Sull'interpolazione di operatori nonlineari noncompatti in spazi di funzioni integrabili

Usando la misura di non compattezza di Hausdorff, si dimostra un teorema elementare di interpolazione per gli operatori nonlineari noncompatti tra retticoli di Banach di funzioni misurabili. Il teorema astratto viene illustrato tramite un'equazione integrale nonlineare del tipo di Hammerstein. An elementary interpolation theorem involving the Hausdorff measure of noncompactness for nonlinear operators in Banach lattices of measurable functions is proved. The abstract theorem is illustrated by a nonlinear noncompact integral operator equation of Hammerstein type.

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(Conferenza tenuta l'11 ottobre 1988)     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

Eigenvalue intervals and the existence of finitely many positive eigenfunctions for semi-positone Hammerstein integral equations are obtained. The positive characteristic values and their upper and lower bounds of the corresponding linear Hammerstein integral operators are studied. Applications of the results are given to third-order differential equations with three-point boundary conditions.     

Krasnoselskii‐type fixed point theorems with applications to Hammerstein integral equations in spaces

In this paper, we study fixed point theorems and new variants of some nonlinear altenatives of Krasnoselskii type in Banach spaces by using measures of weak noncompactness. Then we give an application to solve a nonlinear Hammerstein integral equation in L¹ spaces.     

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin-Kravchenko-Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S¹, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S¹-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111-127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.     

Approximating solutions of Hammerstein type equations in Banach spaces

Abstract

Let X be a real Banach space and X^? be its dual. Let F: X → X^? and K: X^? → X be Lipschitz monotone mappings. In this paper an explicit iterative scheme is constructed for approximating solutions of the Hammerstein type equation, 0 = u + KF u, when they exist. Strong convergence of the scheme is obtained under appropriate conditions. Our results improve and unify many of the results in the literature.     

Convergence of hybrid viscosity and steepest-descent methods for pseudocontractive mappings and nonlinear Hammerstein equations

In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Measures of weak noncompactness,nonlinear leray-schauder alternatives in banach algebras satisfying condition (p) and an application

In this paper, we establish some new nonlinear Leray-Schauder alternatives for the sum and the product of weakly sequentially continuous operators in Banach algebras satisfying certain sequential condition (P). The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. As an application, our results are used to prove the existence of solutions for a nonlinear integral equation.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

Locally defined operators and a partial solution of a conjecture

We prove a weaker form of conjecture concerning the representation theorem for locally defined operators acting between some spaces of differentiable functions presented by K. Lichawski, J. Matkowski and J. Mi? [K. Lichawski, J. Matkowski, J. Mi?, Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math. 37 (1989) 315-325].As a corollary we obtain the representation formula for continuous locally defined operators mapping the Banach space C^m(A) into C^k(A), for k≥2 where C^m(A) is the space of m-times Whitney continuously differentiable functions on a perfect set .     

Dixmier trace and the Fock space

We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal–Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. Along the road, analogous results for the Weyl pseudodifferential operators are also obtained.     

ON STRICT SINGULARITY OF LOCALLY INTEGRAL OPERATORS ON BANACH FUNCTION SPACES

Abstract

In this note we generalize a result of L. Weis and V. V. Shevchik about strict singularity of the Fourier transform to a large class of locally integral operators on function spaces.     

Strong convergence theorem for approximation of solutions of equations of Hammerstein type

Let H$H$ be a real Hilbert space. Let K,F:H→H$K,F:H\to H$ be bounded, continuous and monotone mappings. Suppose that u^∗∈H$u^{\ast }\in H$ is a solution to the Hammerstein equation u+KFu=0$u+KFu=0$. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Furthermore, we give some examples to show that our result is interdisciplinary in nature, covers a large variety of areas and should be of much interest to a wide audience.