Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

Abstract

It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in 𝐹 to a certain variational inequality, where 𝐹∶=∩^∞_𝑛=0𝐹(𝑇_𝑛) stands for the common fixed-point set of the family of multivalued nonexpansive mapping {𝑇_𝑛}.

1. Introduction

Let 𝑋 be a Banach space with dual 𝑋^∗, and let 𝐾 be a nonempty subset of 𝑋. A gauge function is a continuous strictly increasing function 𝜑∶𝑅⁺→𝑅⁺ such that 𝜑(0)=0 and lim_𝑡→∞𝜑(𝑡)=∞. The duality mapping 𝐽_𝜑∶𝑋→𝑋^∗ associated with a gauge function 𝜑 is defined by 𝐽_𝜑(𝑥)∶={𝑓∈𝑋^∗∶⟨𝑥,𝑓⟩=‖𝑥‖‖𝑓‖,‖𝑓‖=𝜑(‖𝑥‖)}, 𝑥∈𝑋, where ⟨⋅,⋅⟩ denotes the generalized duality pairing. In the particular case 𝜑(𝑡)=𝑡, the duality map 𝐽=𝐽_𝜑 is called the normalized duality map. We note that 𝐽_𝜑(𝑥)=(𝜑(‖𝑥‖)/‖𝑥‖)𝐽(𝑥). It is known that if 𝑋 is smooth, then 𝐽_𝜑 is single valued and norm to weak* continuous (see [1]). When {𝑥_𝑛} is a sequence in 𝑋, then 𝑥_𝑛→𝑥(𝑥_𝑛⇀𝑥,𝑥_𝑛⇁𝑥) will denote strong (weak, weak*) convergence of the sequence {𝑥_𝑛} to 𝑥. s

Following Browder [2], we say that a Banach space 𝑋 has the weakly continuous duality mapping if there exists a gauge function 𝜑 for which the duality map 𝐽_𝜑 is single valued and weak to weak* sequentially continuous, that is, if {𝑥_𝑛} is a sequence in 𝑋 weakly convergent to a point 𝑥, then the sequence {𝐽_𝜑(𝑥_𝑛)} converges weak* to 𝐽_𝜑(𝑥). It is known that 𝑙_𝑝(1<𝑝<1) spaces have a weakly continuous duality mapping 𝐽_𝜑 with a gauge 𝜑(𝑡)=𝑡^𝑝−1. Setting Φ(𝑡)=^𝑡₀𝜑(𝜏)𝑑𝜏,𝑡≥0, (1.1) it is easy to see that Φ(𝑡) is a convex function and 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖), for 𝑥∈𝑋, where 𝜕 denotes the subdifferential in the sense of convex analysis. We will denote by 2^𝑋 the family of all subsets of 𝑋, by CB(𝑋) the family of all nonempty closed bounded subsets of 𝑋, and by 𝐶(𝑋) the family of all nonempty compact subsets of 𝑋. A multivalued mapping 𝑇∶𝐾→2^𝑋 is said to be nonexpansive (resp., contractive) if 𝐻≤(),𝑇𝑥,𝑇𝑦𝑥−𝑦,𝑥,𝑦∈𝐾,resp.,𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑥−𝑦,forsome𝑘∈0,1 (1.2) where 𝐻(⋅,⋅) denotes the Hausdorff metric on CB(𝑋) defined by 𝐻(𝐴,𝐵)∶=maxsup_𝑥∈𝐴inf_𝑦∈𝐵𝑥−𝑦,sup_𝑦∈𝐵inf_𝑥∈𝐴𝑥−𝑦,𝐴,𝐵∈CB(𝑋). (1.3) Since Banach's contraction mapping principle was extended nicely to multivalued mappings by Nadler in 1969 (see [3]), many authors have studied the fixed-point theory for multivalued mappings.

