We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed points of demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces. We have applied certain results due to Sahu (2005) to obtain conditions for existence—and to introduce an asymptotic iterative process for construction—of fixed points of these hemicontractions with respect to a new auxiliary operator.

1. Introduction

In this paper, we have applied certain results due to Sahu [1] on nearly contraction mapping principle to obtain conditions for existence of fixed points of certain hemicontinuous mappings and introduced an asymptotic iterative process for construction of fixed points of these hemicontinuous mappings with respect to a new auxiliary operator. Our results are important generalizations and an extension of important and fundamental aspect of a branch of asymptotic theory of fixed points of non-Lipschitzian nonlinear mappings in real Banach spaces.

Let E and Y be real Banach spaces andK⊆Ea nonempty subset ofE. A mappingT:K→Yis said to be (see, e.g., [2])

demicontinuous if whenever a sequence{xn}⊂Xconverges strongly tox∈Xit implies that the sequence{Txn}converges weakly toTx∈Y;

hemicontinuous if whenever a sequence{xn}⊂Xconverges stronly on a line tox∈Xit implies that the sequence{Txn}converges weakly toTx∈Y, that is, T(x0+tnx)⇀Tx0astn→0.

Asymptotic fixed point theory which has been studied by so many authors [1, 3–6] has a fundamental role in nonlinear functional analysis concerning existence and construction of fixed points of Lipschitzian mappings, L-uniformly Lipschitzian mappings, and non-Lipschitzian mappings among other classes of operators (see, e.g., [5, 7–9]). A very important branch of the theory of asymptotic fixed point relates to the important class of asymptotically nonexpansive mappings which have been studied by various authors in specific types of Banach spaces.

Motivated by the need to relax continuity condition inherent in asymptotic nonexpansiveness of asymptotically nonexpansive mappings in certain applications, Sahu [1] considered and introduced the nearly contraction mapping principle into the study of asymptotic fixed point theory concerning nearly Lipschitzian mappings and obtained the following results among others.

Lemma 1.

LetCbe a nonempty subset of a Banach space, and letT:C→Cbe hemicontinuous. Suppose thatTnu=pasn→∞for someu,p∈C.Then, p is an element ofFix(T), the set of fixed point ofT.

Theorem 2.

LetCbe a nonempty closed subset of a Banach spaceXandT:C→Ca demicontinuous nearly Lipschitzian mapping with sequence{(an,η(Tn))}. Supposeη∞(T)=lim¯n→∞[η(Tn)]1/n<1. Then, we have the following:

Thas unique fixed pointx*∈C;

for eachx0∈C, the sequence{Tnx0} converges strongly tox*;

∥Tnx0-x*∥≤(∥x0-Tx0∥M)∑i=n∞η(Ti)for alln∈ℕ.

The aim of this work is appling Lemma 1 to obtain conditions for existence and uniqueness of asymptotic fixed point of a new auxiliary operator and appling Theorem 2 on the auxiliary operator to obtain an extension and a generalization of Theorem 2 which is a fundamental extension of important classical and related results.

2. Preliminaries

LetKbe a nonempty subset of a Banach spaceEandT:K→Ka nonlinear mapping. The mapping T is said to be Lipschitzian if for each n∈ℕ there exists a constantLn>0such that ∥Tnx-Tny∥≤Ln∥x-y∥ for allx,y∈K. A Lipschitzian mapping is called uniformly L-Lipschitzian if Ln=L for all n∈ℕ and asymptotically nonexpansive if limn→∞Ln=1.

Next, let K be a nonempty subset of a Banach spaceEand{an}a fixed sequence in[0,∞)with an→0 asn→∞. A mapping T:K→K is called nearly Lipschitzian mapping with respect to {an} if for each n∈ℕ there exists a constantLn≥0such that
(1)∥Tnx-Tny∥≤Ln(∥x-y∥+an).
The infimumη(Tn)=sup{∥Tnx-Tny∥/(∥x-y∥+an);x,y∈K,x≠y}of constants Ln for which (1) holds is called nearly Lipschitzian constant. Nearly Lipschitzian operators with sequences {(an,η(Tn))} are classified in [1, 2] as shown below:

nearly contraction ifη(Tn)<1for alln∈ℕ;

nearly nonexpansive ifη(Tn)≤1for alln∈ℕ;

nearly asymptotically nonexpansive ifη(Tn)≥1for alln∈ℕ and limn→∞η(Tn)≤1;

