Integral Equations Of Hammerstein Type Research Articles

An iterative method for solving nonlinear integral equations of Hammerstein type

Suppose H is a real Hilbert space and F,K:H→H are continuous bounded monotone maps with D(K)=D(F)=H. Assume that the Hammerstein equation u+KFu=0 has a solution. An explicit iteration process is proved to converge strongly to a solution of this equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorem complements the Galerkin method of Brézis and Browder to provide methods for approximating solutions of nonlinear integral equations of Hammerstein type.

Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems

A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.

Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods

New discrete Adomian decomposition methods are presented by using some identified Clenshaw-Curtis quadrature rules. We investigate two mixed quadrature rules one of precision five and the other of precision seven. The first rule is formed by using the

Positive solutions of semi-positone Hammerstein integral equations and applications

Existence of solutions for semi-positone integral equations of Hammerstein type is obtained by using the fixed point index method. Many boundary value problems of differential equations can be transformed into these integral equations. A specific example is given.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

Approximation of solutions of generalized equations of Hammerstein type

Let E be a 2-uniformly smooth real Banach space. For each i=1,2,…,m, let Fi,Ki:E→E be bounded and accretive mappings. Assume that the generalized Hammerstein equation u+∑i=1mKiFiu=0 has a solution in E. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation. Our result improves, generalizes and extends the results of Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, Solution of nonlinear integral equations of Hammerstein type, Nonlinear Anal. 74 (2011) 4293–4299] and Chidume and Shehu [C.E. Chidume, Y. Shehu, Approximation of solutions of generalized equations of Hammerstein type, Comput. Math. Appl. 63 (2012) 966–974].

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.

On Discrete Adomian Decomposition Method with Chebyshev Abscissa for Nonlinear Integral Equations of Hammerstein Type

We study approximate solutions of a nonlinear integral equation of Hammerstein type. We describe the principle of discrete Adomian decomposition method (DADM). DADM is considered in the case we evaluate numerical integration by using Chebyshev roots. This technique gives an accurate solutions as will shown by illustrate examples.

Existence of Integrable Solutions of an Integral Equation of Hammerstein Type on an Unbounded Interval

In this paper we investigate the existence of integrable solutions of a nonlinear integral equation of Hammerstein type on an unbounded interval. Our analysis relies on a Krasnosel’skii type fixed point theorem and uses the technique of measure of weak noncompactness.

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 + K F u = 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.

Semilocal convergence of Stirling's method under Hölder continuous first derivative in Banach spaces

The aim of this paper is to establish the semilocal convergence of Stirling's method used to find fixed points in Banach spaces assuming the Hölder continuity condition on the first Fréchet derivative of nonlinear operators. This condition generalizes the Lipschtiz continuity condition used earlier for the convergence. Also, the Hölder continuity condition holds on some problems, where the Lipschiz continuity condition fails. The R-order of convergence and a priori error bounds are also derived. On comparison with Newton's method, larger domains of existence and uniqueness of fixed points are obtained. An integral equation of Hammerstein type of second kind is solved to show the efficiency of our convergence analysis.

Numerical iterative method for Volterra equations of the convolution type

The objective of this paper is to present an algorithm from which a rapidly convergent solution is obtained for Volterra integral equations of Hammerstein type. Such equations are often encountered when describing the response of viscoelastic materials where the time dependency of the material properties is often expressed in the form of a convolution integral. Frequently, singularity is encountered and often ignored when dealing with the constitutive equations of viscoelastic materials. In this paper, the singularity is incorporated in the solution and the iterative scheme used to solve the equation converges within six iterations to a typical toleration error of 10−5. The algorithm is applied to the strain response of a polymer under impulsive (constant) loading and the results show excellent correlation between the experimental and analytical solution. Copyright © 2010 John Wiley & Sons, Ltd.

Convergence of Stirling's method under weak differentiability condition

The aim of this paper is to establish the semilocal convergence analysis of Stirling's method used to find fixed points of nonlinear operator equations in Banach spaces. This is done by using recurrence relations under weak Holder continuity condition on the first Frechet derivative of the involved operator. The existence and uniqueness regions for a fixed point are obtained. The efficacy of our work is demonstrated by solving an integral equation of Hammerstein type and comparing the results obtained by Newton's method. It is found that our approach gives better existence and uniqueness regions for a fixed point. Copyright © 2010 John Wiley & Sons, Ltd.