In this paper, we construct two viscosity approximation sequences for a family of multivalued nonexpansive mappings in Banach spaces. Let 𝐾 be a nonempty closed convex subset of Banach space 𝑋 and let 𝑇_𝑛∶𝐾→𝐶(𝐾), 𝑛=1,2… be a family of multivalued nonexpansive mapping, 𝑓∶𝐾→𝐾 is a contraction mapping with constant 𝛼∈(0,1). Let 𝛼_𝑛∈(0,1), 𝛽_𝑛∈(0,1). For any given 𝑥₀∈𝐾, let 𝑦₀∈𝑇₀𝑥₀ such that 𝑥₁=𝛼₀𝑓𝑥₀+1−𝛼₀𝑦₀. (1.4) From Nadler Theorem (see [3]), we can choose 𝑦₁∈𝑇₁𝑥₁ such that 𝑦₀−𝑦₁≤𝐻𝑇₀𝑥₀,𝑇₁𝑥₁. (1.5) Inductively, we can get the sequence {𝑥_𝑛} as follows:𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛∈𝑁, (1.6) where, for each 𝑛∈𝑁, 𝑦_𝑛∈𝑇_𝑛𝑥_𝑛 such that 𝑦_𝑛+1−𝑦_𝑛≤𝐻𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛. (1.7) Similarly, we also have the following multivalued version of the modified Mann iteration: 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑦_𝑛, (1.8) and 𝑦_𝑛∈𝑇_𝑛𝑥_𝑛 such that ‖𝑦_𝑛+1−𝑦_𝑛‖≤𝐻(𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛). Then, {𝑥_𝑛} is said to satisfy Condition (A^′) if for any subsequence 𝑥_𝑛𝑘⇀𝑥 and 𝑑(𝑥_𝑛+1,𝑇_𝑛(𝑥_𝑛))→0 implies that 𝑥∈𝐹, where 𝐹∶=∩^∞_𝑖=0𝐹(𝑇_𝑛)≠∅ is the common fixed-point set of the family of multivalued mapping {𝑇_𝑛}. We give an example of a family of multivalued nonexpansive mappings with Condition (A^′) as follows.

Example 1.1. Take 𝑋=𝑅 and 𝑇_𝑛=𝑇 (for all 𝑛≥0), where 𝑇 is defined by 𝑇(𝑥)={0}1,𝑥≤1,𝑥−2,12−𝑥,otherwise. (1.9) Let 𝑓∶𝑅→{0} and 𝛼_𝑛=1/𝑛, 𝑛≥2, then 𝐹={0} and the iteration (1.6), reduced to 𝑥_𝑛+1=11−𝑦𝑛+2^𝑛,∀𝑛≥0, (1.10) where 𝑦_𝑛∈𝑇𝑥_𝑛, and it satisfies Condition (A^′). In fact, if 𝑥₀≤1, then (for all 𝑛∈ℕ,𝑛>0)𝑥_𝑛=0 and Condition (A^′) is automatically satisfied. If 𝑥₀>1, then there exists an integer 𝑝≥2, such that 𝑥₀∈𝑝𝑝+14−12,𝑝+1𝑝+24−12,𝑥_𝑝−1=1𝑝𝑥₀−𝑝𝑝−14. (1.11) Then, 𝑦_𝑝∈𝑇𝑥_𝑝−1={0}; hence, 𝑥_𝑛=0(for all 𝑛≥𝑝), from which we deduce that Condition (A^′) is satisfied.

2. Preliminaries

Let 𝐾⊂𝑋 be a closed convex and 𝑄 a mapping of 𝑋 onto 𝐾, then 𝑄 is said to be sunny if 𝑄(𝑄(𝑥)+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥) for all 𝑥∈𝑋 and 𝑡≥0. A mapping 𝑄 of 𝑋 into 𝑋 is said to be a retraction if 𝑄²=𝑄. A subset 𝐾 of 𝑋 is said to be a sunny nonexpansive retract of 𝑋 if there exists a sunny nonexpansive retraction of 𝑋 onto 𝐾, and it is said to be a nonexpansive retract of 𝑋. If 𝑋=𝐻, the metric projection 𝑃 is a sunny nonexpansive retraction from 𝐻 to any closed convex subset of 𝐻. The following Lemmas will be useful in this paper.

