Convergence of Path and Approximation of Common Element of Null Spaces of Countably Infinite Family of 𝑚-Accretive Mappings in Uniformly Convex Banach Spaces

Abstract

We prove path convergence theorems and introduce a new iterative sequence for a countably infinite family of 𝑚-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.

1. Introduction

Let 𝐸 be a real Banach space with dual 𝐸^∗. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥∶={𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²}, (1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing between members of 𝐸 and 𝐸^∗. It is well known that if 𝐸^∗ is strictly convex, then 𝐽 is single-valued (see, e.g., [1, 2]). In the sequel, we will denote the single-valued normalized duality mapping by 𝑗.

A mapping 𝐴∶𝐷(𝐴)⊆𝐸→𝐸 is called accretive if for all 𝑥,𝑦∈𝐷(𝐴) there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥0. (1.2) By the result of Kato [3], (1.2) is equivalent to ‖𝑥−𝑦‖≤‖𝑥−𝑦+𝑠(𝐴𝑥−𝐴𝑦)‖,∀𝑠>0. (1.3) If 𝐸 is a Hilbert space, accretive operators are also called monotone. An operator 𝐴 is called m-accretive if it is accretive and 𝑅(𝐼+𝑟𝐴), range of (𝐼+𝑟𝐴), is 𝐸 for all 𝑟>0; and 𝐴 is said to satisfy the range condition if cl(𝐷(𝐴))⊆𝑅(𝐼+𝑟𝐴), for all 𝑟>0, where cl(𝐷(𝐴)) denotes the closure of the domain of 𝐴. It is easy to see that every 𝑚-accretive operator satisfies the range condition. An operator 𝐴 is said to be maximal accretive if it is accretive and the inclusion 𝐺(𝐴)⊆𝐺(𝐵) implies 𝐺(𝐴)=𝐺(𝐵), where 𝐺(𝐴) denotes the graph of 𝐴 and 𝐵 is an accretive operator.

A mapping 𝑇∶𝐷(𝑇)⊆𝐸→𝑅(𝑇)⊆𝐸 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐷(𝑇). (1.4) It is not difficult to deduce from (1.3) that a mapping 𝐴 is accretive if and only if its resolvent 𝐽_𝑟∶=(𝐼+𝑟𝐴)⁻¹, for all 𝑟>0, is nonexpansive and single valued on the range of (𝐼+𝑟𝐴). Thus, in particular, 𝐽_𝐴=𝐽₁∶=(𝐼+𝐴)⁻¹ is nonexpansive and single valued on the range of (𝐼+𝐴). Furthermore, 𝐹(𝐽_𝐴)∶=𝐴⁻¹(0)∶={𝑥∈𝐷(𝐴)∶𝐴𝑥=0}. For more details see, for example, [4, 5].

Closely related to the class of accretive operators is the class of pseudocontractive maps. An operator 𝑇 with domain 𝐷(𝑇) in 𝐸 and range 𝑅(𝑇) in 𝐸 is called pseudocontractive if 𝐴∶=𝐼−𝑇 is accretive. The importance of these operators in application is well known (see, e.g., [6–9] and the references contained therein).

It is well known that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings (see, e.g., [4]). Construction of fixed points of nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [10]).

Iterative approximation of fixed points and zeros of nonlinear operators have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems (see, e.g., [11–15]). The iterative scheme 𝑥₀∈𝐸,𝑥_𝑛+1=𝐽_𝑟𝑛𝑥_𝑛,𝑛≥0, (1.5) (where 𝐽_𝑟𝑛 is the resolvent of an 𝑚-accretive operator 𝐴) for example, has been extensively studied over the past forty years or so for construction of zeros of accretive operators (see, e.g., [16–20]).

Kim and Xu [21] introduced a modification of Mann iterative scheme in a reflexive Banach space having weakly continuous duality mapping for finding a zero of an 𝑚-accretive operator 𝐴 as follows: 𝑥₀=𝑢∈𝐸,𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑛≥0. (1.6) They proved that the sequence {𝑥_𝑛}^∞_𝑛=0 generated by (1.6) converges to a zero of 𝑚-accretive operator 𝐴 under the following conditions:

(i)lim_𝑛→∞𝛼_𝑛=0,(ii)∑^∞_𝑛=0𝛼_𝑛=∞ (equivalently, ∏^∞_𝑛=0(1−𝛼_𝑛)=0),(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞; 𝑟_𝑛≥𝜖 for some 𝜖>0 and for all 𝑛≥0, (iv)∑^∞_𝑛=1|1−𝑟_𝑛−1/𝑟_𝑛|<∞,(v)𝑟_𝑛≥𝜖 for some 𝜖>0 and for all 𝑛≥0 and ∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞.

In 2007, Qin and Su [22] also considered the following iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping: 𝑥₀=𝑢∈𝐶,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑦_𝑛𝑛≥0, (1.7) where {𝛼_𝑛}^∞_𝑛=0 and {𝛽_𝑛}^∞_𝑛=0 are sequences in (0,1). They proved that the sequence {𝑥_𝑛}^∞_𝑛=0 generated by (1.7) converges strongly to a zero of 𝑚-accretive operator 𝐴 provided that {𝛼_𝑛}^∞_𝑛=0 and {𝛽_𝑛}^∞_𝑛=0 satisfy conditions (i), (ii), and (iii), and {𝑟_𝑛}^∞_𝑛=0 satisfies condition (v).

Chen and Zhu [23] considered the following viscosity iterative scheme for resolvent 𝐽_𝑟𝑛 of 𝑚-accretive mapping 𝐴: 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑛≥0, (1.8) where 𝑓 is a contraction mapping defined on 𝐶. Under the assumption that {𝑟_𝑛}^∞_𝑛=0 satisfies condition (v), Chen and Zhu [23] proved in a reflexive Banach space having weakly sequentially continuous duality mapping that the sequence {𝑥_𝑛}^∞_𝑛=0 generated by (1.8) converges strongly to a zero of 𝐴, which solves a certain variational inequality.

