Weak Contractions, Common Fixed Points, and Invariant Approximations

Abstract

The existence of common fixed points is established for the mappings, where is -weak contraction on a nonempty subset of a Banach space. As application, some results on the invariant best approximation are proved. Our results unify and substantially improve several recent results given by some authors.

1. Introduction and Preliminaries

Let be a subset of a normed space . The set

(1.1)

is called the set of best approximants to out of where

(1.2)

We denote and (resp., ) by the set of positive integers and the closure (resp., weak closure) of a set in , respectively. Let be mappings. The set of fixed points of is denoted by . A point is a coincidence point (resp., common fixed point) of and if (resp., ). The set of coincidence points of and is denoted by

The pair is said to be

(1)commuting [1] if for all

(2)compatible [2, 3] if whenever is a sequence such that for some in ,

(3)weakly compatible if they commute at their coincidence points, that is, if whenever ,

(4)a Banach operator pair if the set is -invariant, namely, .

Obviously, the commuting pair is a Banach operator pair, but converse is not true in general (see [4, 5].) If is a Banach operator pair, then needs not be a Banach operator pair (see [4, Example 1]).

The set is said to be -starshaped with if the segment joining to is contained in for all The mapping defined on a -starshaped set is said to be affine if

(1.3)

Suppose that the set is -starshaped with and is both - and -invariant. Then and are said to be

(5)-commuting [3, 6] if for all , where where ,

(6)pointwise -subweakly commuting [7] if, for given there exists a real number such that ,

(7)-subweakly commuting on [8] if, for all there exists a real number such that .

In 1963, Meinardus [9] employed Schauder's fixed point theorem to prove a result regarding invariant approximation. Further, some generalizations of the result of Meinardus were obtained by Habiniak [10], Jungck and Sessa [11], and Singh [12].

Since then, Al-Thagafi [13] extended these works and proved some results on invariant approximations for commuting mappings. Hussain and Jungck [8], Hussain [5], Jungck and Hussain [3], O'Regan and Hussain [7], Pathak and Hussain [14], and Pathak et al. [15] extended the work of Al-Thagafi [13] for more general noncommuting mappings.

Recently, Chen and Li [4] introduced the class of Banach operator pairs as a new class of noncommuting mappings and it has been further studied by Hussain [5], Khan and Akbar [16], and Pathak and Hussain [14].

In this paper, we extend and improve the recent common fixed point and invariant approximation results of Al-Thagafi [13], Al-Thagafi and Shahzad [17], Berinde [18], Chen and Li [4], Habiniak [10], Jungck and Sessa [11], Pathak and Hussain [14], and Singh [12] to the class of -weak contractions. The applications of the fixed point theorems are remarkable in diverse disciplines of mathematics, statistics, engineering, and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and others (see [14, 15, 19, 20]).

2. Main Results

Let be a metric space. A mapping is called a weak contraction if there exist two constants and such that

(2.1)

Remark 2.1.

Due to the symmetry of the distance, the weak contraction condition (2.1) includes the following:

(2.2)

which is obtained from (2.1) by formally replacing , by , , respectively, and then interchanging and .

Consequently, in order to check the weak contraction of , it is necessary to check both (2.1) and (2.2). Obviously, a Banach contraction satisfies (2.1) and hence is a weak contraction. Some examples of weak contractions are given in [18, 21, 22]. The next example shows that a weak contraction needs not to be continuous.

Example 2.2 (see [18, 22]).

Let be the unit interval with the usual norm and let be given by for all and . Then satisfies the inequality (2.1) with and and has a unique fixed point , but is not continuous.

Let be a self-mapping on . A mapping is said to be -weak contraction or -weak contraction if there exist two constants and such that

(2.3)

Berinde [18] introduced the notion of a -weak contraction and proved that a lot of the well-known contractive conditions do imply the -weak contraction. The concept of -weak contraction does not ask to be less than as happens in many kinds of fixed point theorems for the contractive conditions that involve one or more of the displacements . For more details, we refer to [18, 21] and references cited in these papers.

The following result is a consequence of the main theorem of Berinde [18].

Lemma 2.3.

Let be a nonempty subset of a metric space and let be a self-mapping of . Assume that , is complete, and is a -weak contraction. Then is nonempty.

Theorem 2.4.