Lemma 2.1 (see [4]). Let 𝐾 be a nonempty convex subset of a smooth Banach space 𝑋, let 𝐽∶𝑋→𝑋^∗ be the (normalized) duality mapping of 𝑋, and let 𝑄∶𝑋→𝐾 be a retraction, then the following are equivalent: (1)⟨𝑥−𝑃𝑥,𝑗(𝑦−𝑃𝑥)⟩≤0 for all 𝑥∈𝑋 and 𝑦∈𝐾,(2)𝑄 is both sunny and nonexpansive. We note that Lemma 2.1 still holds if the normalized duality map 𝐽 is replaced with the general duality map 𝐽_𝜑, where 𝜑 is a gauge function.

Lemma 2.2 (see [5]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛽_𝑛,𝑛≥0, (2.1) where {𝛾_𝑛}⊂(0,1) and {𝛽_𝑛} is a real number sequence such that (i)∑^∞_𝑛=0𝛾_𝑛=∞,(ii)either limsup_𝑛→∞(𝛽_𝑛/𝛾_𝑛)≤0 or ∑^∞_𝑛=0|𝛽_𝑛|<∞, then {𝑎_𝑛} converges to zero, as 𝑛→∞.

Lemma 2.3 (see [1]). Let 𝑋 be a real Banach space, then for all 𝑥,𝑦∈𝑋, one gets that Φ‖𝑥+𝑦‖≤Φ(‖𝑥‖)+⟨𝑦,𝑗_𝜑𝑥+𝑦⟩,∀𝑗_𝜑∈𝐽_𝜑. (2.2)

Lemma 2.4 (see [6]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 such that 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛,𝑛≥0, (2.3) where {𝛾_𝑛} is a sequence in [0,1] such that 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1. (2.4) Assume that limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0, then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

3. Main Results

Theorem 3.1. Let 𝑋 be a reflexive Banach space with weakly sequentially continuous duality mapping 𝐽_𝜑 for some gauge 𝜑, let 𝐾 be a nonempty closed convex subset of 𝑋, and let 𝑇_𝑛∶𝐾→𝐶(𝐾), 𝑛=0,1,2…, be a family of multivalued nonexpansive mappings with 𝐹≠∅ which is sunny nonexpansive retract of 𝐾 with 𝑄 a nonexpansive retraction. Furthermore, 𝑇_𝑛(𝑝)={𝑝} for any fixed-point 𝑝∈𝐹, {𝑥_𝑛} is defined by (1.6), and 𝛼_𝑛∈(0,1) satisfies the following conditions: (1)𝛼_𝑛→0 as 𝑛→∞, (2)∑^∞_𝑛=0𝛼_𝑛=∞,(3){𝑥_𝑛} satisfies Condition (𝐴′). Then, {𝑥_𝑛} converges strongly to a common fixed-point 𝑥=𝑄(𝑓(𝑥)) of a family 𝑇_𝑛, 𝑛=0,1,2…, as 𝑛→∞. Moreover, 𝑥 is the unique solution in 𝐹 to the variational inequality 𝑓(𝑥)−𝑥,𝑗_𝜑𝑦−𝑥≤0,∀𝑦∈𝐹. (3.1)

