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

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

Mixed Approximation for Nonexpansive Mappings in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2010/763207 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

Qing-Bang Zhang ◽

Fu-Quan Xia ◽

Ming-Jie Liu

Keyword(s):

Banach Space ◽

Fixed Point ◽

Nonexpansive Mapping ◽

Real Banach Space ◽

Nonexpansive Mappings ◽

Viscosity Approximation ◽

Uniformly Gâteaux Differentiable Norm ◽

Type Approximation ◽

Differentiable Norm ◽

Convergence Of The Scheme

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.

Strong convergence and control condition of modified Halpern iterations in Banach spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/29728 ◽

2006 ◽

Vol 2006 ◽

pp. 1-10

Author(s):

Yonghong Yao ◽

Rudong Chen ◽

Haiyun Zhou

Keyword(s):

Banach Space ◽

Fixed Point ◽

Strong Convergence ◽

Convex Subset ◽

Real Banach Space ◽

Nonempty Closed Convex Subset ◽

Unique Element ◽

Uniformly Gâteaux Differentiable Norm ◽

Differentiable Norm ◽

And Control

LetCbe a nonempty closed convex subset of a real Banach spaceXwhich has a uniformly Gâteaux differentiable norm. LetT∈ΓCandf∈ΠC. Assume that{xt}converges strongly to a fixed pointzofTast→0, wherextis the unique element ofCwhich satisfiesxt=tf(xt)+(1−t)Txt. Let{αn}and{βn}be two real sequences in(0,1)which satisfy the following conditions:(C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitraryx0∈C, let the sequence{xn}be defined iteratively byyn=αnf(xn)+(1−αn)Txn,n≥0,xn+1=βnxn+(1−βn)yn,n≥0. Then{xn}converges strongly to a fixed point ofT.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.756-759.3628 ◽

2013 ◽

Vol 756-759 ◽

pp. 3628-3633

Author(s):

Yuan Heng Wang ◽

Wei Wei Sun

Keyword(s):

Banach Space ◽

Fixed Point ◽

Nonexpansive Mapping ◽

Strong Convergence ◽

Iterative Algorithm ◽

Real Banach Space ◽

Nonexpansive Mappings ◽

Iterative Sequence ◽

Asymptotically Nonexpansive ◽

Differentiable Norm

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

The smooth variational principle and generic differentiability

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700028902 ◽

1991 ◽

Vol 43 (1) ◽

pp. 169-175 ◽

Cited By ~ 4

Author(s):

Pando Grigorov Georgiev

Keyword(s):

Banach Space ◽

Variational Principle ◽

Continuous Function ◽

Uniformly Gâteaux Differentiable Norm ◽

Differentiable Norm ◽

Generic Differentiability

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

Creative Mathematics and Informatics ◽

10.37193/cmi.2016.01.14 ◽

2016 ◽

Vol 25 (1) ◽

pp. 107-120

Author(s):

T. M. M. SOW ◽

◽

C. DIOP ◽

N. DJITTE ◽

◽

...

Keyword(s):

Banach Space ◽

Strong Convergence ◽

Banach Spaces ◽

Iterative Process ◽

Real Banach Space ◽

Monotone Mappings ◽

Uniformly Convex ◽

Hammerstein Equation ◽

Uniformly Smooth ◽

Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

Creative Mathematics and Informatics ◽

10.37193/cmi.2016.01.02 ◽

2016 ◽

Vol 25 (1) ◽

pp. 107-120

Author(s):

T. M. M. SOW ◽

◽

C. DIOP ◽

N. DJITTE ◽

◽

...

Keyword(s):

Banach Space ◽

Strong Convergence ◽

Banach Spaces ◽

Iterative Process ◽

Real Banach Space ◽

Monotone Mappings ◽

Uniformly Convex ◽

Hammerstein Equation ◽

Uniformly Smooth ◽

Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

ISRN Mathematical Analysis ◽

10.5402/2012/169751 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12 ◽

Cited By ~ 6

Author(s):

C. E. Chidume ◽

N. Djitté

Keyword(s):

Hilbert Space ◽

Galerkin Method ◽

Integral Equations ◽

Real Hilbert Space ◽

Iteration Process ◽

Nonlinear Integral Equations ◽

Hammerstein Equation ◽

Monotone Maps ◽

Continuity Assumption ◽

The Galerkin Method

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.

An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Journal of Applied Mathematics ◽

10.1155/2012/341953 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Youli Yu

Keyword(s):

Banach Space ◽

Fixed Point ◽

Reflexive Banach Space ◽

Contractive Mapping ◽

Bounded Subset ◽

Point Property ◽

Uniformly Gâteaux Differentiable Norm ◽

Pseudocontractive Mappings ◽

Differentiable Norm ◽

Uniformly Continuous

LetEbe a real reflexive Banach space with a uniformly Gâteaux differentiable norm. LetKbe a nonempty bounded closed convex subset ofE,and every nonempty closed convex bounded subset ofKhas the fixed point property for non-expansive self-mappings. Letf:K→Ka contractive mapping andT:K→Kbe a uniformly continuous pseudocontractive mapping withF(T)≠∅. Let{λn}⊂(0,1/2)be a sequence satisfying the following conditions: (i)limn→∞λn=0; (ii)∑n=0∞λn=∞. Define the sequence{xn}inKbyx0∈K,xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for alln≥0. Under some appropriate assumptions, we prove that the sequence{xn}converges strongly to a fixed pointp∈F(T)which is the unique solution of the following variational inequality:〈f(p)−p,j(z−p)〉≤0, for allz∈F(T).

Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

Abstract and Applied Analysis ◽

10.1155/2020/6150398 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Thanomsak Laokul

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Convergence Theorem ◽

Convex Banach Space ◽

Uniformly Convex Banach Space ◽

Multivalued Mappings ◽

Uniformly Convex ◽

Uniformly Gâteaux Differentiable Norm ◽

Differentiable Norm

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).