Let be a nonempty subset of a metric space and let be self-mappings of Assume that is nonempty, , is complete, and is an -weak contraction. Then .

Proof.

Since is a closed subset of , is complete. Further, by the -weak contraction of , for all , we have

(2.4)

Hence is a -weak contraction on and . Therefore, by Lemma 2.3, has a fixed point in and so .

Corollary 2.5.

Let be a nonempty subset of a metric space and let be a Banach operator pair on . Assume that is complete, is -weak contraction, and is nonempty and closed. Then .

In Theorem 2.4 and Corollary 2.5, if , then we easily obtain the following result, which improves Lemma 3.1 of Chen and Li [4].

Corollary 2.6 (see [17, Theorem 2.2]).

Let be a nonempty subset of a metric space and let be self-mappings of Assume that is nonempty, , is complete, and is an -contraction. Then is a singleton.

The following result properly contains [4, Theorems 3.2-3.3] and improves [13, Theorem 2.2], [10, Theorem 4], and [11, Theorem 6].

Theorem 2.7.

Let be a nonempty subset of a normed (resp., Banach) space and let be self-mappings of Suppose that is -starshaped, (resp., ), is compact (resp., is weakly compact, and either is demiclosed at or satisfies Opial's condition, where stands for the identity mapping), and there exists a constant such that

(2.5)

Then .

Proof.

For each , define by for all and a fixed sequence of real numbers converging to . Since is -starshaped and (resp., ), we have (resp., ) for each . Also, by the inequality (2.5),

(2.6)

for all , , and Thus, for , is a -weak contraction, where .

If is compact, then, for each , is compact and hence complete. By Theorem 2.4, for each , there exists such that The compactness of implies that there exists a subsequence of such that as . Since is a sequence in and , we have . Further, it follows that

(2.7)

Moreover, we have

(2.8)

Taking the limit as we get and so .

Next, the weak compactness of implies that is weakly compact and hence complete due to completeness of (see [3]). From Theorem 2.4, for each , there exists such that The weak compactness of implies that there is a subsequence of converging weakly to as . Since is a sequence in , we have . Also, we have as . If is demiclosed at , then and so

If then we have

(2.9)

which is a contradiction. Thus and hence This completes the proof.

Obviously, -nonexpansive mappings satisfy the inequality (2.5) and so we obtain the following.

Corollary 2.8 (see [17, Theorem 2.4]).

Let be a nonempty subset of a normed (resp., Banach) space and let , be self-mappings of Suppose that is -starshaped, (resp., ), is compact, (resp., is weakly compact, and either is demiclosed at or satisfies Opial's condition), and is -nonexpansive on . Then .

Corollary 2.9 (see [4, Theorems 3.2-3.3]).

Let be a nonempty subset of a normed (resp., Banach) space and let be self-mappings of Suppose that is -starshaped and closed (resp., weakly closed), is compact (resp., is weakly compact, and either is demiclosed at or satisfies Opial's condition), is a Banach operator pair, and is -nonexpansive on . Then .

Corollary 2.10 (see [13, Theorem 2.1]).

Let be a nonempty closed and -starshaped subset of a normed space and let , be self-mappings of such that . Suppose that commutes with and . If is compact, is continuous, linear, and is -nonexpansive on , then .

Let where

Corollary 2.11.

Let be a normed (resp., Banach) space and let be self-mappings of If , , is -starshaped, (resp., ), is compact, (resp., is weakly compact, and is demiclosed at ). If the inequality (2.5) holds for all then .

Corollary 2.12.

Let be a normed (resp., Banach) space and let be self-mappings of If , , is -starshaped, (resp., ), is compact, (resp., is weakly compact, and is demiclosed at ). If the inequality (2.5) holds for all then .

Corollary 2.13 (see [11, Theorem 7]).

Let , be self-mappings of a Banach space with and with . Suppose that is q-starshaped with , , and is affine, continuous in the weak and strong topology on . If and are commuting on and is -nonexpansive on , then provided either (i) is weakly compact and is demiclosed or (ii) is weakly compact and satisfies Opial's condition.

Remark 2.14.

Corollary 2.5 in [17] and Theorems 4.1-4.2 of Chen and Li [4] are special cases of Corollaries 2.11-2.12

We denote by the class of closed convex subsets of containing . For any , we define It is clear that (see [8, 13]).

