Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when λ > λ₁, the problem has extremal solutions of constant sign and when λ > λ₂ it has also a nodal (sign-changing) solution. Here λ₁ < λ₂ are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p = 2) we produce two nodal solutions.     

Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when ?? > ??₁, the problem has extremal solutions of constant sign and when ?? > ??₂ it has also a nodal (sign-changing) solution. Here ??₁?<???₂ are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p?=?2) we produce two nodal solutions.     

On KP-II type equations on cylinders

In this article we study the generalized dispersion version of the Kadomtsev–Petviashvili II equation, on T×R$\mathbb{T}\times \mathbb{R}$ and T×R²$\mathbb{T}\times {\mathbb{R}}^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.     

On hypergeometric type partial differential equations

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 1, pp. 82–90, January–March, 1995.     

On trigonometric functional equations of rectangular type

Summary Letf, G₁ × G₂ C, where G _i (i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) (1) andf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) (2) are determined assuming thatf satisfies the conditionf(x ₁y₁z₁, x₂) = f(x₁z₁y₁, x₂), f(x₁, x₂y₂z₂) = f(x₁, x₂z₂y₂) (C) for allx _i, y_i, x_i G_i (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x ₁ + y₁, x₂ + y₂) + f(x₁ + y₁, x₂ – y₂) + f(x₁ – y₁, x₂ + y₂) + f(x₁ – y₁, x₂ – y₂) = 4f(x₁, x₂) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968.     

On nonlinear Volterra equations of convolution type

Weakly nonlinear and strongly nonlinear convolution-type Volterra equations u(x) = (K * ϕ(u))(x) are studied in new classes of weakly synchronous and quasiconcave functions f under assumptions less restrictive than the classical ones. Existence and uniqueness theorems, as well as theorems on the absence of solutions, are proved. Smoothness issues for solutions of both weakly nonlinear and strongly nonlinear equations are considered. An integral inequality is obtained for the weight function of a metric ensuring that a nonlinear operator is a contraction, and a number of other results are obtained.     

On a class of equations of monge-ampere type

We prove that the Dirichlet problem is solvable in a generalized sense for a class of nonlinear elliptic equations and related equations of Monge-Ampère type. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 89–106.     

关于Fermat 型函数方程的亚纯解

本文研究了三项Fermat 型函数方程亚纯解的存在性问题, 证明了: 如果(1/n) + (1/m) + (1/k) < (25/72) ; 则不存在非常数亚纯函数f, g, h 满足函数方程fⁿ + g^m + h^k ≡ 1:     

On a class of difference equations of monotone type

Of concern is the existence and uniqueness of the solution to a class of abstract second-order difference equations. They are the discrete version of some evolution equations which are intensely studied. Some asymptotic behavior results are established. The periodic solutions are also investigated. We use the theory of the maximal monotone operators in Hilbert spaces. An application to a partial differential equation is given.     

On alienation of two functional equations of quadratic type

Aequationes mathematicae - &nbsp; We deal with an alienation problem for an Euler–Lagrange type functional equation $$begin{aligned} f(alpha x + beta y) + f(alpha x - beta y) =...     

On approximate solution for operator equations of Hammerstein type

The aim of this paper is to investigate a method of approximating a solution of the operator equation of Hammerstein type x + KF(x) = f by solutions of similar finite-dimensional problems which contain operators better than K and F. Conditions of convergence and convergence rate are given and an iteration method to solve the approximative equation is proposed and applied to a concrete example.     

On multi-dimensional integral equations of convolution type with shift

The criterion of invertibility or Fredholmness of some multi-dimensional integral equations with Carleman type shifts are given. The investigation is based on some Banach space approach to equations with an involutive operator. A modified version of this approach is also presented in the paper.This approach is applied to multi-dimensional convolution type equations when the kernels may be integrable or of singular Calderon-Zygmund-Mikhlin type and shift generated by a linear transformation in the Euclidean space satisfying the generalized Carleman condition. The convolution type equations are also specially considered in the two-dimensional case in a sector on the plane symmetric with respect to one of the axes and the corresponding reflection shift. Another application deals with multi-dimensional equations with homogeneous kernels and the shift .