Recently, Jung [24] introduced the following viscosity iterative method: 𝑥₀∈𝐶,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛𝑛≥0. (1.9) Under certain appropriate conditions on the parameters {𝛼_𝑛}^∞_𝑛=0, {𝛽_𝑛}^∞_𝑛=0, {𝑟_𝑛}^∞_𝑛=0 and the sequence {𝑥_𝑛}^∞_𝑛=0; Jung [24] established strong convergence of the sequence {𝑥_𝑛}^∞_𝑛=0 generated by (1.9) to a zero of 𝐴, which is a unique solution of a certain variational inequality problem, in either a reflexive Banach space having a weakly sequentially continuous duality mapping or a reflexive Banach space having a uniformly Gâteaux differentiable norm such that every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings.

In [5], Zegeye and Shahzad proved the following theorem.

Theorem 1 ZS. Let 𝐸 be a strictly convex reflexive real Banach space which has uniformly Gâteaux differentiable norm and let 𝐾 be a nonempty closed convex subset of 𝐸. Assume that every nonempty closed convex and bounded subset of 𝐸 has the fixed point property for nonexpansive mappings. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…,𝑟 be a finite family of 𝑚-accretive mappings with ⋂^𝑟_𝑖=1𝐴_𝑖⁻¹(0)≠∅. For given 𝑢,𝑥₁∈𝐾, let {𝑥_𝑛}_𝑛≥1 be generated by the algorithm 𝑥_𝑛+1=𝜃_𝑛𝑢+(1−𝜃_𝑛)𝑆_𝑟𝑥_𝑛,∀𝑛≥1, (1.10) where 𝑆_𝑟=𝑎₀𝐼+𝑎₁𝐽_𝐴1+⋯+𝑎_𝑟𝐽_𝐴𝑟, with 𝐽_𝐴𝑖=(𝐼+𝐴_𝑖)⁻¹, 0<𝑎_𝑖<1, 𝑖=1,…,𝑟, ∑^𝑟_𝑖=1𝑎_𝑖=1, and {𝜃_𝑛}_𝑛≥1 is a sequence in (0,1) satisfying the following conditions:(i)lim_𝑛→∞𝜃_𝑛=0; (ii)∑^∞_𝑛=1𝜃_𝑛=∞; (iii)∑^∞_𝑛=1|𝜃_𝑛−𝜃_𝑛−1|<∞ or lim_𝑛→∞(|𝜃_𝑛−𝜃_𝑛−1|/𝜃_𝑛)=0. Then, {𝑥_𝑛}_𝑛≥1 converges strongly to a common solution of the equation 𝐴_𝑖𝑥=0 for 𝑖=1,2,…,𝑟.

Motivated by the results of the authors mentioned above, it is our purpose in this paper to prove new path convergnce theorems and introduce a new iteration process for a countably infinite family of 𝑚-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach spaces. As a result, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.

2. Preliminaries

Let 𝐸 be a real normed linear space. Let 𝑆∶={𝑥∈𝐸∶‖𝑥‖=1}. 𝐸 is said to have a Gâteaux differentiable norm (and 𝐸 is called smooth) if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (2.1) exists for each 𝑥,𝑦∈𝑆; 𝐸 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆 the limit is attained uniformly for 𝑥∈𝑆. Furthermore, 𝐸 is said to be uniformly smooth if the limit exists uniformly for (𝑥,𝑦)∈𝑆×𝑆.

Let 𝐸 be a real normed linear space. The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶[0,2]→[0,1] defined by 𝛿_𝐸(𝜖)=inf{1−‖𝑥+𝑦‖2∶‖𝑥‖=‖𝑦‖=1,𝜖=‖𝑥−𝑦‖}. (2.2) The space 𝐸 is said to be uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2]. 𝐸 is said to be strictly convex if for all 𝑥,𝑦∈𝐸 such that ‖𝑥‖=‖𝑦‖=1 and for all 𝜆∈(0,1) we have ‖𝜆𝑥+(1−𝜆)𝑦‖<1. It is well known that every uniformly convex Banach space is strictly convex.

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at 𝑝 if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that 𝑥_𝑛⇀𝑥∈𝐷(𝑇) and 𝑇𝑥_𝑛→𝑝, then 𝑇𝑥=𝑝.

A mapping 𝑇∶𝐷(𝑇)⊆𝐸→𝐸 is said to be demicompact at ℎ if for any bounded sequence {𝑥_𝑛} in 𝐷(𝑇) such that (𝑥_𝑛−𝑇𝑥_𝑛)→ℎ as 𝑛→∞, there exists a subsequence say {𝑥_𝑛𝑗} of {𝑥_𝑛} and 𝑥^∗∈𝐷(𝑇) such that {𝑥_𝑛𝑗} converges strongly to 𝑥^∗ and 𝑥^∗−𝑇𝑥^∗=ℎ.

We need the following lemmas in the sequel.

Lemma 2.1. Let 𝐸 be a real normed space, then ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩, (2.3) for all 𝑥,𝑦∈𝐸 and for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2.2 (Lemma  3 of Bruck [25]). Let 𝐾 be a nonempty closed convex subset of a strictly convex real Banach space 𝐸. Let {𝑇_𝑖}^∞_𝑖=1 be a sequence of nonexpansive mappings from 𝐾 to 𝐸 such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝜆_𝑖}^∞_𝑖=1 be a sequence of positive numbers such that ∑^∞_𝑖=1𝜆_𝑖=1, then a mapping 𝐺 on 𝐾 defined by 𝐺𝑥∶=∑^∞_𝑖=1𝜆_𝑖𝑇_𝑖𝑥 for all 𝑥∈𝐾 is well defined, nonexpansive, and 𝐹(𝐺)=⋂^∞_𝑖=1𝐹(𝑇_𝑖).

Lemma 2.3 (Xu [26]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤(1−𝛼_𝑛)𝑎_𝑛+𝛼_𝑛𝜎_𝑛,𝑛≥1, (2.4) where {𝛼_𝑛}^∞_𝑛=1⊂[0,1] and {𝜎_𝑛}^∞_𝑛=1 is a sequence in ℝ satisfying (i) ∑𝛼_𝑛=∞; (ii) limsup𝜎_𝑛≤0. Then, 𝑎_𝑛→0 as 𝑛→∞.

Lemma 2.4 (Suzuki [27]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥1 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.5 (Cioranescu [28]). Let 𝐴 be a continuous accretive operator defined on a real Banach space 𝐸 with 𝐷(𝐴)=𝐸. Then, 𝐴 is 𝑚-accretive.

