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Some common fixed-point and invariant approximation results with generalized almost contractions
Fixed Point Theory and Applications volume 2014, Article number: 23 (2014)
Abstract
In this paper, the concept of a generalized almost -contraction is introduced and we establish some common fixed-point results for the noncommuting generalized almost -contraction in the setup of metric spaces and normed linear spaces, where the set of fixed points of f and g need not be starshaped. As applications, invariant approximation results are proved. Supporting examples are also given.
1 Introduction
The classical Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. It has various applications to operator theory, variational analysis, and approximation theory, so it has been extended in many ways (see, e.g., [1–30]).
In 2004, Berinde [1] defined the notion of a weak contraction mapping, which is more general than a contraction mapping. However, in [2] Berinde renamed it as an almost contraction, which is more appropriate.
Definition 1.1 Let () be a complete metric space. A map is called an almost contraction if there exist a constant and some such that
Berinde [1] proved some fixed-point theorems for almost contractions in a complete metric space which generalized the results of Kannan [3], Chatterjea [4], and Zamfirescu [5].
In 2008, Babu et al. [6] defined the class of mappings satisfying ‘condition (B)’ as follows.
Definition 1.2 Let be a metric space. A map is said to satisfy ‘condition (B)’ if there exist a constant and some such that
for all .
They prove that any map T satisfying ‘condition (B)’ has a unique fixed point in complete metric spaces. They also discuss quasi-contraction, almost contraction, and the class of mappings that satisfy ‘condition (B)’ in detail.
Afterwards Berinde [7] generalized the above definition and proved the following fixed-point result.
Theorem 1.3 Let be a complete metric space and let be a mapping for which there exist and some such that for all
where
Then T has a unique fixed point.
The contractive condition (1.3) is termed as generalized almost contraction.
Recently, Abbas and Ilić in [15] introduced the following definition.
Definition 1.4 Let T and f be two self-maps of a metric space . A map T is called a generalized almost f-contraction if there exist and some such that
where
If , then condition (1.3) can be obtained as particular case of condition (1.4). However, in [15] Abbas and Ilić obtained various common fixed-point and invariant approximation results for such mappings under the assumption of weak compatibility of maps.
Recently, Chen and Li [10] introduced the class of Banach operator pairs, as a new class of noncommuting mappings and obtained some common fixed-point and invariant approximation results for this class of maps. This class of noncommuting maps is different from the class of noncommuting maps (viz. R-subcommuting, R-sub-weakly commuting, -commuting, compatible, weakly compatible etc.) studied in [11–13, 15, 17–19, 27–29]. So, it has been further studied by various authors (see, e.g., [16, 21, 22, 24]).
In this article, we introduce the class of generalized almost -contraction and consequently establish some common fixed-point results for the noncommuting generalized almost -contraction in the framework of metric spaces and normed linear spaces, where the set of fixed points of f and g need not be starshaped. As an application, invariant approximation results are proved. The proved results generalize and extend the corresponding results of Chen and Li [10], Al-Thagafi and Shahzad [16], Akbar et al. [22], Chandok and Narang [24], Al-Thagafi [25] and Jungck and Sessa [26], Shahzad [28] to the class of generalized almost -contractions.
2 Preliminaries
First, we introduce some well-known notations and definitions that will be needed in the sequel.
Let be a metric space, M be a subset of X and f, T be self-maps of M. A point is a coincidence point (common fixed point) of f and T if (). The set of coincidence points of f and T is denoted by and the set of fixed points of f is denoted by . The pair is called
-
(1)
commuting if for all ,
-
(2)
compatible [8] if whenever is a sequence in M such that for some ,
-
(3)
weakly compatible [9] if for all ,
-
(4)
a Banach operator pair [10] if the set is T-invariant, namely .
Obviously, a commuting pair is a Banach operator pair but not conversely. If is a Banach operator pair, then need not be Banach operator pair (see [10]).
Let M be a subset of a normed space . The set is called the set of best approximants to out of M, where . We denote by â„• and () the set of positive integers and the closure (weak closure) of a set M in X, respectively.
The set M is said to be (a) q-starshaped if there exists such that the line segment joining q to x is contained in M for all ; (b) convex if for all . The map f defined on a set M is called
-
(1)
affine [11] if M is convex and , for all ,
-
(2)
q-affine [11] if M is q-starshaped and , for all .
Suppose that M is q-starshaped with and is both T- and f-invariant. Then T and f are called
-
(1)
-commuting [11] if for all , where where ,
-
(2)
R-subcommuting on M [12] if, for all , there exists a real number such that , ,
-
(3)
R-sub-weakly commuting on M [13] if, for all , there exists a real number such that .
