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.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

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.     

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.     

Positive solution to the nonlinear abstract Hammerstein equation and applications to {\phi} -Laplacian BVPs

We provide in this paper existence results for positive solution to the abstract Hammerstein equation NFu = u where N : E → E is a completely continuous operator, F : C → C is a continuous and bounded map and C is a cone in the Banach space E. The obtained results are used to prove existence results for positive solution to ${\phi}$ -Laplacian boundary value problem.     

Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X*. Let F: X → X* and K: X* → X be bounded monotone mappings such that the Hammerstein equation u + KFu = 0 has a solution. An explicit iteration sequence is constructed and proved to converge strongly to a solution of this equation.     

One-Dimensional Diffusions and Approximation

Consider the Voronovskaja operator A of a sequence of positive linear operators and let u(t, x) be the solution of the Cauchy problem for A. In the spirit of Altomare’s theory this solution can be studied by using the semigroup (T(t))_{t ≥ 0} generated by A and represented in terms of the operators L_n.One associates to A a stochastic equation; its solution can be also used in order to represent u(t, x).The relations between all these objects are described in the case of the operator A associated with some Meyer-König and Zeller type operators.     

Mann iteration of Cesàro means for asymptotically non-expansive mappings

Let K be a nonempty closed convex subset of a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm. Suppose that T:K→K is an asymptotically non-expansive mapping and for arbitrary initial value x₀∈K, we will introduce the Mann iteration of its Cesàro means:

     

Bifurcation points of Hammerstein equations

We apply global bifurcation theorems to systems of nonlinear integral equations of Hammerstein type involving a scalar parameter. To this end, we give sufficient conditions for the continuous dependence, compactness, Fréchet differentiability, and asymptotic linearity of the corresponding operators, which are more general than in the classical setting. These properties are ensured only after passing to some equivalent operator equation which typically contains fractional powers of the linear part. Finally, we show that the abstract hypotheses on the operators correspond to natural hypotheses on the kernel function and the nonlinearity in the Hammerstein equation under consideration.     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.     

On Stabilized Point Spectra of Multivalued Systems

Let H be an infinite dimensional Hilbert space. Denote by Λ (E, F) the set of all for which the multivalued system 0 ∈ (F − λ E) (x) admits a nonzero solution x ∈ H. One says that Λ (E, F) is the point spectrum of the pair (E, F). It is well known that Λ (E, F) does not behave in a stable manner with respect to perturbations in the argument (E, F). The purpose of this note is to study the outer-semicontinuous hull (or graph-closure) of the mapping Λ.     

Convergence of the variants of the Chebyshev–Halley iteration family under the Hölder condition of the first derivative

The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order p$p$, a convergence criterion of order 1+p$1+p$ for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0. Let . We proved that the sequence {x_n} generated by the iterative method x_n+1=α_nγf(x_n)+(I−μα_nF)Tx_n converges strongly to a fixed point , which solves the variational inequality , for x∈F_ix(T).     

On the method of alternating resolvents

The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeros of A and B. In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

A Dodds–Fremlin Property for Asplund and Radon–Nikodým Operators

Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator.     

Hammerstein Integral Equations with Nontrivial Solutions

By means of critical point theory, existence theorems for nontrivial solutions to the Hammerstein equation x = KFx are given, where K is a compact linear integral operator and F is a nonlinear superposition operator. To this end, appropriate conditions on the spectrum of the linear parte are combined with growth and representation conditions on the nonlinear part to ensure the applicability of the mountain — pass lemma. The abstract existence theorems are applied to nonlinear elliptic equations and systems subject to Dirichlet boundary conditions.     

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)