Lemma 2.6 (C. E. Chidume and C. O. Chidume [29]). Let 𝐾 be a nonempty closed convex subset of a real Banach space 𝐸. For arbitrary 𝑟>0, let 𝐵_𝑟(0)∶={𝑥∈𝐸∶‖𝑥‖≤1}. Then, there exists a continuous strictly increasing function 𝑔∶[0,∞)→[0,∞), 𝑔(0)=0 such that for every 𝑥,𝑦∈𝐵_𝑟(0) and for 𝑝∈(1,∞), the following inequality holds: 4.2^𝑝1𝑔(2‖𝑥+𝑦‖)≤(𝑝.2^𝑝−4)‖𝑥‖^𝑝+𝑝.2^𝑝⟨𝑦,𝑗_𝑝(𝑥)⟩+4‖𝑦‖^𝑝. (2.5)

3. Path Convergence Theorems

We begin with the following lemma.

Lemma 3.1. Let 𝐾 be a nonempty closed convex subset of a reflexive strictly convex Banach space 𝐸. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping. Let {𝑥_𝑛}^∞_𝑛=1, a bounded sequence in 𝐾, be an approximate fixed point sequence of 𝑇, that is lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. Let 𝜑(𝑥)=𝜇_𝑛‖𝑥_𝑛−𝑥‖², for all 𝑥∈𝐾 and let Γ={𝑥∈𝐾∩𝐵∶𝜑(𝑥)=min_𝑧∈𝐾𝜑(𝑧)}, where 𝐵 is any bounded closed convex nonempty subset of 𝐸 such that 𝑥_𝑛∈𝐵 for all 𝑛∈ℕ. Then 𝑇 has a fixed point in Γ, provided that 𝐹(𝑇)≠∅.

Proof. Since 𝐸 is a reflexive Banach space, then Γ is a bounded closed convex nonempty subset of 𝐸. Since lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0, we have that for all 𝑥∈Γ, 𝜑(𝑇𝑥)=𝜇_𝑛‖𝑥_𝑛−𝑇𝑥‖²≤𝜇_𝑛(‖𝑥_𝑛−𝑇𝑥_𝑛‖+‖𝑇𝑥_𝑛−𝑇𝑥‖)²≤𝜇_𝑛‖𝑥_𝑛−𝑥‖²=𝜑(𝑥). (3.1) Hence, 𝑇𝑥∈Γ, for all 𝑥∈Γ, that is, Γ is invariant under 𝑇. Let 𝑥^∗∈𝐹(𝑇). Then since every closed convex nonempty subset of a reflexive strictly convex Banach space is a Chebyshev set (see, e.g., [30, Corollary  5.1.19]), there exists a unique 𝑢^∗∈Γ such that ‖𝑥^∗−𝑢^∗‖=inf_𝑧∈Γ‖𝑥^∗−𝑧‖, (3.2) but 𝑥^∗=𝑇𝑥^∗ and 𝑇𝑢^∗∈Γ. Thus, ‖𝑥^∗−𝑇𝑢^∗‖=‖𝑇𝑥^∗−𝑇𝑢^∗‖≤‖𝑥^∗−𝑢^∗‖. (3.3) So, 𝑇𝑢^∗=𝑢^∗. Hence, 𝐹(𝑇)∩Γ≠∅. This completes the proof.

Proposition 3.2. Let 𝐾 be a nonempty closed convex subset of a real Banach space 𝐸. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of 𝑚-accretive mappings and define 𝐽_𝐴𝑖∶=(𝐼+𝐴_𝑖)⁻¹, 𝑖=1,2,…. Let {𝛼_𝑛}^∞_𝑛=1, {𝜎_𝑖,𝑛}^∞_𝑛=1, 𝑖=1,2,… be sequences in (0,1) such that ∑^∞_𝑖=1𝜎_𝑖,𝑛=(1−𝛼_𝑛). Fix 𝛿∈[𝛾₁,𝛾₂], for some 𝛾₁,𝛾₂∈(0,1). For arbitrary fixed 𝑢∈𝐾, define a map 𝑇_𝑛∶𝐾→𝐾 by 𝑇_𝑛𝑥=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥+𝛿𝐽_𝐴𝑖𝑥),∀𝑥∈𝐾. (3.4) Then, 𝑇_𝑛 is a strict contraction on 𝐾.

Proof. Let 𝑥,𝑦∈𝐾, then ‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖=‖^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)(𝑥−𝑦)+𝛿(𝐽_𝐴𝑖𝑥−𝐽_𝐴𝑖≤𝑦))‖_∞𝑖=1𝜎_𝑖,𝑛((1−𝛿)‖𝑥−𝑦‖+𝛿‖𝐽_𝐴𝑖𝑥−𝐽_𝐴𝑖𝑦‖)≤(1−𝛼_𝑛)‖𝑥−𝑦‖. (3.5)

Thus, for each 𝑛∈ℕ, there is a unique 𝑧_𝑛∈𝐾 satisfying 𝑧_𝑛=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛). (3.6)

Lemma 3.3. Let 𝐾 be a nonempty closed convex subset of a real Banach space 𝐸. For each 𝑖≥1, let 𝐴_𝑖∶𝐾→𝐸 be a countably infinite family of 𝑚-accretive mappings. For 𝑛∈ℕ, let {𝑧_𝑛} be a sequence satisfying (3.6) and assume ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Then, {𝑧_𝑛} is bounded.

Proof. Let 𝑥^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)=⋂^∞_𝑖=1𝐹(𝐽_𝐴𝑖). Then, using (3.6), we obtain ‖𝑧_𝑛−𝑥^∗‖²=⟨𝛼_𝑛(𝑢−𝑥^∗)+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛−𝑥^∗),𝑗(𝑧_𝑛−𝑥^∗)⟩≤𝛼_𝑛⟨𝑢−𝑥^∗,𝑗(𝑧_𝑛−𝑥^∗)⟩+^∞_𝑖=1𝜎_𝑖,𝑛‖𝑧_𝑛−𝑥^∗‖²=𝛼_𝑛⟨𝑢−𝑥^∗,𝑗(𝑧_𝑛−𝑥^∗)⟩+(1−𝛼_𝑛)‖𝑧_𝑛−𝑥^∗‖², (3.7) which implies that ‖𝑧_𝑛−𝑥^∗‖≤‖𝑢−𝑥^∗‖. Thus, {𝑧_𝑛} is bounded.