A Banach space X is said to satisfy Opial’s condition if, whenever is a sequence in X such that converges weakly to , the inequality
holds for all . A Hilbert space and the space () satisfy Opial’s condition. The map is said to be demiclosed at zero if, whenever is a sequence in M such that converges weakly to and converges to 0, then .
The following important extension of the concept of starshapedness was defined by Naimpally et al. [14] and has been studied by many authors.
Definition 2.1 A subset M of a linear space X is said to have property (N) with respect to T if
-
(1)
,
-
(2)
, for some and a fixed sequence of real numbers () converging to 1 and for each .
It is to be noted that each T-invariant q-starshaped set has property (N) but converse does not hold in general. This is shown by the following example.
Example 2.2 Let be the set of real numbers and be endowed with the usual norm. Define for each . Then clearly M is not q-starshaped but has property (N) with respect to T, for , .
3 Main results
First we introduce the notion of a generalized almost -contraction.
Definition 3.1 Let be a metric space and f, g be self-maps of X. A mapping is said to be a generalized almost -contraction if there exist and some such that
where
and
If , then Definition 1.4 is a particular case of Definition 3.1. If (identity operator), then equation (1.3) can be obtained as a special case of equation (3.1).
Here we observe that if T satisfies ‘condition (B)’ then T is a generalized almost contraction but its converse need not be true. This is shown by the following example.
Example 3.2 Let be endowed with the Euclidean metric . We define a mapping by
Then T is a generalized almost contraction with and . But T does not satisfy ‘condition (B)’ at , for any and .
In (3.1) if , then T is called a generalized -contraction. Obviously, a generalized -contraction implies a generalized almost -contraction, but the converse is not true in general.
Example 3.3 Let with the usual metric and be given by for all . Also define a mapping as
Then T is a generalized almost -contraction with any and . But T is not a generalized -contraction at , or , for any .
The following lemma is a particular case of the main theorem of Abbas and Ilić [15].
Lemma 3.4 Let M be a nonempty subset of a metric space , and T be a self-map of M. Assume that , is complete, and T is a generalized almost contraction. Then is singleton.
Now, we start with the following common fixed-point result, which will be used in sequel.
Theorem 3.5 Let M be a nonempty subset of a metric space , and T, f and g be self-maps of M. Assume that is nonempty, , is complete, and T is a generalized almost -contraction. Then is singleton.
Proof The completeness of implies that of . Further, by a generalized almost -contraction of T, for all , we have
Hence T is a generalized almost contraction mapping on and . By Lemma 3.4, T has a unique fixed point z in and consequently is singleton. □
Corollary 3.6 Let M be a nonempty subset of a metric space , and T, f and g be self-maps of M such that and are Banach operator pairs on M. Assume that is complete, T is a generalized almost -contraction and is nonempty and closed. Then is singleton.
In Theorem 3.5 if we take , then we easily obtain the following result, which improves and extends Lemma 3.1 of Chen and Li [10] and Theorem 2.2 of Al-Thagafi and Shahzad [16].
Corollary 3.7 Let M be a nonempty subset of a metric space , and T, f, and g be self-maps on M. Assume that is nonempty, , is complete, and T is a generalized -contraction. Then is singleton.
Remark 3.8 By comparing Theorem 2.1 of Shahzad [17] with Corollary 3.7 (when ), their assumptions that M is closed, , T is continuous and is R-weakly commuting pair on M are replaced with ‘ is nonempty, ’.
Theorem 3.9 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. If has the property (N) with respect to T, (respectively, ), and there exists a constant such that
where
and
then , provided is compact (respectively, is weakly compact) and T is continuous (respectively, is demiclosed at 0, where I stands for identity map).
Proof As and has the property (N) with respect to T, for each , we can define by for all and a fixed sequence of real numbers () converging to 1. Since has the property (N) with respect to T, and (respectively, ), we have (respectively, ) for each . Also, by the inequality (3.2),
where
and
for all , , and . Thus, for each , is a generalized -almost contraction.
If is compact, then, for each , is compact and hence complete. By Theorem 3.5, for each , there is a unique in M such that . The compactness of implies that there exists a subsequence of such that . Since is a sequence in and , we have . Moreover,
As T is continuous on M, we have . Thus .