A nonlinear integral equation of Hammerstein type with a noncompact operator

We consider a homogeneous nonlinear integral equation of Hammerstein type which has an important application in the kinetic theory of gases. We prove a positive solution of this equation exists and describe its asymptotic behaviour at infinity.Bibliography: 6 titles.

Indentation of a round punch into a rough elastic half-space

We consider the axisymmetric contact problem on the indentation of a round punch with plane base into an elastic half-space. Roughness is taken into account in the framework of the well-known model in which the local surface displacements are proportional to some power of the contact pressures at the same point. To study the problem, we use the theory of nonlinear integral equations of Hammerstein type together with the mathematical technique of orthogonal Legendre polynomials and the contraction mapping principle. We obtain an approximate analytic solution of the problem, numerically analyze the results, and reveal typical laws of variation of the main mechanical variables.

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations u i ( t ) = ∫ 0 T g i ( t , s ) f i ( s , u 1 ( s ) + ρ 1 ( s ) , u 2 ( s ) + ρ 2 ( s ) , … , u n ( s ) + ρ n ( s ) ) d s , t ∈ [ 0 , T ] , 1 ≤ i ≤ n where T > 0 is fixed, ρ i ’s are given functions and the nonlinearities f i ( t , x 1 , x 2 , … , x n ) can be singular at t = 0 and x j = 0 where j ∈ { 1 , 2 , ⋯ , n } . Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i ( t ) ≥ 0 for t ∈ [ 0 , T ] and 1 ≤ i ≤ n , where θ i ∈ { 1 , − 1 } is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.

AN EXTENSION OF FIXED POINT THEOREMS CONCERNING CONE EXPANSION AND COMPRESSION AND ITS APPLICATION

The famous Guo-Krasnosel'skii fixed point theorems concerning cone expansion and compression of norm type and order type are extended, respectively. As an application, the existence of multiple positive solutions for systems of Hammerstein type integral equations is considered.

UNIFYING A MULTITUDE OF COMMON FIXED POINT THEOREMS EMPLOYING AN IMPLICIT RELATION

A general common fixed point theorem for two pairs of weakly compatible mappings using an implicit function is proved without any continuity requirement which generalizes the result due to Popa (20, Theorem 3). In process, several previously known results due to Fisher, Kannan, Jeong and Rhoades, Imdad and Ali, Imdad and Khan, Khan, Shahzad and others are derived as special cases. Some related results and illustrative examples are also discussed. As an application of our main result, we prove an existence theorem for the solution of simultaneous Hammerstein type integral equations. As established in Jungck (12), a common fixed point theorem in metric spaces generally involves conditions on contraction, commutativity and continuity of the involved mappings besides a suitable containment of range of one mapping into the range of other. To prove a new metrical common fixed point theorem one is always required to improve one or more of these conditions. Of all these four conditions, the means of improving contraction condition to prove new results on fixed and common fixed point is much discussed which continue to attract the attention of the researchers of this domain. But in this paper we use an implicit function due to Popa (20). The tradition of improving commutativity condition in common fixed point theorems was initiated by Sessa (21). Inspired by the definition of weak com- mutativity of Sessa (21), researchers of this domain, introduced several def- initions of weak commutativity such as: Compatible mappings, Compatible mappings of type (A), Compatible mappings of type (B), Compatible map- pings of type (P), Compatible mappings of type (C), Biased maps, R-weakly commuting mappings and some others whose lucid comparison and illustration

Relaxing convergence conditions for Stirling's method

The aim of this paper is to study the convergence of Stirling's method used for finding fixed points of nonlinear operator equations assuming that the first Frechet derivative of the nonlinear operator be ω-conditioned. The ω-condition relaxes the Lipschitz/Holder condition on the first Frechet derivative used earlier for the convergence. The existence and uniqueness regions are derived for the fixed points. Finally, the efficacy of our convergence analysis is shown by working out an integral equation of Hammerstein type of second kind. The results obtained show that our approach finds better results when compared with the results obtained by Newton's method. Copyright © 2009 John Wiley & Sons, Ltd.

Toeplitz matrix method and nonlinear integral equation of Hammerstein type

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated.