Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T, where T is κ-strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ_i-strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.     

Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces

Let C be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let {T_n} be a family of mappings of C into itself such that the set of all common fixed points of {T_n} is nonempty. We consider a sequence {x_n} generated by the hybrid method by generalized projection in mathematical programming. We give conditions on {T_n} under which {x_n} converges strongly to a common fixed point of {T_n} and generalize the results given in [12], [14], [13] and [11].     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

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.     

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<∞.     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

The common minimal-norm fixed point of a finite family of nonexpansive mappings

Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space H. Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element.     

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     

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:

     

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.     

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by

Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings

Let E be a real Banach space, K a closed convex nonempty subset of E. Let be m total asymptotically nonexpansive mappings. An iterative sequence for approximation of common fixed points (assuming existence) of T₁,T₂,…,Tm is constructed; necessary and sufficient conditions for the convergence of the scheme to a common fixed point of the mappings are given. Furthermore, in the case that E is uniformly convex, a sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is established.     

Path convergence,approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Let E$E$ be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K$K$ be a nonempty closed convex subset of E$E$, and let T:K?E$T:K?E$ be a continuous pseudocontraction which satisfies the weakly inward condition. For f:K?K$f:K?K$ any contraction map on K$K$, and every nonempty closed convex and bounded subset of K$K$ having the fixed point property for nonexpansive self-mappings, it is shown that the path x→_xt,t∈[0,1)$x\to x_t,t\in [0,1)$, in K$K$, defined by _xt=tT_xt+(1−t)f(_xt)$x_t=tT{x}_t+(1-t)f\left(x_t\right)$ is continuous and strongly converges to the fixed point of T$T$, which is the unique solution of some co-variational inequality. If, in particular, T$T$ is a Lipschitz pseudocontractive self-mapping of K$K$, it is also shown, under appropriate conditions on the sequences of real numbers {_αn},{_μn}$\left\{{\alpha }_n\right\},\left\{{\mu }_n\right\}$, that the iteration process: _z1∈K$z_1\in K$, _zn+1=_μn(_αnT_zn+(1−_αn)_zn)+(1−_μn)f(_zn),n∈N$z_{n+1}={\mu }_n({\alpha }_{n}T{z}_n+(1-{\alpha }_n)z_n)+(1-{\mu }_n)f\left(z_n\right),n\in \mathbb{N}$, strongly converges to the fixed point of T$T$, which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.     

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 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).     

非线性算子具误差的隐迭代程序的强收敛性

设K是一致凸Banach空间中的非空闭凸子集,T_i:K→K(i=1,2,…,N)是有限族完全渐近非扩张映象.对任意的x_0∈K,具误差的隐迭代序列{x_n}为:x_n=α_nx_n-1+β_nT_n~kx_n+γ_nu_n,n≥1,其中{α_n},{β_n},{γ_n}■[0,1]满足α_n+β_n+γ_n=1,{u_n}是K中的有界序列.在一定的条件下,该文建立了隐迭代序列{x_n}的强收敛性.得到隐迭代序列{x_n}强收敛于有限族完全渐近非扩张映象公共不动点的充要条件.所得结果改进和推广了Shahzad与Zegeye,Zhou与Chang,Chang,Tan,Lee与Chan等人的相应结果.     

A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems

We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.     

Iterative construction of fixed points of strictly pseudocontractive mappings

Let (E, ‖ ? ‖) be a smooth Banach space over the real field and A a nonempty closed bounded convex subset of E. Suppose T : A → A is a uniformly continuous strictly pseudocontractive selfmapping of A. Then, if [math001]satisfies [math001]the iteration process [math001] and [math001] converges strongly to the unique fixed point x of T. This is an improvement of a result of C.E. Chidume who established strong convergence of (x ⁿ to x in case E is L ^p or l ^p with [math001] making essential use of the inepuality [math001] which is kown to hold in these spaces for all x and y     

A new iterative algorithm for common solutions of a finite family of accretive operators

The purpose in this paper is to prove a theorem of strong convergence to a common solution for a finite family of accretive operators in a strictly convex Banach space by means of a new iterative algorithm, which is a generalization and extension of the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Zegeye and Shahzad [H. Zegeye, N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal. 66 (2007) 1161–1169]. Further using the result, the theorem of strong convergence to a common fixed point is discussed for a finite family of pseudocontractive mappings under certain conditions.     

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

In this paper, we continue to study weak convergence problems for the implicit iteration process for a finite family of Lipschitzian continuous pseudocontractions in general Banach spaces. The results presented in this paper improve and extend the corresponding ones of Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Chen et al. [R. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709] and others.