# Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

Authors: Mathew Aibinu; Surendra Thakur; Sibusiso Moyo
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point ofλ-strictly pseudocontractive mappings, solution of α-inverse-strongly monotone mappings, and solution of integral equations of Fredholm type.


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## Body

## 1. Introduction

The Viscosity Approximation Method (VAM) for solving nonlinear operator equations has recently attracted much attention. In 1996, Attouch [1] considered the viscosity solutions of minimization problems. In 2000, Moudafi [2] introduced an explicit viscosity method for nonexpansive mappings. The iterative explicit viscosity sequence xnn=1∞ is defined by
(1)xn+1=αnfxn+1−αnTxn,n∈ℕ,where αnn=1∞⊂0,1,f is a contraction on K and the nonexpansive mapping T:K⟶K is also defined on K, which is a nonempty closed convex subset of a Hilbert space H. The sequence (1) converges strongly to a fixed point of a nonexpansive mapping T under suitable conditions. Xu et al. [3] proposed the concept of the implicit midpoint rule
(2)xn+1=αnfxn+1−αnTxn+xn+12,n∈ℕ,where αnn=1∞, T, and f remain as defined in (1). Under certain conditions, they established that the implicit midpoint sequence (2) converges to a fixed point p of T which also solves the variational inequality
(3)I−fp,x−p≥0,∀x∈FT,where , is the inner product. Aibinu et al. [4] studied the convergence of the sequence (2) in uniformly smooth Banach spaces. Ke and Ma [5] introduced generalized viscosity implicit rules which extend the results of Xu et al. [3]. The generalized viscosity implicit procedures are given by
(4)xn+1=αnfxn+1−αnTδnxn+1−δnxn+1,n∈ℕ,(5)yn+1=αnfyn+βnyn+γnTδnyn+1−δnyn+1,n∈ℕ,where δnn=1∞⊂0,1,αnn=1∞,βnn=1∞,γnn=1∞⊂0,1 with αn+βn+γn=1. Replacement of strict contractions in (5) by the generalized contractions and extension to uniformly smooth Banach spaces was considered by Yan et al. [6]. Under certain conditions imposed on the parameters involved, the sequence xnn=1∞ converges strongly to a fixed point p of the nonexpansive mapping T, which is also the unique solution of the variational inequality
(6)I−fp,Jx−p≥0,forallx∈FT,where J is a normalized duality mapping and , is the duality pairing. Aibinu and Kim [7] used an analytical method to compare the rate of convergence of the sequences (4) and (5). Due to the important roles which nonlinear operator equations play in modeling many phenomena in scientific fields, research on the solution and application of nonlinear operator equations are ever green (see e.g., [8–12]).Inspired by the previous works in this direction, this paper studies an implicit iterative sequence that involves the generalized contraction and which is effective for obtaining the solutions of various nonlinear operator equations. Precisely, for a nonempty closed convex subsetK of a uniformly smooth Banach space E and real sequences αnin=1∞i=13⊂0,1 and δnn=1∞⊂0,1 such that ∑i=13αni=1, an implicit iterative scheme is defined from an arbitrary x1∈K by
(7)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,where T:K⟶K is a nonexpansive mapping and f is a generalized contraction mapping. The sequence is shown to converge strongly to a fixed point of a nonexpansive mapping in Banach spaces. The adopted technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The results are applied to obtain a fixed point of λ-strictly pseudocontractive mapping, solution of α-inverse-strongly monotone mapping, and solution of the integral equation of Fredholm type. An example of real sequences which satisfy the stated conditions of our iteration is given.