Examples and a short survey of these classes of nearly Lipschitzian operators are listed above, and related operators are illustrated in [1] (pp. 655–656) where it is remarked that ifKis bounded then the asymptotically nonexpansive mappingTis a nearly nonexpansive mapping. Also, it is observed therein that a nearly asymptotically nonexpansive mapping reduces to asymptotically nonexpansive type ifKis bounded. For details authors are referred to Agarwal et al. [2] pp. 259–263, especially the bibliographic notes and remarks there in.

3. Main Results

Our main results depend on Lemma 1 and the following new important inequality, needed in the sequel, which we shall prove using archimedean property. We are still sharpening an estimate for the parameterτin Lemma 3 below.

Lemma 3.

LetVbe a normed linear space over, a scaler field 𝔽 (𝔽 is real or complex). Then, for all distinct pointsx,y∈Vthere existsτ∈ℝ such that
(2)∥αx-βy∥≤[2|α|+τ|β|]∥x-y∥
for allα,β∈𝔽.

Proof.

As mentioned above, the proof is a consequence of Archimedean property of real numbers that ifaandbare positive real numbers then a<nb for some n∈ℕ. Since x≠y, we have
(3)(α+β)(x-y)=αx-βy+βx-αx(α+β)(x-y)=αx-βy-(αy-βx)⟹|α+β|∥x-y∥=∥αx-βy-(αy-βx)∥⟹|α+β|∥x-y∥≥∥αx-βy∥-∥αy-βx∥⟹∥αx-βy∥≤|α+β|∥x-y∥+∥βx-αy∥⟹∥αx-βy∥=|α+β|∥x-y∥+∥(α+β1)x-αy∥(forsomeβ1∈𝔽)≤|α+β|∥x-y∥+|α|∥x-y∥+|β1|∥x∥≤(2|α|+τ|β|)∥x-y∥.
Equation (3) follows from Archimedean property while boundedness is inferred from the fact that β1=β-α for arbitrary α,β∈𝔽.

Remark 4.

It is important to make the following observations.

Ifα,β∈ℝ then (3) reduces to
(4)∥αx-βy∥≤2(|α|+|β|)∥x-y∥,

as verified below: sinceα,β∈ℝ, if on the contrary (4) is not satisfied then from (3) we have |β1|∥x∥>|β|∥x-y∥ which end up with a contradiction demonstrated below.

Suppose|β1|∥x∥>|β|∥x-y∥. Settingα,β>0such that β-α>0 yields α>1 whenever α,β>0 such that β-α>0 which is a contradiction.

It is important to observe that ifxandywere not distinct in Lemma 3 thenα=βwould be a valid and natural constraint. However, for x=y the problem is trivial.

Lemma 5.

LetKbe a nonempty subset of a Banach space, and letT:K→Kbe a nearly Lipschitzian map with sequence{(an,η(Tn))}such that η∞(T)=lim¯n→∞[η(Tn)]1/n<1.Then the auxiliary operator S:ℕ×K→Kdefined byS(n,x)=Tn-1x+∥Tnx-Tn-1x∥xhas a fixed point inE.

Proof.

Given thatS(n,x)=Tn-1x+∥Tnx-Tn-1x∥x where T is a nearly Lipschitzian map with sequence {(an,η(Tn))}, we have
(5)∥S(n,x)-S(n+1,x)∥≤∥Tnx-Tn-1x∥+∥∥Tnx-Tn-1x∥x-∥Tn+1x-Tnx∥x∥≤η(Tn-1)(∥Tx-x∥+an-1)+∥∥T(Tnx)-T(Tn-1x)∥x-∥Tnx-Tn-1x∥x∥≤η(Tn-1)(∥Tx-x∥+an-1)+∥η(T)(∥Tnx-Tn-1x∥+a1)x-∥Tnx-Tn-1x∥x∥≤η(Tn-1)(∥Tx-x∥+an-1)+∥[η(T)a1+(η(T)-1)∥Tnx-Tn-1x∥]x∥≤η(Tn-1)(∥Tx-x∥+an-1)+[η(T)a1+η(Tn-1)(η(T)-1)(∥Tx-x∥+an-1)]×∥x∥.
This gives ∥Snx-Sn+1x∥≤η(T)a1∥x∥+[1+(η(T)-1)∥x∥]η(Tn-1)(∥Tx-x∥+an-1) which yields ∥Snx-Sn+1x∥≤η(T)a1∥x∥+[1+(η(T)-1)∥x∥](d0x+M)η(Tn-1)where
(6)dnx=∥Tn+1x-Tnx∥.
Using the hypothesis
(7)η∞(T)=lim-n→∞[η(Tn)]1/n<1
together with the Root Test for convergence of series of real numbers, we obtain ∑n=1∞∥Snx-Sn+1x∥<∞ which means the sequence {S(n,x)} is a Cauchy sequence and so has a limit pointx*inE.

