Nearly Contraction Mapping Principle for Fixed Points of Hemicontinuous Mappings

Abstract

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]) (i)demicontinuous if whenever a sequence{𝑥_𝑛}⊂𝑋converges strongly to𝑥∈𝑋it implies that the sequence{𝑇𝑥_𝑛}converges weakly to𝑇𝑥∈𝑌;(ii)hemicontinuous if whenever a sequence{𝑥_𝑛}⊂𝑋converges stronly on a line to𝑥∈𝑋it implies that the sequence{𝑇𝑥_𝑛}converges weakly to𝑇𝑥∈𝑌, that is, 𝑇(𝑥₀+𝑡_𝑛𝑥)⇀𝑇𝑥₀as𝑡_𝑛→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 that𝑇^𝑛𝑢=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{(𝑎_𝑛,𝜂(𝑇^𝑛))}. Suppose𝜂_∞(𝑇)=lim_𝑛→∞[𝜂(𝑇^𝑛)]^1/𝑛<1. Then, we have the following: (a)𝑇has unique fixed point𝑥^∗∈𝐶;(b)for each𝑥₀∈𝐶, the sequence{𝑇^𝑛𝑥₀} converges strongly to𝑥^∗;(c)‖𝑇^𝑛𝑥₀−𝑥^∗‖≤(‖𝑥₀−𝑇𝑥₀‖𝑀)∑^∞_𝑖=𝑛𝜂(𝑇^𝑖)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 constant𝐿_𝑛>0such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿_𝑛‖𝑥−𝑦‖ for all𝑥,𝑦∈𝐾. A Lipschitzian mapping is called uniformly 𝐿-Lipschitzian if 𝐿_𝑛=𝐿 for all 𝑛∈ℕ and asymptotically nonexpansive if lim_𝑛→∞𝐿_𝑛=1.

Next, let 𝐾 be a nonempty subset of a Banach space𝐸and{𝑎_𝑛}a fixed sequence in[0,∞)with 𝑎_𝑛→0 as𝑛→∞. A mapping 𝑇∶𝐾→𝐾 is called nearly Lipschitzian mapping with respect to {𝑎_𝑛} if for each 𝑛∈ℕ there exists a constant𝐿_𝑛≥0such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿_𝑛𝑥−𝑦+𝑎_𝑛. (1) The infimum𝜂(𝑇^𝑛)=sup{‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖/(‖𝑥−𝑦‖+𝑎_𝑛);𝑥,𝑦∈𝐾,𝑥≠𝑦}of constants 𝐿_𝑛 for which (1) holds is called nearly Lipschitzian constant. Nearly Lipschitzian operators with sequences {(𝑎_𝑛,𝜂(𝑇^𝑛))} are classified in [1, 2] as shown below: (a)nearly contraction if𝜂(𝑇^𝑛)<1for all𝑛∈ℕ;(b)nearly nonexpansive if𝜂(𝑇^𝑛)≤1for all𝑛∈ℕ;(c)nearly asymptotically nonexpansive if𝜂(𝑇^𝑛)≥1for all𝑛∈ℕ and lim_𝑛→∞𝜂(𝑇^𝑛)≤1;(d)nearly uniformly 𝐿-Lipschitzian if𝜂(𝑇^𝑛)≤𝐿for all𝑛∈ℕ;(e)nearly uniformly 𝐿-contraction if𝜂(𝑇^𝑛)≤𝐿<1for all𝑛∈ℕ.

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 𝛼𝑥−𝛽𝑦≤2|𝛼|+𝜏𝛽𝑥−𝑦 (2) 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 ⟹=≥−⟹≤+=+𝛼+𝛽𝑥−𝑦=𝛼𝑥−𝛽𝑦+𝛽𝑥−𝛼𝑥=𝛼𝑥−𝛽𝑦−𝛼𝑦−𝛽𝑥𝛼+𝛽𝑥−𝑦𝛼𝑥−𝛽𝑦−𝛼𝑦−𝛽𝑥𝛼𝑥−𝛽𝑦𝛼𝑦−𝛽𝑥𝛼𝑥−𝛽𝑦𝛼+𝛽𝑥−𝑦𝛽𝑥−𝛼𝑦𝛼+𝛽𝑥−𝑦𝛼+𝛽₁𝑥−𝛼𝑦forsome𝛽₁≤+|𝛼+𝛽∈𝔽𝛼+𝛽𝑥−𝑦|𝑥−𝑦₁𝑥‖≤2|𝛼|𝛽.‖+𝜏𝑥−𝑦 (3) 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. (1) If𝛼,𝛽∈ℝ then (3) reduces to 𝛼𝑥−𝛽𝑦≤2|𝛼|+𝛽𝑥−𝑦, (4) 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. (2) 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{(𝑎_𝑛,𝜂(𝑇^𝑛))}such that 𝜂_∞(𝑇)=lim_𝑛→∞[𝜂(𝑇^𝑛)]^1/𝑛<1.Then the auxiliary operator 𝑆∶ℕ×𝐾→𝐾defined by𝑆(𝑛,𝑥)=𝑇^𝑛−1𝑥+‖𝑇^𝑛𝑥−𝑇^𝑛−1𝑥‖𝑥has a fixed point in𝐸.