## 2. Preliminaries

Definition 1.
LetEbe a real Banach space with dualE∗and denote the norm onEby.. The normalized duality mappingJ:E⟶2E∗is defined as(8)Jx=f∈E∗:x,f=xf,x=f,where .,. is the duality pairing between E and E∗.Definition 2.
LetBE≔x∈E:x=1. Eis said to be smooth (or Gâteaux differentiable) if the limit(9)limt→0+x+ty−xtexists for each x,y∈BE.E is said to have uniformly Gâteaux differentiable norm if for each y∈BE, the limit is attained uniformly for x∈BE and uniformly smooth if it is smooth and the limit is attained uniformly for each x,y∈BE. Also, E is said to be uniformly smooth if
(10)limτ→0ρEττ=0.
It is well known thatρE is nondecreasing. E is said to be q-uniformly smooth if there exist a constant c>0 and a real number q>1 such that ρEτ≤cτq.Lp, lp, and Wpm1<p<∞ are typical examples of such spaces, where
(11)Lp,lp,Wpmis2−uniformlysmoothif2≤p<∞,p−uniformlysmoothif1<p<2.
Recall that ifE is smooth, then J is single valued and onto if E is reflexive. Furthermore, the normalized duality mapping J is uniformly continuous on bounded subsets of E from the strong topology of E to the weak-star topology of E∗ if E is a Banach space with a uniformly Gâteaux differentiable norm.Definition 3.
LetE,d be a metric space and K a subset of E with f:K⟶K a mapping defined on K.(i)
f is said to be Lipschitzian if there exists a constant c>0, such that for all x,y∈K(12)dfx,fy≤cdx,y.f is called nonexpansive if c=1, and it is a contraction if c∈0,1. A contraction mapping f will be referred to as c-contraction mapping(ii)
f is said to be a Meir-Keeler contraction if for each ε>0 there exists δ=δε>0 such that for each x,y∈K, with ε≤dx,y<ε+δ, we have dfx,fy<ε(iii)
Letℕ be the set of all positive integers and ℝ+ the set of all positive real numbers. A mapping ψ:ℝ+⟶ℝ+ is said to be an L-function if ψ0=0,ψt>0 for all t>0 and for every s>0, there exists u>s such that ψt≤s for each t∈s,u(iv)
f:E⟶E is called a ψ,L-contraction if ψ:ℝ+⟶ℝ+ is an L-function and dfx,fy<ψdx,y, for all x,y∈E,x≠yThroughout this paper, the generalized contraction mappings will refer to Meir-Keeler orψ,L-contraction contractions. It is assumed that the L-function from the definition of ψ,L-contraction is continuous, strictly increasing, and limt→∞ϕt=∞, where ϕt=t−ψt for all t∈ℝ+. Whenever there is no confusion, ϕt and ψt will be written as ϕt and ψt, respectively.We have the following interesting results about the Meir-Keeler contraction.Proposition 4 Meir and Keeler [13].
LetE,dbe a complete metric space and letfbe a Meir-Keeler contraction onE.Then,fhas a unique fixed point inE.Proposition 5 Suzuki [14].
LetEbe a Banach space,Ka convex subset ofEandf:K⟶Ka Meir-Keeler contraction. Then,∀ε>0,there existsc∈0,1such that(13)fu−fv≤cu−v,
for allu,v∈K with u−v≥ε.Proposition 6 Lim [15].
LetE,dbe a metric space andf:E⟶Ebe a mapping. The following assertions are equivalent:(i)
f is a Meir-Keeler type mapping(ii)
there exists anL-function ψ:ℝ+⟶ℝ+ such that f is a ψ,L-contractionProposition 7 Lim [15].
LetKbe a nonempty convex subset of a Banach spaceE,T:K⟶Ka nonexpansive mapping andf:K⟶Ka Meir-Keeler contraction. Then,TfandfTare Meir-Keeler contractions.The following lemmas are needed in the sequel.Lemma 8 Suzuki [16].
Letunn=1∞andvnn=1∞be bounded sequences in a Banach spaceEandtnn=1∞be a sequence in0,1with0<liminfn→∞tn≤limsupn→∞tn<1.Suppose thatun+1=1−tnun+tnvnfor alln≥0andlimsupn→∞un+1−un−vn+1−vn≤0.Then,limn→∞un−vn=0.Lemma 9 Sunthrayuth and Kumam [17].
LetKbe a nonempty closed and convex subset of a uniformly smooth Banach spaceE. LetT:K⟶Kbe a nonexpansive mapping such thatFT≠∅andf:K⟶Kbe a generalized contraction mapping. Assume thatxtdefined byxt=tfxt+1−tTxtfort∈0,1converges strongly top∈FTast⟶0.Suppose thatxnis a bounded sequence such thatxn−Txn⟶0asn⟶∞.Then,(14)limsupn→∞fp−p,Jxn−p≤0.Lemma 10 Sunthrayuth and Kumam [17].
LetKbe a nonempty closed and convex subset of a uniformly smooth Banach spaceE.LetT:K⟶Kbe a nonexpansive mapping such thatFT≠∅andf:K⟶Kbe a generalized contraction mapping. Then,xtdefined byxt=tfxt+1−tTxtfort∈0,1converges strongly top∈FT,which solves the following variational inequality:(15)fp−p,Jz−p≤0,∀z∈FT.Lemma 11 Xu [18].
Letanbe a sequence of nonnegative real numbers satisfying the following relations:(16)an+1≤1−αnan+αnσn+γn,n∈ℕ,where
(i)
αn⊂0,1, ∑n=1∞αn=∞(ii)
limsupσn≤0(iii)
γn≥0, ∑n=1∞γn<∞
Then,limn→∞an=0.