Lemma 3.4. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸. For each 𝑖≥1, let 𝐴_𝑖∶𝐾→𝐸 be a countably infinite family of 𝑚-accretive mappings such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Let {𝛼_𝑛} be a sequence in (0,1) such that lim_𝑛→∞(𝛼_𝑛/𝜎_𝑖,𝑛)=0, for all 𝑖≥1, ∑^∞_𝑖=1𝜎_𝑖,𝑛=(1−𝛼_𝑛). Let {𝑧_𝑛} be a sequence satisfying (3.6). Then, lim_𝑛→∞‖𝑧_𝑛−𝐽_𝐴𝑖𝑧_𝑛‖=0, for all 𝑖≥1. Furthermore, if {𝜆_𝑖}^∞_𝑖=1 is a sequence in (0,1) such that ∑^∞_𝑖=1𝜆_𝑖=1; lim_𝑛→∞∑^∞_𝑖=1|𝜎_𝑖,𝑛−𝜆_𝑖|=0 and define 𝐺∶=(1−𝛿)𝐼+𝛿𝑇, where 𝑇∶=∑^∞_𝑖=1𝜆_𝑖𝐽_𝐴𝑖, then lim_𝑛→∞‖𝑧_𝑛−𝐺𝑧_𝑛‖=0.

Proof. We start by showing that lim_𝑛→∞‖𝑧_𝑛−𝐽_𝐴𝑖𝑧_𝑛‖=0, for all 𝑖≥1. For this, let 𝑆_𝑖∶=(1−𝛿)𝐼+𝛿𝐽_𝐴𝑖, where 𝐼 is the identity operator on 𝐾. Since {𝑧_𝑛}^∞_𝑛=1 is bounded, then for each 𝑖≥1 and 𝑥^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0), we have the following using (2.5): 4.2^𝑝1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖)=4.2^𝑝1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑥^∗+𝑥^∗−𝑧_𝑛‖)≤(𝑝.2^𝑝−4)‖𝑥^∗−𝑧_𝑛‖^𝑝+𝑝.2^𝑝⟨𝑆_𝑖𝑧_𝑛−𝑥^∗,𝑗_𝑝(𝑥^∗−𝑧_𝑛)⟩+4‖𝑆_𝑖𝑧_𝑛−𝑥^∗‖^𝑝≤(𝑝.2^𝑝−4)‖𝑥^∗−𝑧_𝑛‖^𝑝+𝑝.2^𝑝⟨𝑆_𝑖𝑧_𝑛−𝑧_𝑛+𝑧_𝑛−𝑥^∗,𝑗_𝑝(𝑥^∗−𝑧_𝑛)⟩+4‖𝑆_𝑖𝑧_𝑛−𝑥^∗‖^𝑝=(𝑝.2^𝑝−4)‖𝑥^∗−𝑧_𝑛‖^𝑝+𝑝.2^𝑝⟨𝑆_𝑖𝑧_𝑛−𝑧_𝑛,𝑗_𝑝(𝑥^∗−𝑧_𝑛)⟩−𝑝.2^𝑝⟨𝑥^∗−𝑧_𝑛,𝑗_𝑝(𝑥^∗−𝑧_𝑛)⟩+4‖𝑆_𝑖𝑧_𝑛−𝑥^∗‖^𝑝≤𝑝.2^𝑝⟨𝑧_𝑛−𝑆_𝑖𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩. (3.8) Hence, 4𝑝1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖)≤⟨𝑧_𝑛−𝑆_𝑖𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩, (3.9) and so, 4𝑝^∞_𝑖=1𝜎_𝑖,𝑛1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛4‖)≤𝑝^∞_𝑖=1𝜎_𝑖,𝑛⟨𝑧_𝑛−𝑆_𝑖𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩. (3.10) Using (3.6), we have ⟨𝑧_𝑛−𝑥^∗,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩=𝛼_𝑛⟨𝑢−𝑥^∗,𝑗_𝑝(𝑧_𝑛−𝑥^∗+)⟩_∞𝑖=1𝜎_𝑖,𝑛⟨𝑆_𝑖𝑧_𝑛−𝑧_𝑛+𝑧_𝑛−𝑥^∗,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩=𝛼_𝑛⟨𝑢−𝑥^∗,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩+^∞_𝑖=1𝜎_𝑖,𝑛⟨𝑆_𝑖𝑧_𝑛−𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩+(1−𝛼_𝑛)⟨𝑧_𝑛−𝑥^∗,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩, (3.11) which implies ^∞_𝑖=1𝜎_𝑖,𝑛⟨𝑧_𝑛−𝑆_𝑖𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩=𝛼_𝑛⟨𝑢−𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩. (3.12) Using this and (3.10), we get 4𝑝^∞_𝑖=1𝜎_𝑖,𝑛1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖)≤𝛼_𝑛⟨𝑢−𝑧_𝑛,𝑗_𝑝(𝑧_𝑛−𝑥^∗)⟩. (3.13) Since {𝑧_𝑛} is bounded, we have that ^∞_𝑖=1𝜎_𝑖,𝑛1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖)≤𝛼_𝑛𝑀, (3.14) for some constant 𝑀>0. This yields 1𝑔(2‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛𝛼‖)≤^𝑛𝜎_𝑖,𝑛𝑀. (3.15) Thus, since 𝑔 is continuous, strictly increasing, 𝑔(0)=0, and lim_𝑛→∞(𝛼_𝑛/𝜎_𝑖,𝑛)=0, for all 𝑖≥1, we have 12𝑔(2lim_𝑛→∞‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖)=0. (3.16) So, lim_𝑛→∞‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖=0, for all 𝑖≥1, but ‖𝑆_𝑖𝑧_𝑛−𝑧_𝑛‖=‖(1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛−𝑧_𝑛‖=‖𝛿(𝐽_𝐴𝑖𝑧_𝑛−𝑧_𝑛)‖=𝛿‖𝐽_𝐴𝑖𝑧_𝑛−𝑧_𝑛‖. (3.17) Thus, lim_𝑛→∞‖𝐽_𝐴𝑖𝑧_𝑛−𝑧_𝑛‖=0,∀𝑖≥1. (3.18) Next, we show that lim_𝑛→∞‖𝑧_𝑛−𝐺𝑧_𝑛‖=0. Observe that 𝑧_𝑛−𝐺𝑧_𝑛=𝛼_𝑛𝑢+^∞_𝑖=1(𝜎_𝑖,𝑛−𝜆_𝑖)[(1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛]. (3.19) So, ‖𝑧_𝑛−𝐺𝑧_𝑛‖≤𝛼_𝑛‖𝑢‖+𝑀^∞_𝑖=1|𝜎_𝑖,𝑛−𝜆_𝑖| (3.20) for some 𝑀>0. Hence, lim_𝑛→∞‖𝑧_𝑛−𝐺𝑧_𝑛‖=0. (3.21) This completes the proof.