Proof. First, we show the uniqueness of the solution to the variational inequality (3.1) in 𝑋. In fact, let 𝑦∈𝐹 be another solution of (3.1) in 𝐹, then we have 𝑓(𝑥)−𝑥,𝑗_𝜑𝑦−𝑥≤0,𝑓𝑦−𝑦,𝑗_𝜑𝑥−𝑦≤0. (3.2) From (3.2), we have that (1−𝛼)𝜑𝑥−𝑦𝑥−𝑦≤0. (3.3) We must have 𝑥=𝑦 and the uniqueness is proved. Let 𝑝∈𝐹, then, from iteration (1.6), we obtain that 𝑥_𝑛+1−𝑝≤𝑥_𝑛+1−𝛼_𝑛𝑓𝑝−1−𝛼_𝑛𝑝+𝛼_𝑛𝑓𝑝+1−𝛼_𝑛=𝛼𝑝−𝑝_𝑛𝑓𝑥_𝑛−𝑓𝑝+1−𝛼_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝≤𝛼_𝑛𝛼𝑥_𝑛−𝑝+1−𝛼_𝑛𝐻𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑝+𝛼_𝑛𝑓𝑝≤()𝛼−𝑝1−1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝. (3.4) Using an induction, we obtain ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₀−𝑝‖,(1/(1−𝛼))‖𝑓(𝑝)−𝑝‖}, for all integers 𝑛, thus, {𝑥_𝑛} is bounded and so are {𝑇_𝑛𝑥_𝑛} and {𝑓(𝑥_𝑛)}. This implies that 𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑦_𝑛⟶0as𝑛⟶∞. (3.5) Next, we will show that limsup_𝑛→∞⟨𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥⟩≤0. (3.6) Since 𝑋 is reflexive and {𝑥_𝑛} is bounded, we may assume that 𝑥_𝑛𝑘⇀𝑞 such that limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥=limsup_𝑘→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛𝑘−𝑥. (3.7) From (3.5) and {𝑥_𝑛} satisfying Condition (A^′), we obtain that 𝑞∈𝐹. On the other hand, we notice that the assumption that the duality mapping 𝐽_𝜑 is weakly continuous implies that 𝑋 is smooth; from Lemma 2.1, we have limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥=limsup_𝑘→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛𝑘−𝑥=⟨𝑓(𝑥)−𝑥,𝑗_𝜑𝑞−𝑥⟩=⟨𝑄(𝑥)−𝑥,𝑗_𝜑𝑞−𝑥⟩≤0. (3.8) Finally, we will show that 𝑥_𝑛→𝑥 as 𝑛→∞. From iteration (1.6) and Lemma 2.3, we get that Φ‖𝑥_𝑛+1−𝑥‖≤Φ𝛼_𝑛𝑓𝑥_𝑛−𝑓(𝑥)+1−𝛼_𝑛𝑦_𝑛−𝑥+𝛼_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥≤Φ𝛼_𝑛𝛼‖𝑥_𝑛−𝑥‖+1−𝛼_𝑛𝐻𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥+𝛼_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥≤1−𝛼_𝑛(1−𝛼)Φ𝑥_𝑛−𝑥+𝛼_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥. (3.9) Lemma 2.2 gives that 𝑥_𝑛→𝑥 as 𝑛→∞. Moreover, 𝑥 satisfying the variational inequality follows from the property of 𝑄.

Let 𝑓≡𝑢∈𝐾 in iteration (1.6) be a constant mapping, then 𝑥=𝑄𝑢. In fact, we have the following corollary.

Corollary 3.2. Let {𝑥_𝑛} and 𝑇_𝑛 be as in Theorem 3.1, 𝑓≡𝑢∈𝐾, then {𝑥_𝑛} converges strongly to a common fixed-point 𝑥=𝑄(𝑢) of a family 𝑇_𝑛, 𝑛=0,1,2…, as 𝑛→∞. Moreover, 𝑥 is the unique solution in 𝐹 to the variational inequality 𝑢−𝑄(𝑢),𝑗_𝜑𝑦−𝑄(𝑢)≤0,∀𝑦∈𝐹. (3.10) If 𝑋=𝐻, then the condition that 𝐹 is a sunny nonexpansive retract of 𝐾 in Theorem 3.1 is not necessary, and one has the following Corollary.

Corollary 3.3. Let 𝐻 be a Hilbert space with weakly sequentially continuous duality mapping 𝐽_𝜑 for some gauge 𝜑, and let {𝑥_𝑛} and 𝑇_𝑛 be as in Theorem 3.1, then {𝑥_𝑛} converges strongly to a common fixed-point 𝑥=𝑃_𝐹𝑓(𝑥) of a family of 𝑇_𝑛, 𝑛=0,1,2…, where 𝑃_𝐹 is the metric projection from 𝐾 onto 𝐹.