## 3. Main Results

Assumption 12.
LetKbe a nonempty closed convex subset of a uniformly smooth Banach spaceEandf:K⟶Ka generalized contraction mapping. LetTbe a nonexpansive self-mapping defined onKwithFT≠∅.The real sequencesαnin=1∞i=13⊂0,1andδnn=1∞⊂0,1are assumed to satisfy the following conditions:(i)
∑i=13αni=1(ii)
limn→∞1−αn3δn−αn2=0,∑n=1∞1−αn3δn−αn2=∞(iii)
0<liminfn→∞αn2≤limsupn→∞αn2<1(iv)
limn→∞αn3=0,∑n=1∞αn31−δn<∞(v)
0<ε≤δn≤δn+1≤δ<1, for all n∈ℕ
Under the conditions (i)-(v) of Assumption12 stated above, this study establishes the convergence of the iterative scheme (7).
Firstly, it is shown that for allω∈K, the mapping defined by
(17)u↦Tωu≔αn1fω+αn2ω+αn3T1−δnfω+δnu,
for allu∈K, where αnn=1∞i=13⊂0,1,δnn=1∞⊂0,1, is a contraction with δ∈0,1 a contractive constant.
Indeed, for allu,v∈K,(18)Tωu−Tωv=αn3T1−δnfω+δnu−T1−δnfω+δnv≤αn31−δnfω+δnu−1−δnfω−δnv≤αn3δnu−v≤δnu−v≤δu−v.Therefore,Tω is a contraction. By Banach’s contraction mapping principle, Tω has a fixed point.The proof of the following lemmas which are useful in establishing the main result are given.Lemma 13.
LetKbe a nonempty closed convex subset of a uniformly smooth Banach spaceEandf:K⟶Ka generalized contraction mapping. LetTbe a nonexpansive self-mapping defined onKwithFT≠∅.For an arbitraryx1∈K, define the iterative sequencexnn=1∞by (7). Then, the sequencexnn=1∞is bounded under the conditions (i)-(v) of Assumption 12.Proof.
It is needed to show that the sequencexnn=1∞ is bounded. For p∈FT,(19)xn+1−p=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1−p≤αn1fxn−p+αn2xn−p+αn3T1−δnfxn+δnxn+1−p≤αn1fxn−fp+αn1fp−p+αn2xn−p+αn31−δnfxn+δnxn+1−p=αn1fxn−fp+αn1fp−p+αn2xn−p+αn31−δnfxn−p+δnxn+1−p≤αn1fxn−fp+αn1fp−p+αn2xn−p+αn31−δnfxn−fp+αn31−δnfp−p+αn3δnxn+1−p≤αn1ψxn−p+αn1fp−p+αn2xn−p+αn31−δnψxn−p+αn31−δnfp−p+αn3δnxn+1−p=αn1ψ+αn2+αn31−δnψxn−p+αn1+αn31−δnfp−p+αn3δnxn+1−p=αn1+αn3ψ+αn2−αn3δnψxn−p+αn1+αn3−αn3δnfp−p+αn3δnxn+1−p=1−αn2ψ+αn2−αn3δnψxn−p+1−αn2−αn3δnfp−p+αn3δnxn+1−p=ψ+αn21−ψ−αn3δnψxn−p+1−αn2−αn3δnfp−p+αn3δnxn+1−p.
Therefore,(20)xn+1−p≤ψ+αn21−ψ−αn3δnψ1−αn3δnxn−p+1−αn2−αn3δn1−αn3δnfp−p+1−αn2−αn3δn1−αn3δnfp−p=1−1−αn2−αn3δnϕ1−αn3δnxn−p+1−αn2−αn3δnϕ1−αn3δnϕ−1fp−p≤maxxn−p,ϕ−1fp−p.
Then by induction, we have(21)xn+1−p≤maxx1−p,ϕ−1fp−p.
Forp∈FT,(22)fxn≤fxn−fp+fp≤ψxn−p+fp≤maxψx1−p,ψϕ−1fp−p+fpbyinduction.
So,xnn=1∞ is bounded. Also,
(23)T1−δnfxn+δnxn+1=T1−δnfxn+δnxn+1−p+p≤T1−δnfxn+δnxn+1−Tp+p≤1−δnfxn+δnxn+1−p+p≤1−δnfxn−p+δnxn+1−p+p≤1−δnfxn−fp+1−δnfp−p+δnxn+1−p+p≤1−δnψxn−p+δnxn+1−p+1−δnfp−p+p≤1−εψxn−p+δxn+1−p+1−εfp−p+p.
Therefore,(24)T1−δnfxn+δnxn+1≤1+δ−εψmaxxn−p,ϕ−1fp−p+1−εfp−p+pbyinduction.
Hence,T1−δnfxn+δnxn+1n=1∞ is bounded.Lemma 14.
LetKbe a nonempty closed convex subset of a uniformly smooth Banach spaceEandf:K⟶Ka generalized contraction mapping. LetTbe a nonexpansive self-mapping defined onKwithFT≠∅.Supposeδnn=1∞is a real sequences in0,1andxnn=1∞⊂K.Setyn=1−δnfxn+δnxn+1,then(25)Tyn+1−Tyn≤1−δn+1ψxn+1−xn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1.Proof.
(26)Tyn+1−Tyn=T1−δn+1fxn+1+δn+1xn+2−T1−δnfxn+δnxn+1≤1−δn+1fxn+1+δn+1xn+2−1−δnfxn−δnxn+1=1−δn+1fxn+1−1−δn+1fxn+1−δn+1fxn−1−δnfxn+δn+1xn+2−δn+1xn+1+δn+1xn+1−δnxn+1=1−δn+1fxn+1−fxn−δn+1−δnfxn+δn+1xn+2−xn+1+δn+1−δnxn+1=1−δn+1fxn+1−fxn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1≤1−δn+1fxn+1−fxn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1≤1−δn+1ψxn+1−xn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1.Theorem 15.