Theorem 3.5. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸 with uniformly Gâteaux differentiable norm. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of 𝑚-accretive mappings such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Let {𝑧_𝑛} be a sequence satisfying (3.6). Let {𝜆_𝑖}^∞_𝑖=1 be a sequence in (0,1) such that ∑^∞_𝑖=1𝜆_𝑖=1 and lim_𝑛→∞∑^∞_𝑖=1|𝜎_𝑖,𝑛−𝜆_𝑖|=0. Let 𝐺∶=(1−𝛿)𝐼+𝛿𝑇, where 𝑇∶=∑^∞_𝑖=1𝜆_𝑖𝐽_𝐴𝑖. Then, {𝑧_𝑛} converges strongly to an element in ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0).

Proof. Observe that by Lemma 2.2, 𝑇∶=∑^∞_𝑖=1𝜆_𝑖𝐽_𝐴𝑖 is well defined, nonexpansive, and 𝐹(𝑇)=⋂^∞_𝑖=1𝐹(𝐽_𝐴𝑖)=⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Furthermore, it is easy to see that 𝐺 is nonexpansive and that 𝐹(𝐺)=𝐹(𝑇)=⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Now, since {𝑧_𝑛} is bounded and lim_𝑛→∞‖𝐺𝑧_𝑛−𝑧_𝑛‖=0, we have by Lemma 3.1 that there exists a unique 𝑧^∗ in the set Ω^∗∶={𝑥∈𝐾∩𝐵^∗∶𝜇_𝑛‖𝑧_𝑛−𝑥‖²=min_𝑦∈𝐾‖𝑧_𝑛−𝑦‖} such that 𝐺𝑧^∗=𝑧^∗, where 𝐵^∗ is a bounded closed convex nonempty subset of 𝐸 such that 𝑢,𝑧_𝑛∈𝐵^∗ for all 𝑛∈ℕ. Thus, 𝑧^∗∈𝐹(𝐺)=⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Let 𝑡∈(0,1), then by convexity of 𝐾∩𝐵^∗, we have that (1−𝑡)𝑧^∗+𝑡𝑢∈𝐾∩𝐵^∗. Thus, 𝜇_𝑛‖𝑧_𝑛−𝑧^∗‖²≤𝜇_𝑛‖𝑧_𝑛−((1−𝑡)𝑧^∗+𝑡𝑢)‖²=𝜇_𝑛‖𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗)‖². Moreover, we have, by Lemma 2.1 that ‖𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗)‖²≤‖𝑧_𝑛−𝑧^∗‖²−2𝑡⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗))⟩. (3.22) This implies that 𝜇_𝑛⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗))⟩≤0. Furthermore, since 𝐸 has uniformly Gâteaux differentiable norm, we obtain that lim_𝑡→0(⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩−⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗))⟩)=0. (3.23) Thus, given 𝜖>0, there exists 𝛿_𝜖>0 such that for all 𝑡∈(0,𝛿_𝜖) and for all 𝑛∈ℕ, ⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩<𝜖+⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗−𝑡(𝑢−𝑧^∗))⟩. (3.24) Taking Banach limit on both sides of this inequality, we obtain 𝜇_𝑛⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩≤𝜖; (3.25) and since 𝜖>0 is arbitrary, we have that 𝜇_𝑛⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩≤0. (3.26) Now, using (3.6), we have that ‖𝑧_𝑛−𝑧^∗‖²=⟨𝛼_𝑛(𝑢−𝑧^∗)+^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛)−𝑧^∗),𝑗(𝑧_𝑛−𝑧^∗)⟩≤𝛼_𝑛⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩+(1−𝛼_𝑛)‖𝑧_𝑛−𝑧^∗‖². (3.27) So, ‖𝑧_𝑛−𝑧^∗‖²≤⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩. (3.28) Again, taking Banach limit, we obtain 𝜇_𝑛‖𝑧_𝑛−𝑧^∗‖²≤𝜇_𝑛⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛−𝑧^∗)⟩≤0, (3.29) so that 𝜇_𝑛‖𝑧_𝑛−𝑧^∗‖²=0. Hence, there exists a subsequence {𝑧_𝑛𝑙}^∞_𝑙=1 of {𝑧_𝑛}^∞_𝑛=1 such that lim_𝑙→∞𝑧_𝑛𝑙=𝑧^∗. We now show that {𝑧_𝑛}^∞_𝑛=1 actually converges to 𝑧^∗. Suppose there is another subsequence {𝑧_𝑛𝑘}^∞_𝑘=1 of {𝑧_𝑛}^∞_𝑛=1 such that lim_𝑘→∞𝑧_𝑛𝑘=𝑢^∗. Then, since lim_𝑛→∞‖𝐽_𝐴𝑖𝑧_𝑛−𝑧_𝑛‖=0 and 𝐽_𝐴𝑖 is continuous for all 𝑖∈ℕ, we have that 𝑢^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Claim 1 (𝑢^∗=𝑧^∗). Suppose for contradiction that 𝑢^∗≠𝑧^∗, then ‖𝑢^∗−𝑧^∗‖>0, but using (3.6), we have that ‖𝑧_𝑛𝑙−𝑢^∗‖²=⟨𝛼_𝑛𝑙(𝑢−𝑢^∗)+^∞_𝑙=1𝜎_{𝑖,𝑛𝑙}(((1−𝛿)𝑧_𝑛𝑙+𝛿𝐽_𝐴𝑖𝑧_𝑛𝑙)−𝑢^∗),𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩=𝛼_𝑛𝑙⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+𝛼_𝑛𝑙⟨𝑧^∗−𝑧_𝑛𝑙,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+𝛼_𝑛𝑙‖𝑧_𝑛𝑙−𝑢^∗‖²+^∞_𝑖=1𝜎_{𝑖,𝑛𝑙}⟨(1−𝛿)𝑧_𝑛𝑙+𝛿𝐽_𝐴𝑖𝑧_𝑛𝑙−𝑢^∗,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩≤𝛼_𝑛𝑙⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+𝛼_𝑛𝑙⟨𝑧^∗−𝑧_𝑛𝑙,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+𝛼_𝑛𝑙‖𝑧_𝑛𝑙−𝑢^∗‖²+(1−𝛿)(1−𝛼_𝑛𝑙)‖𝑧_𝑛𝑙−𝑢^∗‖²+𝛿(1−𝛼_𝑛𝑙)‖𝑧_𝑛𝑙−𝑢^∗‖²=𝛼_𝑛𝑙⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+𝛼_𝑛𝑙⟨𝑧^∗−𝑧_𝑛𝑙,𝑗(𝑧_𝑛𝑙−𝑢^∗)⟩+‖𝑧_𝑛𝑙−𝑢^∗‖². (3.30) Thus, ⟨𝑢−𝑧^∗,𝑗(𝑢^∗−𝑧_𝑛𝑙)⟩≤‖𝑧_𝑛𝑙−𝑢^∗‖‖𝑧_𝑛𝑙−𝑧^∗‖. (3.31) Using the fact that {𝑧_𝑛}^∞_𝑛=1 is bounded and that 𝐸 has a uniformly Gâteaux differentiable norm, we obtain from (3.31) that ⟨𝑢−𝑧^∗,𝑗(𝑢^∗−𝑧^∗)⟩≤0. (3.32) Similarly, we also obtain that ⟨𝑢−𝑢^∗,𝑗(𝑧^∗−𝑢^∗)⟩≤0 or rather ⟨𝑢^∗−𝑢,𝑗(𝑢^∗−𝑧^∗)⟩≤0. (3.33) Adding (3.32) and (3.33), we have that ‖𝑧^∗−𝑢^∗‖≤0, a contradiction. Thus, 𝑧^∗=𝑢^∗. Hence, {𝑧_𝑛}^∞_𝑛=1 converges strongly to 𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). This completes the proof.