We are left to show that the limitx*of{xn}={S(n,x)}is a fixed point ofS(n,·), for all n∈ℕ. To achieve this, it suffices to prove thatS(n,·)is continuous which follows an application of Lemma 3, namely. Letx,y∈K, then
(8)∥S(·,x)-S(·,y)∥=∥x+∥Tx-x∥x-y-∥Ty-y∥y∥≤∥x-y∥+∥∥Tx-x∥x-∥Ty-y∥y∥=∥x-y∥+∥d1xx-d1yy∥≤[d1x+τd1y+1]∥x-y∥(byLemma3and(8))
for some positive real number τ. So given any ϵ, we have δ=ϵ/(d1x+τd1y+1) such that ∥Sx-Sy∥<ϵ whenever ∥x-y∥<δ for some τ>0. Therefore, S is continuous in x and so limn→∞xn=limn→∞S(n,x)=S(n,limn→∞xn)=x*.

To apply Lemma 1, we need its extension for hemicontinuous mappings given in the following form.

Lemma 6.

Let K be a nonempty subset of a Banach space, and letT:K→Kbe hemicontinuous nearly Lipschitzian mapping. Suppose that Tnu=pas n→∞ for some u,p∈K. Then, pis an element of Fix(T).

Proof.

Consider the following operator 𝒮:ℕ×E→E defined by
(9)𝒮(n,u)=u+∥Tnu-Tn-1u∥Tn-1u.
Clearly, 𝒮 restricted to K reduces to the auxiliary operator S above at the fixed point of 𝒮. We will show that given that T is hemicontinuous then 𝒮 is a selfmap of K for all n, that is, 𝒮(·,x):K→K since K is closed.

Clearly, 𝒮 restricted to K and T have common fixed point set, that is, Fix(T)=Fix(S)(providedThas a fixed point) and S(n+1,u)=u+∥Tn+1u-Tnu∥Tnu=u+dnTnu. But from the last proof, we verified thatSis a continuous mapping onKand has asymptotic fixed point x*∈E. Also, by hemicontinuity of T and continuity of 𝒮 the sequencefxng=f𝒮(n;x0g) converges strongly to x which means that f𝒮(n;xn)g converges weakly to 𝒮(n;p) which means 𝒮 is demicontinuous onK.

By Lemma 1, we have thatp∈Fix(𝒮)=Fix(T).

Theorem 7.

LetKbe a nonempty closed subset of a Banach spaceEandT:K→Ka hemicontinuous nearly Lipschitzian mapping with sequence {(an,η(Tn))}. Suppose η∞(T)=lim¯n→∞[η(Tn)]1/n<1. Then, we have the following:

Thas unique fixed pointp∈K;

for eachx0∈K, the sequence{Tnx0}converges strongly top;

∥S(n,x0)-p∥≤[2M+(η(T)+1)∥x0-Tx0∥]∑i=n∞η(Ti) for all n∈ℕ where M=supn∈ℕan and S(n,x)=Tn-1x+∥Tnx-Tn-1x∥x.

Proof.

By Lemma 6, the auxiliary operator given by Su=u+∥Tu-u∥u is a selfmap of K, and together with Lemma 5 we conclude thatShas a fixed point inKwhich is also a fixed point ofT.To prove (a), we are left to show that the fixed point is unique. The proof of uniqueness and for (b) and (c) follow from the fact that 𝒮 is demicontinuous contraction so that Theorem 2 applies.

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