LetK be a nonempty closed convex subset of a uniformly smooth Banach space E and f:K⟶K a generalized contraction mapping. Let T be a nonexpansive self-mapping defined on K with FT≠∅. Assume that the conditions (i)-(v) of Assumption 12 are satisfied. Then, the iterative sequence xnn=1∞ which is defined from an arbitrary x1∈K by (7) converges strongly to a fixed point p of T, which solves the variational inequality (6), given by
(27)I−fp,Jx−p≥0,forallx∈FT.Proof.
Settingzn=xn+1−αn2xn/1−αn2 and yn=1−δnfxn+δnxn+1, one can obtain that
(28)zn+1−zn=xn+2−αn+12xn+11−αn+12−xn+1−αn2xn1−αn2=αn+11fxn+1+αn+13Tyn+11−αn+12−αn1fxn+αn3Tyn1−αn2=αn+111−αn+12fxn+1−fxn+αn+111−αn+12−αn11−αn2fxn+αn+131−αn+12Tyn+1−Tyn+αn+131−αn+12−αn31−αn2Tyn=αn+111−αn+12fxn+1−fxn−αn+131−αn+12−αn31−αn2fxn+αn+131−αn+12Tyn+1−Tyn+αn+131−αn+12−αn31−αn2Tyn=αn+111−αn+12fxn+1−fxn+αn+131−αn+12−αn31−αn2Tyn−fxn+αn+131−αn+12Tyn+1−Tyn.
LetM1=supnTyn−fxn, M2=supnxn+1−fxn, and M=maxM1,M2. Then,
(29)zn+1−zn≤αn+111−αn+12fxn+1−fxn+αn+131−αn+12−αn31−αn2Tyn−fxn+αn+131−αn+12Tyn+1−Tyn≤αn+111−αn+12ψxn+1−xn+αn+131−αn+12−αn31−αn2Tyn−fxn+αn+131−αn+121−δn+1ψxn+1−xn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1by25=αn+11ψ+αn+131−δn+1ψ1−αn+12xn+1−xn+αn+131−αn+12−αn31−αn2+αn+13δn+1−δn1−αn+12M+αn+13δn+11−αn+12xn+2−xn+1.
It is needed to evaluatexn+2−xn+1.(30)xn+2−xn+1=αn+11fxn+1+αn+12xn+1+αn+13Tyn+1−αn1fxn+αn2xn+αn3Tyn=αn+11fxn+1−fxn+αn+12xn+1−xn+αn+13Tyn+1−Tyn+αn+11−αn1fxn+αn+12−αn2xn+αn+13−αn3Tyn=αn+11fxn+1−fxn+αn+12xn+1−xn+αn+13Tyn+1−Tyn+αn2−αn+12+αn3−αn+13fxn+αn+12−αn2xn+αn+13−αn3Tyn=αn+11fxn+1−fxn+αn+12xn+1−xn+αn+13Tyn+1−Tyn+αn+12−αn2xn−fxn+αn+13−αn3Tyn−fxn.
This leads to(31)xn+2−xn+1≤αn+11ψxn+1−xn+αn+12xn+1−xn+αn+13Tyn+1−Tyn+αn+12−αn2xn−fxn+αn+13−αn3Tyn−fxn≤αn+11ψxn+1−xn+αn+12xn+1−xn+αn+131−δn+1ψxn+1−xn+δn+1−δnxn+1−fxn+δn+1xn+2−xn+1by25+αn+12−αn2xn−fxn+αn+13−αn3Tyn−fxn=αn+12+αn+13+αn+11ψ−αn+13δn+1ψxn+1−xn+αn+13δn+1xn+2−xn+1+αn+12−αn2+αn+13−αn3+αn+13δn+1−δnM=αn+12+1−αn+12ψ−αn+13δn+1ψxn+1−xn+αn+13δn+1xn+2−xn+1+αn+12−αn2+αn+13−αn3+αn+13δn+1−δnM=ψ+αn+121−ψ−αn+13δn+1ψxn+1−xn+αn+13δn+1xn+2−xn+1+αn+12−αn2+αn+13−αn3+αn+13δn+1−δnM=αn+121−ψ+1−αn+13δn+1ψxn+1−xn+αn+13δn+1xn+2−xn+1+αn+12−αn2+αn+13−αn3+αn+13δn+1−δnM.
Letdn=∣αn+12−αn2∣+∣αn+13−αn3∣+αn+13δn+1−δn. Therefore,
(32)xn+2−xn+1≤αn+121−ψ+1−αn+13δn+1ψ1−αn+13δn+1xn+1−xn+dnM1−αn+13δn+1.
LetSn=αn+13/1−αn+12−αn3/1−αn2+αn+13δn+1−δn/1−αn+12 and substitute (32) into (29) to obtain
(33)zn+1−zn≤αn+11ψ+αn+131−δn+1ψ1−αn+12+αn+13δn+11−αn+12×αn+121−ψ+1−αn+13δn+1ψ1−αn+13δn+1xn+1−xn+SnM+αn+13δn+11−αn+12×dnM1−αn+13δn+1=αn+11ψ+αn+131−δn+1ψ−αn+13δn+1αn+11ψ+αn+131−δn+1ψ1−αn+121−αn+13δn+1+αn+13δn+1αn+121−ψ+1−αn+13δn+1ψ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=αn+11ψ+αn+131−δn+1ψ−αn+13δn+1αn+11ψ+αn+13ψ−αn+13δn+1ψ1−αn+121−αn+13δn+1+αn+13δn+1αn+12−αn+12ψ+ψ−αn+13δn+1ψ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=αn+11ψ+αn+131−δn+1ψ−αn+13δn+11−αn+12ψ−αn+13δn+1ψ1−αn+121−αn+13δn+1+αn+13δn+1αn+12+1−αn+12ψ−αn+13δn+1ψ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=αn+11ψ+αn+131−δn+1ψ+αn+13δn+1αn+121−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=1−αn+12ψ−αn+13δn+1ψ+αn+13δn+1αn+121−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=1−1−αn+121−ψ−αn+13δn+11−ψ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=1−1−αn+12ϕ−αn+13δn+1ϕ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M=1−1−αn+12−αn+13δn+1ϕ1−αn+121−αn+13δn+1xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M≤1−1−αn+12−αn+13δn+1ϕ1−αn+12xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M.
