Symmetry and nonexistence of positive solutions to an integral system with weighted functions

Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable conditions.It is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 <...     

Classification of solutions for an integral equation

Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x₀ ? ?ⁿ. This solves an open problem posed by Lieb 12 . The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well‐known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.     

Order-continuous extension of a positive operator

For S. L. Sobolev's 80th birthday.     

On an integral operator

In this paper we derive some criteria for univalence of a general integral operator for analytic functions in the open unit disk.     

Existence of positive solutions of mixed fractional integral boundary value problem with p(t)-Laplacian operator

Ricerche di Matematica - In this paper, we investigate a mixed fractional integral boundary value problem with p(t)-Laplacian operator. Firstly, we derive the Green function through the direct...     

The integral operator with logarithmic kernel has only one positive eigenvalue

The aim of this short note is to show that the integral operator with the kernel log(x + y) being indefinite has only one positive eigenvalue.     

Common hermitian and positive solutions to the adjointable operator equations ,

Let be a C^*-algebra. For any Hilbert -modules H and K, let be the set of adjointable operators from H to K. Let H,K,L be Hilbert -modules, and . In this paper, we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions to the equations , and obtain the formulae for the general forms of these solutions. Some results, known for finite matrices and Hilbert space operators, are extended to the adjointable operators acting on Hilbert C^*-modules.     

On the univalence of an integral operator

In this work the author introduces a general integral operator and determines conditions for the univalence of this integral operator.     

The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator

LetE be a Banach lattice which consists of some functionsu: over a fixed domain . This article is concerned with the local stability of a non-zero positive solutionu ^*E to the Hammerstein equationu(x)= x, wherek0, andf:×[0, )[0, ) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and forx,u>0. Under some mild additional hypotheses onE, k, andf it is proved that the spectral radius of the Fréchet derivative atu ^* of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.     

Mapping properties of an integral operator

In this paper, we consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under an integral operator.     

Fixed points of an integral operator

A characterization of the functions fixed by a class of integral operators associated with hyperbolically-harmonic functions is given.     

Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities

In this paper we investigate a system of impulsive integral boundary value problems with sign-changing nonlinearities. Using the fixed point theorem in double cones, we prove the existence of multiple positive solutions.