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In this paper, we introduce the concept of monotone α -nonexpansive mappings in an ordered Banach space E with the partial order ≤, which contains monotone nonexpansive mappings as special case. With the help of the Mann iteration, we show some existence theorems of fixed points of monotone α -nonexpansive mappings in uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of the Mann iteration for finding an order fixed point of monotone α -nonexpansive mappings under the condition lim sup n → ∞ β n ( 1 − β n ) > 0 or lim inf n → ∞ β n ( 1 − β n ) > 0 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, for an -strict pseudocontraction T, we prove strong convergence of the modified Mann’s iteration defined by where and in satisfy: [(i)] with ; [(ii)] and ; [(iii)] . Our results unify and improve some existing results. [Copyright &y& Elsevier]

Abstract: The main objectives of this paper are to employ a new proof technique to prove the strong convergence of and , defined respectively by to some zero of accretive operator in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping ) whenever and are sequences in and satisfying the conditions: , , ,. [Copyright &y& Elsevier]

Abstract: In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed convex subset of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Suppose that is an asymptotically non-expansive mapping and for arbitrary initial value , we will introduce the Mann iteration of its Cesàro means: and prove its strong and weak convergence whenever and is a real sequence in satisfying one of the conditions: either (i) or (ii) or (iii) . [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to study the strong convergence of Halpern iteration for firmly type nonexpansive mappings defined on a Banach space with a uniformly Gâteaux differentiable norm, and the proof techniques are simpler and different from much existing literature also, which gives the affirmative answer of the Halpern open problem for this class of mapping. [Copyright &y& Elsevier]

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by , where f is a Lipschitz strong pseudocontractive mapping and { β n}, { α n} satisfy (i) $$\lim\limits_{n\to\infty}\alpha_n = 0$$; (ii) $$ \sum\limits_{n=1}^\infty \alpha_n = \infty$$; (iii) $$\lim\limits_{n\to\infty}\frac{\beta_n^2}{\alpha_n} = 0$$. [ABSTRACT FROM AUTHOR]

Abstract: An iterative algorithm is proposed for finding a fixed point of a nonexpansive self-mapping of a closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and an estimate of the convergence speed also is given. [Copyright &y& Elsevier]

Abstract: For a countable family of nonexpansive mappings, a strong convergence of Halpern type iteration is shown in order to find a common fixed point of in a reflexive Banach space when satisfies the conditions (C1) and (C2) , and several examples satisfying the condition (B) is given also. We also research the strong convergence of the proximal point algorithm. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings. [Copyright &y& Elsevier]

Abstract: Let be a fixed strongly pseudocontractive mapping and be a continuous pseudocontractive mapping with . The sequence is iteratively defined as follows: where satisfies the condition . We prove that converges strongly to the unique solution to some variational inequality in . Our results develop and complement the corresponding ones by Matsushita–Daishi Kuroiwa [J. Math. Anal. Appl. 294 (2004) 206–214] and Shioji–Takahashi [Arch. Math., 72 (1999) 354–359] and Moore–Nnoli [J. Math. Anal. Appl. 260 (2001) 269–278] and Su–Li [Appl. Math. Comput. 181 (2006) 332–341] and Song–Chen [Appl. Math. Comput. 186 (2007) 1120–1128] and Wangkeeree [Appl. Math. Comput. 201 (2008) 239–249]. [Copyright &y& Elsevier]

Abstract: Many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a Banach space. In this paper, under the hypotheses that is a reflexive and strictly convex Banach spaces with a uniformly Gâeaux differentiable norm, the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self-mappings defined on a closed convex subset of . [Copyright &y& Elsevier]

Abstract: The main aim of this work is to give the equivalence of the convergence between proximal type algorithms. Our convergence analysis covers Rockafellar type proximal point algorithm (with a weak contraction) for finding zeros of accretive operators as well as Browder and Halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in Banach spaces. We also provide simple proofs of some results of Wong, Sahu and Yao [N.C. Wong, D.R. Sahu, J.C. Yao, Solving variational inequalities involving nonexpansive type mappings, Nonlinear Anal. 69 (12) (2008) 4732–4753]. [Copyright &y& Elsevier]

Abstract: Under the assumption that is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping with gauge function , Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that is a reflexive Banach space which has a weakly continuous duality mapping with gauge function , and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem. [Copyright &y& Elsevier]

Abstract: We show strong and weak convergence for Mann iteration of multivalued nonexpansive mappings in a Banach space. Furthermore, we give a strong convergence of the modified Mann iteration which is independent of the convergence of the implicit anchor-like continuous path . [Copyright &y& Elsevier]

In this note, a small gap is corrected in the proof of H.K. Xu [Theorem 3.3, A regularization method for the proximal point algorithm, J. Glob. Optim. 36, 115–125 (2006)], and some strict restriction is removed also. [ABSTRACT FROM AUTHOR]

Abstract: Let E be a uniformly convex Banach space E with a weakly continuous duality mapping and K be a non-empty bounded closed convex subset of E. For an asymptotically non-expansive mapping , an arbitrary initial values and an anchor point , we define iteratively the sequences and as follows:where and satisfy proper conditions. We prove that and converge strongly to some , respectively. [Copyright &y& Elsevier]

Abstract: The aim of this work is to establish the strong convergence of explicit viscosity-like methods for a nonexpansive nonself-mapping without boundary conditions defined in a Banach space. This gets rid of the restriction and dependence on the implicit anchor-like continuous path in the existing literature. [Copyright &y& Elsevier]

Abstract: The aim of this work is to give a new sufficient condition of the strong convergence of the Halpern type iteration for a non-expansive self-mapping defined on a Banach space with a uniformly Gâteaux differentiable norm. Several examples satisfying our condition are presented. Our results not only remove the restriction of the space with the fixed point property for non-expansive self-mappings, but also get rid of the dependence on the convergence of the implicit anchor-like continuous path z t in the proof. [Copyright &y& Elsevier]

In a real Hilbert space H, from an arbitrary initial point x0 ∈ H, an iterative process is defined as follows: [image omitted], [image omitted], n ≥ 0, where [image omitted], [image omitted], (∀ x ∈ H), T, S : H → H are two non-expansive mapping with F(T) ∩ F(S) ≠ ∅ and f (resp. g) : H → H an ηf (resp. ηg)-strongly monotone and kf (resp. kg)-Lipschitzian mapping, {an} ⊂ (0, 1), {bn} ⊂ (0, 1) and {λn} ⊂ [0, 1), {βn} ⊂ [0, 1). Under some suitable conditions, several convergence results of the sequence {xn} are shown. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767–773] in condition , and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder–Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. 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]. [Copyright &y& Elsevier]