It then follows that(34)zn+1−zn−xn+1−xn≤−1−αn+12−αn+13δn+1ϕ1−αn+12xn+1−xn+Sn+dnαn+13δn+11−αn+121−αn+13δn+1M,and thus,
(35)limsupn→∞zn+1−zn−xn+1−xn≤0.
Invoking Lemma8 to obtain that
(36)limn→∞zn−xn=0.
Consequently,(37)xn+1−xn=1−αn2zn+αn2xn−xn=1−αn2zn−1−αn2xn=1−αn2zn−xn≤1−αn2zn−xn⟶0asn⟶∞.
Next is to show thatlimn→∞xn−Txn=0. From (7), it is obtained that
(38)xn−Txn≤xn−xn+1+xn+1−Txn≤xn+1−xn+αn1fxn+αn2xn+αn3Tyn−Txn≤xn+1−xn+αn1fxn−Txn+αn2xn−Txn+αn3Tyn−Txn≤xn+1−xn+αn1fxn−Txn+αn2xn−Txn+αn3yn−xn≤xn+1−xn+αn1fxn−Txn+αn2xn−Txn+αn31−δnfxn+δnxn+1−xn≤xn+1−xn+αn1fxn−Txn+αn2xn−Txn+αn31−δnxn−fxn+αn3δnxn+1−xn=1+αn3δnxn+1−xn+αn1+αn31−δnQ+αn2xn−Txn=1+αn3δnxn+1−xn+1−αn3δn−αn2Q+αn2xn−Txn.
Since0<liminfn→∞αn2≤limsupn→∞αn2<1, let 0<η≤αn2<1, then
(39)xn−Txn≤1+αn3δn1−αn2xn+1−xn+1−αn3δn−αn21−αn2Q≤1+αn3δn1−ηxn+1−xn+1−αn3δn−αn21−ηQ,which goes to zero as n⟶∞ by (37) and condition (ii) of Assumption 12.
Let a sequencext be defined by xt=tfxt+1−tTxt for t∈0,1. It is known by Lemma 10 that xt converges strongly to p∈FT, which solves the variational inequality:
(40)fp−p,Jx−p≤0,∀x∈FT,which is equivalent to
(41)I−fp,Jx−p≥0,∀x∈FT.
It can be shown that(42)limsupn→∞fp−p,Jxn+1−p≤0,where p∈FT is the unique fixed point of the generalized contraction PFTfp (Proposition 7), that is, p=PFTfp.
By (39), limn→∞xn−Txn=0, it follows from Lemma 9 that
(43)limsupn→∞fp−p,Jxn−p≤0.
Due to the continuity of the duality map and the fact thatxn+1−xn⟶0 as n⟶∞ by (37), it is obtained that
(44)limsupn→∞fp−p,Jxn+1−p=limsupn→∞fp−p,Jxn+1−xn+xn−p=limsupn→∞fp−p,Jxn−p≤0.
Lastly, it is shown thatxn⟶p∈FT as n⟶∞.
Suppose that the sequencexnn=1∞ does not converge strongly to p∈FT. Then, there exists ε>0 and a subsequence xnkk=1∞ of xnn=1∞ such that xnk−p≥ε, for all k∈ℕ. Therefore, for this ε, there exists c∈0,1/2 such that
(45)fxnk−fp≤cxnk−p.(46)xnk+1−p2=αnk1fxnk−p,Jxnk+1−p+αnk2xnk−p,Jxnk+1−p+αnk3Tynk−p,Jxnk+1−p=αnk1fxnk−fp,Jxnk+1−p+αn1fp−p,Jxnk+1−p+αnk2xnk−p,Jxnk+1−p+αnk3Tynk−p,Jxnk+1−p≤cαnk1xnk−pxnk+1−p+αn1fp−p,Jxnk+1−p+αnk2xnk−pxnk+1−p+αnk31−δnkfxnk+δnkxnk+1−pxnk+1−p≤cαnk1xnk−pxnk+1−p+αn1fp−p,Jxnk+1−p+αnk2xnk−pxnk+1−p+αnk31−δnkfxnk−pxnk+1−p+αnk3δnkxnk+1−p2≤cαnk1xnk−pxnk+1−p+αn1fp−p,Jxnk+1−p+αnk2xnk−pxnk+1−p+cαnk31−δnkxnk−pxnk+1−p+αnk31−δnkfp−pxnk+1−p+αnk3δnkxnk+1−p2=cαnk1+αnk2+cαnk31−δnkxnk−pxnk+1−p+αn1fp−p,Jxnk+1−p+αnk31−δnkfp−pxnk+1−p+αnk3δnkxnk+1−p2≤12cαnk1+αnk2+cαnk31−δnkxnk−p2+xnk+1−p2+αn1fp−p,Jxnk+1−p+αnk3δnkxnk+1−p2+12αnk31−δnkfp−p2+xnk+1−p2=12cαnk1+αnk31−δnk+αnk2xnk−p2+αn1fp−p,Jxnk+1−p+12cαnk1+αnk31−δnk+αnk2+2αnk3δnk+αnk31−δnkxnk+1−p2+12αnk31−δnkfp−p2=12cαnk1+αnk31−δnk+αnk2xnk−p2+αn1fp−p,Jxnk+1−p+12cαnk1+αnk31−δnk+αnk2+αnk31+δnkxnk+1−p2+12αnk31−δnkfp−p2=12c1−αnk2−αnk3δnk+αnk2xnk−p2+αn1fp−p,Jxnk+1−p+12c1−αnk2−αnk3δnk+αnk2+αnk31+δnkxnk+1−p2+12αnk31−δnkfp−p2.
Observe that(47)2−c1−αnk2−αnk3δnk−αnk2−αnk31+δnk=2−c+cαnk2+cαnk3δnk−αnk2−αnk3−αnk3δnk=2−c−1−cαnk2−1−cαnk3δnk−αnk3=1−c−1−cαnk2−1−cαnk3δnk+1−αnk3=1+1−c1−αnk2−αnk3δnk−αnk3,(48)αnk1=1−αnk2−αnk3≤1−αnk2−αnk3δnksinceδnk∈0,1.
Multiplying (46) by 2 gives
(49)xnk+1−p2≤c1−αnk2−αnk3δnk+αnk21+1−c1−αnk2−αnk3δnk−αnk3xnk−p2+αn11+1−c1−αnk2−αnk3δnk−αnk3fp−p,Jxnk+1−p+αnk31−δnk1+1−c1−αnk2−αnk3δnk−αnk3fp−p2=1−1−2c1−αnk2−αnk3δnk+αnk11+1−c1−αnk2−αnk3δnk−αnk3xnk−p2+αnk11+1−c1−αnk2−αnk3δnk−αnk3fp−p,Jxnk+1−p+αnk31−δnk1+1−c1−αnk2−αnk3δnk−αnk3fp−p2≤1−1−2c1−αnk2−αnk3δnk1+1−c1−αnk2−αnk3δnk−αnk3xnk−p2+1−2c1−αnk2−αnk3δnk1+1−c1−αnk2−αnk3δnk−αnk311−2cfp−p,Jxnk+1−p+αnk31−δnk1+1−c1−αnk2−αnk3δnk−αnk3fp−p2by48.