Proof. Given that𝑆(𝑛,𝑥)=𝑇^𝑛−1𝑥+‖𝑇^𝑛𝑥−𝑇^𝑛−1𝑥‖𝑥 where 𝑇 is a nearly Lipschitzian map with sequence {(𝑎_𝑛,𝜂(𝑇^𝑛))}, we have ‖𝑆()(≤𝑇𝑛,𝑥−𝑆𝑛+1,𝑥)‖_𝑛𝑥−𝑇^𝑛−1𝑥+𝑇^𝑛𝑥−𝑇^𝑛−1𝑥𝑥−𝑇^𝑛+1𝑥−𝑇^𝑛𝑥𝑥≤𝜂𝑇^𝑛−1‖𝑇𝑥−𝑥‖+𝑎_𝑛−1+𝑇𝑇^𝑛𝑥−𝑇𝑇^𝑛−1𝑥𝑥−𝑇^𝑛𝑥−𝑇^𝑛−1𝑥𝑥≤𝜂𝑇^𝑛−1‖𝑇𝑥−𝑥‖+𝑎_𝑛−1+𝜂(𝑇)𝑇^𝑛𝑥−𝑇^𝑛−1𝑥+𝑎₁𝑥−𝑇^𝑛𝑥−𝑇^𝑛−1𝑥𝑥≤𝜂𝑇^𝑛−1‖𝑇𝑥−𝑥‖+𝑎_𝑛−1+𝜂(𝑇)𝑎₁+𝜂(𝑇)−1‖𝑇^𝑛𝑥−𝑇^𝑛−1𝑥𝑇𝑥‖≤𝜂^𝑛−1‖𝑇𝑥−𝑥‖+𝑎_𝑛−1+𝜂(𝑇)𝑎₁+𝜂𝑇^𝑛−1𝜂(𝑇)−1‖𝑇𝑥−𝑥‖+𝑎_𝑛−1×‖𝑥‖. (5) This gives ‖𝑆^𝑛𝑥−𝑆^𝑛+1𝑥‖≤𝜂(𝑇)𝑎₁‖𝑥‖+[1+(𝜂(𝑇)−1)‖𝑥‖]𝜂(𝑇^𝑛−1)(‖𝑇𝑥−𝑥‖+𝑎_𝑛−1) which yields ‖𝑆^𝑛𝑥−𝑆^𝑛+1𝑥‖≤𝜂(𝑇)𝑎₁‖𝑥‖+[1+(𝜂(𝑇)−1)‖𝑥‖](𝑑_0𝑥+𝑀)𝜂(𝑇^𝑛−1)where 𝑑_𝑛𝑥=𝑇^𝑛+1𝑥−𝑇^𝑛𝑥. (6) Using the hypothesis 𝜂_∞(𝑇)=lim_𝑛→∞𝜂𝑇^𝑛1/𝑛<1 (7) together with the Root Test for convergence of series of real numbers, we obtain ∑^∞_𝑛=1‖𝑆^𝑛𝑥−𝑆^𝑛+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{𝑥_𝑛}={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 𝑆()=‖‖𝑦≤+‖𝑦=+𝑑⋅,𝑥−𝑆⋅,𝑦𝑥+𝑇𝑥−𝑥𝑥−𝑦−𝑇𝑦−𝑦𝑥−𝑦‖𝑇𝑥−𝑥𝑥−𝑇𝑦−𝑦𝑥−𝑦_1𝑥𝑥−𝑑_1𝑦𝑦≤𝑑_1𝑥+𝜏𝑑_1𝑦(8)+1𝑥−𝑦byLemma3and (8) for some positive real number 𝜏. So given any 𝜖, we have 𝛿=𝜖/(𝑑_1𝑥+𝜏𝑑_1𝑦+1) such that ‖𝑆𝑥−𝑆𝑦‖<𝜖 whenever ‖𝑥−𝑦‖<𝛿 for some 𝜏>0. Therefore, 𝑆 is continuous in 𝑥 and so lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑆(𝑛,𝑥)=𝑆(𝑛,lim_𝑛→∞𝑥_𝑛)=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 𝑇^𝑛𝑢=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 𝒮(𝑛,𝑢)=𝑢+𝑇^𝑛𝑢−𝑇^𝑛−1𝑢𝑇^𝑛−1𝑢. (9) 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+‖𝑇^𝑛+1𝑢−𝑇^𝑛𝑢‖𝑇^𝑛𝑢=𝑢+𝑑_𝑛𝑇^𝑛𝑢. 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 sequence𝑓𝑥_𝑛𝑔=𝑓𝒮(𝑛;𝑥₀𝑔) converges strongly to 𝑥 which means that 𝑓𝒮(𝑛;𝑥_𝑛)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 {(𝑎_𝑛,𝜂(𝑇^𝑛))}. Suppose 𝜂_∞(𝑇)=lim_𝑛→∞[𝜂(𝑇^𝑛)]^1/𝑛<1. Then, we have the following: (a)𝑇has unique fixed point𝑝∈𝐾;(b)for each𝑥₀∈𝐾, the sequence{𝑇^𝑛𝑥₀}converges strongly to𝑝;(c)‖𝑆(𝑛,𝑥₀)−𝑝‖≤[2𝑀+(𝜂(𝑇)+1)‖𝑥₀−𝑇𝑥₀‖]∑^∞_𝑖=𝑛𝜂(𝑇^𝑖) for all 𝑛∈ℕ where 𝑀=sup_𝑛∈ℕ𝑎_𝑛 and 𝑆(𝑛,𝑥)=𝑇^𝑛−1𝑥+‖𝑇^𝑛𝑥−𝑇^𝑛−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.