Proof. It is well known that 𝐻 is reflexive; by Propositions 2.3 and 2.6(ii) of [7], we get that 𝐹 is closed and convex, and hence the projection mapping 𝑃_𝐹 is sunny nonexpansive retraction mapping, and the result follows from Theorem 3.1.

Corollary 3.4. Let 𝑋 be a real smooth Banach space, let 𝐾 be a nonempty compact subset of 𝑋, and let 𝑇_𝑛 and {𝑥_𝑛} be as in Theorem 3.1, then {𝑥_𝑛} converges strongly to a common fixed-point 𝑥=𝑄(𝑓(𝑥)) of a family of 𝑇_𝑛, 𝑛=0,1,2…, as 𝑛→∞. Moreover, 𝑥 is the unique solution in 𝐹 to the variational inequality 𝑓(𝑥)−𝑥,𝑗_𝜑𝑦−𝑥≤0,∀𝑦∈𝐹. (3.11)

Proof. Following the method of the proof of Theorem 3.1, we get that 𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑦_𝑛⟶0as𝑛⟶∞. (3.12) Next, we will show that limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥≤0. (3.13) Since 𝐾 is compact and {𝑥_𝑛} is bounded, we can assume that there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑞∈𝐾, limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥=lim_𝑘→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛𝑘−𝑥. (3.14) From (3.12) and {𝑥_𝑛} satisfying Condition (A^′), we obtain that 𝑞∈𝐹. On the other hand, from the fact that 𝑋 is smooth, the duality being norm to weak* continuous, and the standard characterization of retraction on 𝐹, we obtain that limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥=lim_𝑘→∞⟨𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛𝑘−𝑥⟩=⟨𝑓(𝑥)−𝑥,𝑗_𝜑𝑞−𝑥⟩=⟨𝑄(𝑥)−𝑥,𝑗_𝜑𝑞−𝑥⟩≤0. (3.15) Now, following the method of the proof of Theorem 3.1, we get the required result.

Theorem 3.5. Let 𝑋 be a reflexive Banach space with weakly sequentially continuous duality mapping 𝐽_𝜑 for some gauge 𝜑, let 𝐾 be a nonempty closed convex subset of 𝑋, and let 𝑇_𝑛∶𝐾→𝐶(𝐾), 𝑛=0,1,2…, be a family of multivalued nonexpansive mappings with 𝐹≠∅ which is sunny nonexpansive retract of 𝐾 with 𝑄 a nonexpansive retraction. 𝐻(𝑇_𝑛+1𝑥,𝑇_𝑛𝑦)≤‖𝑥−𝑦‖ for arbitrary 𝑛∈ℕ. Furthermore, 𝑇_𝑛(𝑝)={𝑝} for any fixed-point 𝑝∈𝐹. {𝑥_𝑛} is defined by (1.8) and 𝛼_𝑛, 𝛽_𝑛 satisfy the following conditions: (i)𝛽_𝑛→0 as 𝑛→∞, (ii)∑^∞_𝑛=0𝛽_𝑛=∞, (iii)0<𝑙𝑖𝑚inf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. If {𝑥_𝑛} satisfies Condition (A^′), then {𝑥_𝑛} converges strongly to a common fixed-point 𝑥=𝑄(𝑓(𝑥)) of a family of 𝑇_𝑛, 𝑛=0,1,2…, as 𝑛→∞. Moreover, 𝑥 is the unique solution in 𝐹 to the variational inequality ⟨𝑓(𝑥)−𝑥,𝑗_𝜑𝑦−𝑥⟩≤0,∀𝑦∈𝐹. (3.16)