Using Lemma11, it shows that xnk⟶p as k⟶∞. A contradiction, hence, xnn=1∞ converges strongly to p∈FT.The next result shows that under suitable conditions, the implicit iterative sequences (5) and (7) converge to the same fixed point.Theorem 16.
LetK be a nonempty closed convex subset of a uniformly smooth Banach space E and f:K⟶K a c-contraction mapping with c∈0,1.LetT be a nonexpansive self-mapping defined on K with FT≠∅. Let αnin=1∞i=13⊂0,1 and δnn=1∞⊂0,1 be real sequences such that ∑i=13αni=1. Given that limn→∞αn3/1−αn2−αn3δn=0, then xnn=1∞ defined by (7) converges to pif and only ifynn=1∞ defined by (5) converges to p.Proof.
Notice that (7) and (5) are, respectively, given by
(50)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,n∈ℕ,yn+1=αnfyn+βnyn+γnTδnyn+1−δnyn+1,n∈ℕ.
It is needed to show thatxn−yn⟶0, as n⟶∞.(51)xn+1−yn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1−αn1fyn+αn2yn+αn3Tδnyn+1−δnyn+1=αn1fxn−fyn+αn2xn−yn+αn3T1−δnfxn+δnxn+1−Tδnyn+1−δnyn+1≤αn1fxn−fyn+αn2xn−yn+αn3T1−δnfxn+δnxn+1−Tδnyn+1−δnyn+1≤αn1cxn−yn+αn2xn−yn+αn31−δnfxn−yn+1+δnxn+1−yn≤αn1cxn−yn+αn2xn−yn+αn31−δnfxn−fyn+fyn−yn+1+αn3δnxn+1−yn+1+yn+1−yn≤αn1cxn−yn+αn2xn−yn+αn31−δncxn−yn+αn31−δnyn+1−fyn+αn3δnxn+1−yn+1+αn3δnyn+1−yn=αn1c+αn31−δnc+αn2xn−yn+αn3δnxn+1−yn+1+αn31−δnyn+1−fyn+αn3δnyn+1−yn.
Sinceynn=1∞ and fynn=1∞ are bounded, let M2=maxsupnyn+1−fyn,supnyn+1−yn. Then,
(52)xn+1−yn+1≤αn1c+αn31−δnc+αn21−αn3δnxn−yn+αn31−αn3δnM2=1−βnxn−yn+αn31−αn2−αn3δn1−cβnM2,where βn=1−αn2−αn3δn1−c/1−αn3δn. From the given condition, it follows that.
limsupn→∞αn3/1−αn2−αn3δn≤0. Apply Lemma 11 to (52) and take γn=0 to get that xn−yn⟶0, as n⟶∞. Next, suppose yn−p⟶0 as n⟶∞. It follows that
(53)xn−p=xn−yn+yn−p≤xn−yn+yn−p⟶0asn⟶∞.
Similarly, supposexn−p⟶0 as n⟶∞. Then,
(54)yn−p=yn−xn+xn−p≤yn−xn+xn−p⟶0asn⟶∞.Hence, the implicit iterative sequences (5) and (7) converge to the same fixed point under suitable conditions.Remark 17.
One can deduce the following results from Theorem15.Corollary 18.
LetK be a nonempty closed convex subset of a uniformly smooth Banach space E and T a nonexpansive self-mapping defined on K with FT≠∅. Assume that the real sequences αnn=1∞⊂0,1 and δnn=1∞⊂0,1 satisfy the conditions:
(i)
limn→∞αn=0(ii)
∑n=1∞αn=∞(iii)
∑n=1∞αn+1−αn<∞(iv)
0<ε≤δn≤δn+1<1
Then, the iterative sequencexnn=1∞ which is defined from an arbitrary x1∈K by
(55)xn+1=αnxn+1−αnT1−δnxn+δnxn+1converges strongly to a fixed point p of T which solves the variational inequality (6).Proof.
The result follows from Theorem15 by simply taking f to be the identity mapping on K.Corollary 19.
LetK be a nonempty closed convex subset of a uniformly smooth Banach space E and T a nonexpansive self-mapping defined on K with FT≠∅. Assume that the real sequence αnn=1∞⊂0,1 satisfies the following conditions:
(i)
limn→∞αn=0(ii)
∑n=1∞αn=∞(iii)
∑n=1∞αn+1−αn<∞
Then, the iterative sequencexnn=1∞ which is defined from an arbitrary x1∈K by
(56)xn+1=αnxn+1−αnTxn+xn+12converges strongly to a fixed point p of T which solves the variational inequality (6).Proof.
The result follows from Theorem15 by simply taking f to be the identity mapping on K and δn=1/2 for all n∈ℕ. Consequently, this improves and extend the results of Alghamdi et al. [19].


