Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

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.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications

In this paper, we prove a strong convergence theorem by the hybrid method for a countable family of relatively nonexpansive mappings in a Banach space. We also establish a new control condition for the sequence of mappings {T_n} which is weaker than the control condition in Lemma 3.1 of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360]. Moreover, we apply our results for finding a common fixed point of two relatively nonexpansive mappings in a Banach space and an element of the set of solutions of an equilibrium problem in a Banach space, respectively. Our results are applicable to a wide class of mappings.     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998     

EXISTENCE OF SOLUTIONS OF A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS IN L^2-SPACES

By using the perturbation results of sums of ranges of accretive mappings of Calvert and Gupta (1978),the abstract results on the existence of solutions of a family of nonlinear boundary value problems in L^2 (Ω) are studied. The equation discussed in this paper and the methods used here are extension and complement to the corresponding results of Wei Li and He Zhen‘s previous papers. Especially,some new techniques are used in this paper.     

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.     

Two results in metric fixed point theory

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition in Banach spaces.     

Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces

In this paper, we establish two coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. The theorems presented extend some results due to ?iri? (2009) [3]. An example is given to illustrate the usability of our results.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

Fixed point iterations for certain classes of nonlinear mappings

Let X = L_p or L_p, 2≤p<∞, and let K be a nonempty closed convex bounded subset of X. It is proved that for some classes of nonlinear mappings T:K → K (more precisely, for T P₂ or C in the terminology of F.E. Browder and W.V. Pretryshyn; and B.E. Rhoades), the iteration process: x₁ ?K,X_n+1 = (1-C_n)x_n+C_n Tx_n, n ≥1,under suitable conditions on K and the real sequence {C_n}_n=1 ^∞ converges strongly to a fixed point of T. While our thorems generalize serveral known results, our method is also of independent interest     

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.     

Four parameter proximal point algorithms

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.     

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.     

A direct construction of very smooth local Navier-Stokes solutions

We will consider Galerkin approximations to the solution of the Navier-Stokes initial boundary-value problem in three dimensions. Uniform convergence (locally in time) will be proved with respect to the same norm (being stronger than theH ²-norm) in which the solution's initial value is bounded. The result is the best possible if we will avoid a nonlinear, nonlocal compatibility condition for the initial value.Dedicated to Professor Robert Finn on the occasion of his 70th birthday     

Critical types of krasnoselskii fixed point theorems in weak topologies

Abstract

In this note, by means of the technique of measures of weak noncom- pactness, we establish a generalized form of fixed point theorem for the sum of T+S in weak topology setups of a metrizable locally convex space, where S is not weakly compact, I?T allows to be noninvertible, and T is not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings.     

Halpern’s iteration in Banach spaces

We prove strong convergence theorems for a sequence which is generated by Halpern’s iteration. We also apply our result for finding zeros of an accretive operator. Our result improves the recent result of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360] by removing some assumptions on the parameters. Finally we discuss the new sufficient condition studied by Song [Y. Song, A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 198 (2) (2008) 721-728; Y. Song, New strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Comput. Math. Appl. 56 (6) (2008) 1473-1478] and correct the main result of Song and Chai [Y. Song, X. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal. 71 (10) (2009) 4500-4506].     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.     

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.