Theorem 3.6. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸 with uniformly Gâteaux differentiable norm. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of 𝑚-accretive mappings such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Let {𝑧_𝑛} be a sequence satisfying (3.6). If at least one of the mappings 𝐽_𝐴𝑖 is demicompact, then {𝑧_𝑛} converges strongly to an element of ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0).

Proof. For fixed 𝑠∈ℕ, let 𝐽_𝐴𝑠 be demicompact. Since lim_𝑛→∞‖𝑧_𝑛−𝐽_𝐴𝑠𝑧_𝑛‖=0, there exists a subsequence say {𝑧_𝑛𝑘} of {𝑧_𝑛} that converges strongly to some point 𝑧^∗∈𝐾. By continuity of 𝐽_𝐴𝑖 and the fact that lim_𝑘→∞‖𝑧_𝑛𝑘−𝐽_𝐴𝑖𝑧_𝑛𝑘‖=0, 𝑖=1,2,…, we have that 𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Assuming that there is another subsequence {𝑧_𝑛𝑗}^∞_𝑗=1 of {𝑧_𝑛} that converges strongly to a point say, 𝑞^∗, then following the argument of the last part of proof of Theorem 3.5, we obtain that {𝑧_𝑛}^∞_𝑛=1 converges strongly to 𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). This completes the proof.

Theorem 3.7. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸 with uniformly Gâteaux differentiable norm and also admits weakly sequential continuous duality mapping. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of 𝑚-accretive mappings such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Let {𝑧_𝑛} be a sequence satisfying (3.6). Then, {𝑧_𝑛} converges strongly to an element of ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0).

Proof. Since {𝑧_𝑛} is bounded, there exists a subsequence say {𝑧_𝑛𝑘} of {𝑧_𝑛} that converges weakly to some point 𝑧^∗∈𝐾. Using the demiclosedness property of (𝐼−𝐽_𝐴𝑖) at 0 for each 𝑖≥1 (since 𝐽_𝐴𝑖 is nonexpansive for each 𝑖∈ℕ, see, e.g., [31]) and the fact that lim_𝑘→∞‖𝑧_𝑛𝑘−𝐽_𝐴𝑖𝑧_𝑛𝑘‖=0, we get that 𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). We also observe from (3.6) that ‖𝑧_𝑛𝑘−𝑧^∗‖²=⟨𝛼_𝑛𝑘𝑢+^∞_𝑖=1𝜎_{𝑖,𝑛𝑘}((1−𝛿)𝑧_𝑛𝑘+𝛿𝐽_𝐴𝑖𝑧_𝑛𝑘)−𝑧^∗,𝑗(𝑧_𝑛𝑘−𝑧^∗)⟩≤𝛼_𝑛𝑘⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑘−𝑧^∗)⟩+^∞_𝑖=1𝜎_{𝑖,𝑛𝑘}‖𝑧_𝑛𝑘−𝑧^∗‖²=𝛼_𝑛𝑘⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑘−𝑧^∗)⟩+(1−𝛼_𝑛𝑘)‖𝑧_𝑛𝑘−𝑧^∗‖². (3.34) This implies that ‖𝑧_𝑛𝑘−𝑧^∗‖²≤⟨𝑢−𝑧^∗,𝑗(𝑧_𝑛𝑘−𝑧^∗)⟩. Using the fact that 𝑗 is weakly sequential continuous, then we have from the last inequality that {𝑧_𝑛𝑘} converges strongly to 𝑧^∗. Then following the argument of the last part of proof of Theorem 3.5, we obtain that {𝑧_𝑛}^∞_𝑛=1 converges strongly to 𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). This completes the proof.

4. Convergence Theorems for Countably Infinite Family of 𝑚-Accretive Mappings

For the rest of this paper, {𝛼_𝑛}^∞_𝑛=1 and {𝜎_𝑖,𝑛}^∞_𝑛=1 are in (0,1) satisfying the following additional conditions:

(i)lim_𝑛→∞𝛼_𝑛=0, (ii)∑^∞_𝑛=1𝛼_𝑛=∞, (iii)lim_𝑛→∞∑^∞_𝑖=1|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|=0.