Theorem 2.15.

Let be self-mappings of a normed (resp., Banach) space . If and such that , is compact (resp., is weakly compact), and for all , then is nonempty closed and convex with . If, in addition, , is -starshaped, (resp., , and is demiclosed at ), and the inequality (2.5) holds for all then .

Proof.

We may assume that . If then . Note that

(2.10)

Thus If is compact, then, by the continuity of the norm, we get for some If we assume that is weakly compact, then, using [23,Lemma 5.5, page 192], we can show the existence of a such that . Thus, in both cases, we have

(2.11)

for all Hence and so is nonempty closed and convex with . The compactness of (resp., the weak compactness of ) implies that is compact (resp., is weakly compact). Therefore, the result now follows from Corollary 2.12. This completes the proof.

Corollary 2.16.

Let be self-mappings of a normed (resp., Banach) space . If and such that , is compact (resp., is weakly compact), and for all , then is nonempty closed and convex with . If, in addition, , is -starshaped and closed (resp., weakly closed and is demiclosed at ), is a Banach operator pair on , and the inequality (2.5) holds for all then .

Remark 2.17.

Theorem 2.15 and Corollary 2.16 extend [13, Theorems 4.1 and 4.2], [17, Theorem 2.6], and [10, Theorem 8].

Banach's Fixed Point Theorem states that if is a complete metric space, is a nonempty closed subset of , and is a self-mapping satisfying the following condition: there exists such that

(2.12)

then has a unique fixed point, say in and the Picard iterative sequence converges to the point for all Since then, Ćirić [24] introduced and studied self-mappings on satisfying the following condition: there exists such that

(2.13)

where

(2.14)

Further, many investigations were developed by Berinde [19], Jungck [1, 2], Hussain and Jungck [8], Hussain and Rhoades [6], O'Regan and Hussain [7], and many other mathematicians (see [14, 25] and references therein). Recently, Jungck and Hussain [3] proved the following extension of the result of Ćirić [24].

Theorem 2.18 (see [3, Theorem 2.1]).

Let be a nonempty subset of a metric space and let be self-mappings of . Assume that , is complete, and , satisfy the following condition: there exists such that

(2.15)

for all . Then .

The following result (Theorem 2.19) properly contains [17, Theorem 3.3], [4, Theorems 3.2-3.3], [5, Theorem 2.11], and [14, Theorem 2.2]. The proof is analogous to the proof of Theorem 2.7. In fact, instead of applying Theorem 2.4, we apply Theorem 2.18 to get the conclusion.

Theorem 2.19.

Let be a nonempty subset of a normed (resp., Banach) space and let be self-mappings of Suppose that is -starshaped, (resp., ), is compact (resp., is weakly compact), and is continuous on (resp., is demiclosed at ). If the following condition holds:

(2.16)

for all then .

Theorem 2.20.

Let , , be self-mappings of a Banach space with and such that . Suppose that , , for all , and is compact. Then one has the following:

(1) is nonempty closed and convex,

(2),

(3) provided is continuous, is -starshaped, , and the pair satisfies the inequality (2.5) for all is -starshaped with , , and the inequality (2.16) holds for all and .

Proof.

• (1) and (2) follow from [5, 8]. By (2), the compactness of implies that and is compact. Theorem 2.7 implies that . Further, is -starshaped with . Therefore, the conclusion now follows from Theorem 2.19 applied to .

Remark 2.21.

• (1) Theorem 2.20 extends [13, Theorem 4.1], [10, Theorem 8], [5, Theorem 2.13], [8, Theorem 2.14], and [14, Theorem 2.11].

• (2) Theorems 2.7–2.16 represent very strong variants of the results in [3, 8, 11, 13] in the sense that the commutativity or compatibility of the mappings and is replaced by the hypothesis that is a Banach operator pair, needs not be linear or affine, and needs not be -nonexpansive.

• (3) The Banach operator pairs are different from those of weakly compatible, -commuting and -subweakly commuting mappings and so our results are different from those in [3, 7, 8, 17]. Consider with the norm for all . Define two self-mappings and on as follows:

  

  (2.17)

Then we have the following:

(2.18)

Thus is a Banach operator pair. It is easy to see that is -weak contraction and , do not commute on the set , and so are not weakly compatible. Clearly, is not affine or linear, is convex and is a common fixed point of and .