## 4. Applications

### 4.1. Application to Fixed Points ofλ-Strictly Pseudocontractive Mappings

LetK be a closed convex subset of a real Banach space E. A mapping S:K⟶K is said to be λ-strictly pseudocontractive mapping if there exists 0≤λ<1 such that
(57)Sx−Sy2≤x−y2−λI−Sx−I−Sy2,∀x,y∈K,where I denotes the identity operator on K.Zhou [20] established the following lemma which gives a relationship between λ-strictly pseudocontractive mappings and nonexpansive mappings.Lemma 20.
LetKbe a nonempty subset of a 2-uniformly smooth Banach spaceE.LetS:K⟶Kbe aλ-strictly pseudocontractive mapping. Forθ∈0,1,define(58)Tx=θx+1−θSx,∀x∈K.Then, asθ∈0,λ/L2 (where L is the 2-uniformly smooth constant of a 2-uniformly smooth Banach space), T:K⟶K is nonexpansive such that FT=FS.The following result is obtained by using Lemma20 and Theorem 15.Corollary 21.
LetK be a nonempty closed convex subset of a 2-uniformly smooth Banach space E and f:K⟶K a generalized contraction mapping. Let S:K⟶Kaλ-strictly pseudocontractive mapping with FT≠∅. Suppose that the conditions (i)-(v) of Assumption 12 are satisfied and T is a mapping from K into itself, defined by Tx=αx+1−θSx,x∈K,θ∈0,1. Then, for an arbitrary x1∈K, the iterative sequence xnn=1∞ defined by
(59)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,foralln∈ℕ,converges strongly to a fixed point p of S, which solves the variational inequality
(60)I−fp,Jx−p≥0,forallx∈FS.


### 4.2. Application to Solution ofα-Inverse-Strongly Monotone Mappings

LetK be a nonempty closed convex subset of a Hilbert space H. The metric projection PK is defined from H onto K by
(61)PKx≔argminy∈Kx−y2,x∈Hand characterized by
(62)PKx≔argminz∈Kx−z2,x∈H.PKx is known as the only point in K that minimizes the objective x−z over z∈K. A mapping A of K into H is called monotone if Au−Av,u−v≥0, for all u,v∈K. The classical variational inequality (VI) problem is to find u∗∈K such that
(63)Au∗,u−u∗≥0,u∈K,where A is a (single-valued) monotone operator in Hilbert space H [21, 22]. In this work, the solution set of (63) is denoted by VIK,A. In the context of the variational inequality problem, (62) implies that
(64)u∈VIK,A⇔u=PKu−γAu,∀γ>0.A is said to be α-inverse-strongly monotone if there exists a positive real number α such that
(65)Au−Av,u−v≥αAu−Av2,for all u,v∈K. If A is an α-inverse-strongly monotone mapping of K to H, it is known that A is 1/α-Lipschitz continuous. Also, we have that for all u,v∈K and γ>0,(66)I−γAu−I−γAv2=u−v−Au−Av2=u−v2−2γu−v,Au−Av+γ2Au−Av2≤u−v2+γγ−2αAu−Av2.Therefore, ifγ≤2α, then I−γA is a nonexpansive mapping of K into K. Consequently, one can apply Theorem 15 to deduce the following result:Corollary 22.
LetK be a nonempty closed convex subset of a real Hilbert space Handf:K⟶K a generalized contractions. Let A be an α-inverse-strongly monotone mapping of K to H with A−10≠∅. Assume that the conditions (i)-(v) of Assumption 12 are satisfied. Then, the iterative sequence xnn=1∞ which is defined from an arbitrary x1∈K by
(67)xn+1=αn1fxn+αn2xn+αn3PKI−γA1−δnfxn+δnxn+1,n∈ℕ,converges strongly to a solution p in A−10, which solves the variational inequality
(68)I−fp,x−p≥0,forallx∈A−10.


### 4.3. Application to Fredholm Integral Equation in Hilbert Spaces

Consider a Fredholm integral equation of the form(69)xt=gt+∫01Φt,s,xsds,t∈0,1,where g is a continuous function on 0,1 and Φ:0,1×0,1×ℝ⟶ℝ is continuous. The existence of solutions of (69) has been studied (see [23] and the references therein). If Φ satisfies the Lipschitz continuity condition
(70)Φt,s,x−Φt,s,y≤x−y,s,t∈0,1,x,y∈ℝ,then equation (69) has at least one solution in the Hilbert space L20,1 ([23], Theorem 3.3). Precisely, define a mapping T:L20,1⟶L20,1 by
(71)Txt=gt+∫01Φt,s,xsds,t∈0,1.It is known thatT is nonexpansive. Indeed, for x,y∈L20,1(72)Tx−Ty2=∫01Txt−Tyt2dt=∫01∫01Φt,s,xs−Φt,s,ysds2dt≤∫01∫01xs−ysds2dt≤∫01xs−ys2ds=x−y2.Thus, finding a solution of integral equation (69) is reduced to finding a fixed point of the nonexpansive mapping T in the Hilbert space L20,1. Consequently, the following result is obtainable.Corollary 23.
LetK be a nonempty closed convex subset of a Hilbert space L20,1,T:K⟶K, defined by (71) with FT≠∅ and f:K⟶K is a generalized contraction. Suppose that the conditions (i)-(v) of Assumption 12 are satisfied. Then, for an arbitrary x1∈K, the iterative sequence xnn=1∞ defined by
(73)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,n∈ℕ,converges strongly to a fixed point p of T, which solves the variational inequality
(74)I−fp,x−p≥0,forallx∈FT.
Examples of real sequences which satisfy the conditions of Assumption12 are
(75)αn1n=1∞=12nn=1∞;αn2n=1∞=1−32nn=1∞;αn3n=1∞=1nn=1∞;δnn=1∞=n2n+1n=1∞.