Theorem 4.1. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸 with uniformly Gâteaux differentiable norm. Let 𝐴_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of 𝑚-accretive mappings such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. For fixed 𝛿∈[𝛾₁,𝛾₂], for some 𝛾₁,𝛾₂∈(0,1), 𝑢∈𝐾, let {𝑥_𝑛}^∞_𝑛=1 be generated by 𝑥₁∈𝐾,𝑥_𝑛+1=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛),𝑛≥1, (4.1) then lim_𝑛→∞∑^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛−𝑥_𝑛)=0. Furthermore, if {𝛼_𝑛}_𝑛≥1 is such that lim_𝑛→∞∑^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛−𝑥_𝑛)𝛼_𝑛=0, (4.2) then {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common zero of {𝐴_𝑖}^∞_𝑖=1.

Proof. Using mathematical induction, it is easy to see that for 𝑥^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0) fixed ‖𝑥_𝑛−𝑥^∗‖≤max{‖𝑢−𝑥^∗‖,‖𝑥₁−𝑥^∗‖},∀𝑛≥1. (4.3) Hence, {𝑥_𝑛}^∞_𝑛=1 is bounded and so {𝐽_𝐴𝑖𝑥_𝑛}^∞_𝑛=1 is also bounded.
Now, define the sequences {𝛽_𝑛}^∞_𝑛=1 and {𝑦_𝑛}^∞_𝑛=1 by 𝛽_𝑛∶=(1−𝛿)𝛼_𝑛+𝛿 and 𝑦_𝑛∶=(𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑥_𝑛)/𝛽_𝑛. Then, 𝑦_𝑛=𝛼_𝑛𝑢+𝛿∑_𝑖≥1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)𝛽_𝑛. (4.4) Observe that {𝑦_𝑛}^∞_𝑛=1 is bounded and that ‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛𝛼‖≤|^𝑛+11−𝛽_𝑛+1−𝛼_𝑛1−𝛽_𝑛|‖𝑢‖+|𝛿(1−𝛼_𝑛+1)𝛽_𝑛+1−1|‖𝑥_𝑛+1−𝑥_𝑛‖+𝛿𝑀𝛽_𝑛+1𝛽_𝑛^∞_𝑖=1|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|+𝛿𝑀𝛽_𝑛+1𝛽_𝑛|𝛽_𝑛+1−𝛽_𝑛|, (4.5) for some 𝑀>0. Thus, limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. (4.6) Hence, by Lemma 2.4, we have lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0. Consequently, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=lim_𝑛→∞𝛽_𝑛‖𝑦_𝑛−𝑥_𝑛‖=0. (4.7) From (4.1), we have that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛(𝑢−𝑥_𝑛)+^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛), (4.8) which implies that ‖∑^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛−𝑥_𝑛)‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+𝛼_𝑛‖𝑢−𝑥_𝑛‖ and thus lim_𝑛→∞‖^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛)‖=0. (4.9) Let {𝑧_𝑛}^∞_𝑛=1 be a sequence satisfying (3.6). Then, by Theorem 3.5, 𝑧_𝑛→𝑧^∗∈⋂^∞_𝑖=1𝐴_𝑖⁻¹(0). Using Lemma 2.1, we have that ‖𝑧_𝑛−𝑥_𝑛‖²≤‖^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑧_𝑛+𝛿𝐽_𝐴𝑖𝑧_𝑛)−((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)+((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛)‖²+2𝛼_𝑛⟨𝑢−𝑥_𝑛,𝑗(𝑧_𝑛−𝑥_𝑛)⟩≤((1−𝛼_𝑛)‖𝑧_𝑛−𝑥_𝑛‖+‖^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛)‖)²+2𝛼_𝑛⟨𝑢−𝑧_𝑛,𝑗(𝑧_𝑛−𝑥_𝑛)⟩. (4.10) This implies that ⟨𝑢−𝑧_𝑛,𝑗(𝑥_𝑛−𝑧_𝑛≤𝛼)⟩_𝑛2‖𝑧_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑛)‖𝑧_𝑛−𝑥_𝑛‖⋅‖∑^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛)‖𝛼_𝑛+‖∑^∞_𝑖=1𝜎_𝑖,𝑛(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑥_𝑛)‖²2𝛼_𝑛 (4.11) and hence, limsup_𝑛→∞⟨𝑢−𝑧_𝑛,𝑗(𝑥_𝑛−𝑧_𝑛)⟩≤0. (4.12) Moreover, we have that ⟨𝑢−𝑧_𝑛,𝑗(𝑥_𝑛−𝑧_𝑛)⟩=⟨𝑢−𝑧^∗,𝑗(𝑥_𝑛−𝑧^∗)⟩+⟨𝑢−𝑧^∗,𝑗(𝑥_𝑛−𝑧_𝑛)−𝑗(𝑥_𝑛−𝑧^∗)⟩+⟨𝑧^∗−𝑧_𝑛,𝑗(𝑥_𝑛−𝑧_𝑛)⟩. (4.13) Using the boundedness of {𝑥_𝑛}^∞_𝑛=1, we have ⟨𝑧^∗−𝑧_𝑛,𝑗(𝑥_𝑛−𝑧_𝑛)⟩→0,as𝑛→∞. (4.14) Also, since 𝑗 is norm-to-weak^∗ uniformly continuous on bounded subsets of 𝐸, we have that ⟨𝑢−𝑧^∗,𝑗(𝑥_𝑛−𝑧_𝑛)−𝑗(𝑥_𝑛−𝑧^∗)⟩→0,as𝑛→∞. (4.15) From (4.12) and (4.13), we obtain that limsup_𝑛→∞⟨𝑢−𝑧^∗,𝑗(𝑥_𝑛−𝑧^∗)⟩≤0. (4.16) Finally, using Lemma 2.1 on (4.1), we get ‖𝑥_𝑛+1−𝑧^∗‖²≤‖^∞_𝑖=1𝜎_𝑖,𝑛+1(((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛)−𝑧^∗)‖²+2𝛼_𝑛⟨𝑢−𝑧^∗,𝑗(𝑥_𝑛+1−𝑧^∗)⟩≤(1−𝛼_𝑛)‖𝑥_𝑛−𝑧^∗‖²+2𝛼_𝑛⟨𝑢−𝑧,𝑗(𝑥_𝑛+1−𝑧^∗)⟩. (4.17) Using (4.16) and Lemma 2.3 in (4.17), we get that {𝑥_𝑛}^∞_𝑛=1 converges strongly to common zero of the family {𝐴_𝑖}^∞_𝑖=1 of 𝑚-accretive operators.

