solutions for a variational relation problem

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.     

On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity

We study the solvability of the Riemann-Hilbert problem for a generalized Cauchy-Riemann system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities; moreover, it does not depend continuously on the coefficients of the equation.     

Continuous dependence on the initial and flux functions for solutions of conservation laws

We prove the continuous dependence on the initial and flux functions for the entropy solutions to the Cauchy problem for conservation laws. Accordingly, we can show that the continuous dependence on the flux function for the entropy solutions depends only on the sup norm, not on the Lipschitz norm.