Next, the weak compactness of implies that is weakly compact and hence complete due to completeness of X. From Theorem 3.5, for each , there is a unique in M such that . The weak compactness of implies that there is a subsequence of such that converges weakly to . Since is a sequence in and , therefore . Also we have as . Further, demiclosedness of at 0 implies , thus . □
Corollary 3.10 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. If has the property (N) with respect to T and is closed (respectively, weakly closed), and are Banach operator pairs and satisfy (3.2) for all . Then , provided is compact (respectively, is weakly compact) and T is continuous (respectively, is demiclosed at 0, where I stands for the identity map).
Remark 3.11 (1) By comparing Theorem 2.2 of Shahzad [17] with the first case of Theorem 3.9 (when , ), their assumptions ‘, M is closed and q-starshaped, f is linear and continuous on M, and is R-sub-weakly commuting pair on M’ are replaced with ‘M is a nonempty subset, has the property (N) with respect to T, ’.
(2) By comparing Theorem 2.2(i) of Hussain and Jungck [18] with the first case of Theorem 3.9 (when ), their assumptions ‘M is complete and q-starshaped, f and g are continuous and affine on M, , , and and are R-sub-weakly commuting pair on M’ are replaced with ‘ has the property (N) with respect to T, ’.
(3) By comparing Theorem 2.2(ii) of Hussain and Jungck [18] with the second case of Theorem 3.9 (when ), their assumptions ‘M is weakly compact and q-starshaped, f and g are affine and continuous on M, , , and and are R-sub-weakly commuting pair on M, and is demiclosed at 0’ are replaced with ‘ is weakly compact, has the property (N) with respect to T, and is demiclosed at 0’.
Remark 3.12 If the contractive condition (3.2) in Theorem 3.9 is replaced with the stronger contractive condition
for all and some , where
and
then continuity of T can be relaxed in the first case of Theorem 3.9.
Proof The proof will be similar to the first case of Theorem 3.9. To prove , instead of continuity of T, using (3.3) we have
where
and
Now taking in (3.4) we can write
This is possible only if as , which implies . □
Let , where .
Corollary 3.13 Let X be a normed (respectively, Banach) space and let T, f, and g be self-maps of X. If and , has the property (N) with respect to T, (respectively, ), is compact (respectively, is weakly compact), T is continuous on D (respectively, is demiclosed at 0, where I stands for identity map) and (3.2) holds for all , then .
Corollary 3.14 Let X be a normed (respectively, Banach) space and let T, f, and g be self-maps of X. If and , has the property (N) with respect to T, (respectively, ), is compact (respectively, is weakly compact), T is continuous on D (respectively, is demiclosed at 0, where I stands for the identity map) and (3.2) holds for all , then .
Remark 3.15 Corollaries 3.13 and 3.14 improve and develop Theorems 2.8-2.11 of Hussain and Jungck [18] and Theorems 3.1-3.4 of Song [19] to the non-starshaped domain.
Denote by the class of closed convex subsets of X containing 0. For , we define . Clearly .
The following invariant approximation result constitutes an extension of Theorem 2.6 of Al-Thagafi and Shahzad [16] and Corollary 2.10 of [29] to a non-starshaped domain.
Theorem 3.16 Let X be a normed (respectively, Banach) space and . If and such that , is compact (respectively, is weakly compact), and for all , then is nonempty, closed, and convex with . If, in addition, D is a subset of , has the property (N) with respect to T, (respectively, ), T is continuous on D (respectively, is demiclosed at 0, where I stands for the identity map) and (3.2) holds for all , then .
Proof We may assume that . If , then and, so
Thus . Assume that is compact, then by the continuity of the norm there exists such that .
If we assume that is weakly compact, then by using Lemma 5.5 of [[20], p.192] we can show the existence of such that . Thus in both cases, we have
for all . It follows that . Thus is nonempty, closed, and convex with . The compactness of (respectively, weak compactness of ) implies that is compact (respectively, is weakly compact). Then by Corollary 3.14, . □
Now, we present some non-trivial examples in support of Theorem 3.9.
Example 3.17 Let be the set of real numbers with the usual norm and . We define mappings by
and , for .
Here we observe that , and is compact. Clearly is not starshaped but has property (N) with respect to T, for and . Further, the mappings T, f, and g satisfy the contractive condition (3.2) and also T is continuous. Hence all the conditions of the first case of Theorem 3.9 are satisfied and consequently T, f, and g have a common fixed point, .
Remark 3.18 In Example 3.17, it is interesting to note that Theorem 2.19 of Hussain and Cho [21], and Corollary 3.10 of Akbar et al. [22] cannot apply, since is not q-starshaped.