## 4.1. Application to Fixed Points ofλ-Strictly Pseudocontractive Mappings

LetK be a closed convex subset of a real Banach space E. A mapping S:K⟶K is said to be λ-strictly pseudocontractive mapping if there exists 0≤λ<1 such that
(57)Sx−Sy2≤x−y2−λI−Sx−I−Sy2,∀x,y∈K,where I denotes the identity operator on K.Zhou [20] established the following lemma which gives a relationship between λ-strictly pseudocontractive mappings and nonexpansive mappings.Lemma 20.
LetKbe a nonempty subset of a 2-uniformly smooth Banach spaceE.LetS:K⟶Kbe aλ-strictly pseudocontractive mapping. Forθ∈0,1,define(58)Tx=θx+1−θSx,∀x∈K.Then, asθ∈0,λ/L2 (where L is the 2-uniformly smooth constant of a 2-uniformly smooth Banach space), T:K⟶K is nonexpansive such that FT=FS.The following result is obtained by using Lemma20 and Theorem 15.Corollary 21.
LetK be a nonempty closed convex subset of a 2-uniformly smooth Banach space E and f:K⟶K a generalized contraction mapping. Let S:K⟶Kaλ-strictly pseudocontractive mapping with FT≠∅. Suppose that the conditions (i)-(v) of Assumption 12 are satisfied and T is a mapping from K into itself, defined by Tx=αx+1−θSx,x∈K,θ∈0,1. Then, for an arbitrary x1∈K, the iterative sequence xnn=1∞ defined by
(59)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,foralln∈ℕ,converges strongly to a fixed point p of S, which solves the variational inequality
(60)I−fp,Jx−p≥0,forallx∈FS.


## 4.2. Application to Solution ofα-Inverse-Strongly Monotone Mappings

LetK be a nonempty closed convex subset of a Hilbert space H. The metric projection PK is defined from H onto K by
(61)PKx≔argminy∈Kx−y2,x∈Hand characterized by
(62)PKx≔argminz∈Kx−z2,x∈H.PKx is known as the only point in K that minimizes the objective x−z over z∈K. A mapping A of K into H is called monotone if Au−Av,u−v≥0, for all u,v∈K. The classical variational inequality (VI) problem is to find u∗∈K such that
(63)Au∗,u−u∗≥0,u∈K,where A is a (single-valued) monotone operator in Hilbert space H [21, 22]. In this work, the solution set of (63) is denoted by VIK,A. In the context of the variational inequality problem, (62) implies that
(64)u∈VIK,A⇔u=PKu−γAu,∀γ>0.A is said to be α-inverse-strongly monotone if there exists a positive real number α such that
(65)Au−Av,u−v≥αAu−Av2,for all u,v∈K. If A is an α-inverse-strongly monotone mapping of K to H, it is known that A is 1/α-Lipschitz continuous. Also, we have that for all u,v∈K and γ>0,(66)I−γAu−I−γAv2=u−v−Au−Av2=u−v2−2γu−v,Au−Av+γ2Au−Av2≤u−v2+γγ−2αAu−Av2.Therefore, ifγ≤2α, then I−γA is a nonexpansive mapping of K into K. Consequently, one can apply Theorem 15 to deduce the following result:Corollary 22.
LetK be a nonempty closed convex subset of a real Hilbert space Handf:K⟶K a generalized contractions. Let A be an α-inverse-strongly monotone mapping of K to H with A−10≠∅. Assume that the conditions (i)-(v) of Assumption 12 are satisfied. Then, the iterative sequence xnn=1∞ which is defined from an arbitrary x1∈K by
(67)xn+1=αn1fxn+αn2xn+αn3PKI−γA1−δnfxn+δnxn+1,n∈ℕ,converges strongly to a solution p in A−10, which solves the variational inequality
(68)I−fp,x−p≥0,forallx∈A−10.


## 4.3. Application to Fredholm Integral Equation in Hilbert Spaces

Consider a Fredholm integral equation of the form(69)xt=gt+∫01Φt,s,xsds,t∈0,1,where g is a continuous function on 0,1 and Φ:0,1×0,1×ℝ⟶ℝ is continuous. The existence of solutions of (69) has been studied (see [23] and the references therein). If Φ satisfies the Lipschitz continuity condition
(70)Φt,s,x−Φt,s,y≤x−y,s,t∈0,1,x,y∈ℝ,then equation (69) has at least one solution in the Hilbert space L20,1 ([23], Theorem 3.3). Precisely, define a mapping T:L20,1⟶L20,1 by
(71)Txt=gt+∫01Φt,s,xsds,t∈0,1.It is known thatT is nonexpansive. Indeed, for x,y∈L20,1(72)Tx−Ty2=∫01Txt−Tyt2dt=∫01∫01Φt,s,xs−Φt,s,ysds2dt≤∫01∫01xs−ysds2dt≤∫01xs−ys2ds=x−y2.Thus, finding a solution of integral equation (69) is reduced to finding a fixed point of the nonexpansive mapping T in the Hilbert space L20,1. Consequently, the following result is obtainable.Corollary 23.
LetK be a nonempty closed convex subset of a Hilbert space L20,1,T:K⟶K, defined by (71) with FT≠∅ and f:K⟶K is a generalized contraction. Suppose that the conditions (i)-(v) of Assumption 12 are satisfied. Then, for an arbitrary x1∈K, the iterative sequence xnn=1∞ defined by
(73)xn+1=αn1fxn+αn2xn+αn3T1−δnfxn+δnxn+1,n∈ℕ,converges strongly to a fixed point p of T, which solves the variational inequality
(74)I−fp,x−p≥0,forallx∈FT.
Examples of real sequences which satisfy the conditions of Assumption12 are
(75)αn1n=1∞=12nn=1∞;αn2n=1∞=1−32nn=1∞;αn3n=1∞=1nn=1∞;δnn=1∞=n2n+1n=1∞.



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