Remark 4.2. If 𝐾 is replaced with 𝐸 in Theorems 3.5, 3.6, 3.7, and 4.1, then by Lemma 2.5, the assumption that 𝐴_𝑖 is 𝑚-accretive for each 𝑖≥1 could be replaced with 𝐴_𝑖 is continuous for each 𝑖≥1.

Hence, we have the following theorem.

Theorem 4.3. Let 𝐸 be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let 𝐴_𝑖∶𝐸→𝐸, 𝑖=1,2,…, be a countably infinite family of continuous accretive operators such that ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)≠∅. For arbitrary but fixed 𝛿∈(0,1), 𝑢∈𝐾, let {𝑥_𝑛}^∞_𝑛=1 be generated by 𝑥₁∈𝐾, 𝑥_𝑛+1=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝐴𝑖𝑥_𝑛),𝑛≥1, (4.18) then, {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common zero of {𝐴_𝑖}^∞_𝑖=1.

Proof. By Lemma 2.5, we have that 𝐴_𝑖 is 𝑚-accretive for each 𝑖≥1. Then, the rest of the proof follows from Theorem 4.1.

We also have the following theorems.

Theorem 4.4. Let 𝐾, 𝐸, 𝐴_𝑖 and {𝑥_𝑛}_𝑛≥1 be as an Theorem 4.1. Suppose that at least one of 𝐽_𝐴𝑖 is demicompact, then {𝑥_𝑛}_𝑛≥1 converges strongly to a common zero of 𝐴_𝑖, 𝑖=1,2,….

Proof. The proof follows as in the proof of Theorem 4.1 but using Theorem 3.6.

Theorem 4.5. Let 𝐾, 𝐸, 𝐴_𝑖 and {𝑥_𝑛}_𝑛≥1 be as an Theorem 4.1. Suppose that, in addition, 𝐸 admits weakly sequential continuous duality mapping, then {𝑥_𝑛}_𝑛≥1 converges strongly to a common zero of 𝐴_𝑖, 𝑖=1,2,….

Proof. The proof follows as in the proof of Theorem 4.1 but using Theorem 3.7.

5. Convergence Theorems for Countably Infinite Family of Pseudocontractive Mappings

Theorem 5.1. Let 𝐾 be a nonempty closed convex subset of a uniformly convex real Banach space 𝐸 with uniformly Gâteaux differentiable norm. Let 𝑇_𝑖∶𝐾→𝐸, 𝑖=1,2,…, be a countably infinite family of pseudocontractive mappings such that for each 𝑖≥1, (𝐼−𝑇_𝑖) is 𝑚-accretive and ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝐽_𝑇𝑖=(2𝐼−𝑇_𝑖)⁻¹, 𝑖≥1. For fixed 𝛿∈[𝛾₁,𝛾₂], for some 𝛾₁,𝛾₂∈(0,1) and 𝑢∈𝐾, let {𝑥_𝑛}^∞_𝑛=1 be generated by 𝑥₁∈𝐾: 𝑥_𝑛+1=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝑇𝑖𝑥_𝑛),𝑛≥1, (5.1) then {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^∞_𝑖=1.

Proof. Put 𝐴_𝑖∶=(𝐼−𝑇_𝑖), 𝑖≥1. It is then obvious that 𝐴_𝑖⁻¹(0)=𝐹(𝑇_𝑖), for all 𝑖∈ℕ and hence ⋂^∞_𝑖=1𝐴_𝑖⁻¹(0)=⋂^∞_𝑖=1𝐹(𝑇_𝑖). Furthermore, 𝐴_𝑖 is 𝑚-accretive for each 𝑖≥1. Thus, we obtain the conclusion from Theorem 4.1 with 𝐽_𝐴𝑖 in the definition of 𝑧_𝑛 replaced with 𝐽_𝑇𝑖.

Theorem 5.2. Let 𝐸 be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. For each 𝑖≥1, let 𝑇_𝑖∶𝐸→𝐸 be a countably infinite family of continuous pseudocontractive mappings such that for each 𝑖≥1 and ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝐽_𝑇𝑖=(2𝐼−𝑇_𝑖)⁻¹, 𝑖≥1. For arbitrary but fixed 𝛿∈[𝛾₁,𝛾₂], for some 𝛾₁,𝛾₂∈(0,1) and 𝑢∈𝐾, let {𝑥_𝑛}^∞_𝑛=1 be generated by 𝑥₁∈𝐾: 𝑥_𝑛+1=𝛼_𝑛𝑢+^∞_𝑖=1𝜎_𝑖,𝑛((1−𝛿)𝑥_𝑛+𝛿𝐽_𝑇𝑖𝑥_𝑛),𝑛≥1, (5.2) then {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^∞_𝑖=1.

Remark 5.3. Theorems similar to Theorems 4.4 and 4.5 could also be obtained for countably infinite family of pseudocontractive mappings.

Remark 5.4. Prototypes for our iteration parameters are 𝛼_𝑛=1𝑛+1,𝜎_𝑖,𝑛=𝑛2^𝑖(𝑛+1),𝜆_𝑖=12^𝑖. (5.3)

Remark 5.5. The addition of bounded error terms in any of our recursion formulas leads to no further generalization.

Remark 5.6. If 𝑓∶𝐾→𝐾 is a contraction map and we replace 𝑢 by 𝑓(𝑥_𝑛) in the recursion formulas of our theorems, we obtain what some authors now call viscosity iteration process. We observe that all our theorems in this paper carry over trivially to the so-called viscosity process. One simply replaces 𝑢 by 𝑓(𝑥_𝑛), repeats the argument of this paper, using the fact that 𝑓 is a contraction map.

Acknowledgments

The first author's research was supported by the Japanese Mori Fellowship of UNESCO at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The second author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.