Proof. We first show that the sequence {𝑥_𝑛} defined by (1.8) is bounded. In fact, take 𝑝∈𝐹, noting that 𝑇_𝑛(𝑝)={𝑝}, we have 𝑥_𝑛+1−𝑝=1−𝛼_𝑛−𝛽_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛=−𝑝1−𝛼_𝑛−𝛽_𝑛𝑦_𝑛−𝑇_𝑛𝑝+𝛼_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛≤−𝑝1−𝛼_𝑛−𝛽_𝑛𝐻𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑝+𝛼_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛≤−𝑝1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛≤−𝑝1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝛼𝑥_𝑛+𝑓𝑝≤()𝛽−𝑝−𝑝1+𝛼−1_𝑛𝑥_𝑛−𝑝+𝛽_𝑛()𝑓𝑝1−𝛼−𝑝.1−𝛼 (3.17) It follows from induction that 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝛼, (3.18) so are {𝑦_𝑛} and {𝑓(𝑥_𝑛)}. Thus, we have that lim_𝑛→∞𝛽_𝑛𝑓𝑥_𝑛−𝑦_𝑛=0. (3.19) Next, we show that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛=0. (3.20) Let 𝜆_𝑛=𝛽_𝑛/(1−𝛼_𝑛) and 𝑧_𝑛=𝜆_𝑛𝑓(𝑥_𝑛)+(1−𝜆_𝑛)𝑦_𝑛, then lim_𝑛→∞𝜆_𝑛=0,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛. (3.21) Therefore, we have for some appropriate constant 𝑀>0 that the following inequality: 𝑧_𝑛+1−𝑧_𝑛=𝜆_𝑛+1𝑓𝑥_𝑛+1+1−𝜆_𝑛+1𝑦_𝑛+1−𝜆_𝑛𝑓𝑥_𝑛+1−𝜆_𝑛𝑦_𝑛≤𝜆_𝑛+1−𝜆_𝑛𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝑦_𝑛+1−𝑦_𝑛+𝜆_𝑛𝑦_𝑛+𝜆_𝑛+1𝑦_𝑛+1≤𝜆_𝑛+1−𝜆_𝑛𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝐻𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+𝜆_𝑛+𝜆_𝑛+1𝑀≤𝜆_𝑛+1−𝜆_𝑛𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+𝜆_𝑛+1𝑀 (3.22) holds. Thus, limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤lim_𝑛→∞(|𝜆_𝑛+1−𝜆_𝑛|‖𝑓(𝑥_𝑛+1)−𝑓(𝑥_𝑛)‖+(𝜆_𝑛+𝜆_𝑛+1)𝑀)=0. By Lemma 2.4, we obtain lim_𝑛→∞𝑥_𝑛−𝑧_𝑛𝑥=0,_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑦_𝑛=𝑥_𝑛−𝑧_𝑛+𝜆_𝑛𝑓𝑥_𝑛−𝑦_𝑛⟶0. (3.23) Therefore, we have 𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑦_𝑛≤𝛽_𝑛𝑓𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑥_𝑛−𝑦_𝑛⟶0. (3.24) Using (3.20) and {𝑥_𝑛} satisfying Condition (A^′), we can use the same argumentation as Theorem 3.1 proves that 𝑥∈𝐹 and limsup_𝑛→∞𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥≤0. (3.25) Finally, we show that 𝑥_𝑛→𝑥 as 𝑛→∞. In fact, from iteration (1.8) and Lemma 2.3, we have Φ𝑥_𝑛+1−𝑥=Φ𝛽_𝑛𝑓𝑥_𝑛+𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑦_𝑛−𝑥=Φ𝛼_𝑛𝑥_𝑛−𝑥+1−𝛼_𝑛−𝛽_𝑛𝑦_𝑛−𝑥+𝛽_𝑛𝑓𝑥_𝑛−𝑓(𝑥)+𝛽_𝑛𝑓(𝑥)−𝑥≤Φ𝛼_𝑛𝑥_𝑛−𝑥+1−𝛼_𝑛−𝛽_𝑛𝐻𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥+𝛼𝛽_𝑛𝑥_𝑛−𝑥+𝛽_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥≤1−(1−𝛼)𝛽_𝑛Φ𝑥_𝑛−𝑥+𝛽_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥. (3.26) From (ii) and (3.25), it then follows that ^∞_𝑛=0(1−𝛼)𝛽_𝑛=∞,limsup_𝑛𝑓(𝑥)−𝑥,𝑗_𝜑𝑥_𝑛+1−𝑥1−𝛼≤0. (3.27) Apply Lemma 2.2 to conclude that 𝑥_𝑛→𝑥.