Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.

Eigenvalues for a combination between local and nonlocal p-Laplacians

In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

Unilateral Global Bifurcation for Eigenvalue Problems with Homogeneous Operator

We focus on the structure of the solution set for the nonlinear equation [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous operators. Under certain hypotheses on [Formula: see text] and [Formula: see text], unilateral global bifurcations for eigenvalue problems are presented. Some applications are illustrated for nonlinear ordinary and partial differential equations. In particular, the existence and multiplicity of one-sign solutions for Monge–Ampère equation is discussed.

Generalizations of Fixed-Point Theorems of Altman and Rothe Types

It is intended to present some extensions of the famous Altman and Rothe types fixed-point theorems. The inequality conditions are relaxed to theα-positive-homogeneous operator. Some new fixed-point theorems are obtained with the help of the theory of topological degree.

Infinite Matrix Transformation of a Class of Operator

On the basis of the uniform convergence theory, this paper discusses the infinite matrix transformation of a class of operator which includes the whole linear operators、homogeneous operator and the many nonlinear operators, and then has obtained the conclusion of the summability of nonlinear operator.

A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications top-Laplacian Boundary Value Problems

We will present a generalization of Mahadevan’s version of the Krein-Rutman theorem for a compact, positively 1-homogeneous operator on a Banach space having the properties of being increasing with respect to a conePand such that there is a nonzerou∈P∖{θ}−Pfor whichMTpu≥ufor some positive constantMand some positive integerp. Moreover, we give some new results on the uniqueness of positive eigenvalue with positive eigenfunction and computation of the fixed point index. As applications, the existence of positive solutions forp-Laplacian boundary-value problems is considered under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.