Example 3.19 Let be the set of real numbers with the usual norm and . Define by
and
Clearly has property (N) with respect to T, for , . Further, , is compact and T, f, and g satisfy the contractive condition (3.2). Hence all the conditions of the first case of Theorem 3.9 are satisfied except the continuity of T. Note that .
Remark 3.20 It is to be noted that the maps T, f, and g given in Example 3.19 do not satisfy the contractive condition (3.3) at the point , .
4 Results with joint contractive family
Dotson [23] proved some results concerning the existence of fixed points of nonexpansive mappings on a certain class of non-convex sets. For proving these results, he extends the concept of starshapedness by introducing the following class of non-convex set.
Let M be a subset of a normed space X and be a family of functions from to M such that for each . The family Γ is said to be contractive if there exists a function such that for all and all , we have
Such a family Γ is said to be jointly continuous (jointly weakly continuous) if in and ( weakly) in M; then ( weakly) in M.
We observe that if M is q-starshaped subset of a normed linear space X and , for each , and , then Γ is a contractive jointly continuous and jointly weakly continuous family with . Thus the class of subsets of X with the property of contractiveness and joint continuity contains the class of starshaped sets which in turns contains the class of convex sets.
We shall denote where .
The following results properly contain Theorems 3.2 and 3.3 of [10], Theorems 1 and 2 of [24] and improves Theorem 2.2 of [25], Theorem 6 of [26].
Theorem 4.1 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f and g be self-maps of M. Suppose is nonempty and has a contractive, jointly continuous (respectively, jointly weakly continuous) family of functions , (respectively, ), and there exists a constant such that
for all , where
and
Then , provided is compact (respectively, is weakly compact) and T is continuous (respectively, T is weakly continuous).
Proof For each natural number n, let . Define by for all . Since has a contractive family and (respectively, ), so for each , (respectively, ).
We have
for each , where , ,
and
Thus, for each , is a generalized almost -contraction.
If is compact, then, for each , is compact and hence complete. By Theorem 3.5, for each , there exists a unique such that . Again the compactness of implies that there exists a subsequence of such that . Since is a sequence in and , we have . Further, the joint continuity of family Γ implies that
By the continuity of T, we obtain . Thus, .
The weak compactness of implies that is weakly compact and hence complete due to completeness of X. From Theorem 3.5 for each , there exists a unique such that . The weak compactness of implies that there is a subsequence of such that converges weakly to as . Since is a sequence in and , we have . By the joint weak continuity of the family we obtain
Since the weak topology is Hausdorff, by weak continuity of T, we have . Thus, . □
Remark 4.2 By comparing Theorem 2.2(i) of Chandok and Narang [27] with the first case of Theorem 4.1 (when ), their assumptions ‘M is complete and has a contractive jointly continuous family Γ with and for , , the pairs and are -commuting and f, g are continuous on M’ are replaced with ‘M is nonempty subset, is nonempty and has a contractive jointly continuous family Γ, and ’.
Corollary 4.3 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. Suppose is q-starshaped, (respectively, ), and there exists a constant such that
for all , where
and
Then , provided is compact (respectively, is weakly compact) and T is continuous (respectively, T is weakly continuous).
Remark 4.4 (1) By comparing Theorem 2.3(i) of Abbas and Ilić [15] with the first case of Corollary 4.3 (when ), their assumptions ‘M is q-starshaped, , f and T are weakly compatible on M’ are replaced with ‘ is q-starshaped, ’.
(2) By comparing Theorem 2.3(ii) of Abbas and Ilić [15] with the second case of Corollary 4.3 (when ), their assumptions ‘M is q-starshaped, , f and T are weakly compatible on M, f is weakly continuous and is demiclosed at 0’ are replaced with ‘ is q-starshaped, and T is weakly continuous’.
(3) By comparing Theorem 2.4 of Song [19] with the first case of Corollary 4.3 (when ), their assumptions ‘M is q-starshaped, , the pairs and are -commuting, f and g are q-affine and continuous on M’ are replaced with ‘ is q-starshaped, ’.
Corollary 4.5 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. If M has a contractive jointly continuous (respectively, jointly weakly continuous) family such that and for all , . Suppose is nonempty, closed (respectively, weakly closed), is compact (respectively, is weakly compact), T is continuous (respectively, weakly continuous), and are Banach operator pair on M and satisfy (4.1). Then .
Proof For each natural number n, define by , for all . Clearly, for each , is a self-map on M. Since is Banach operator pair on M, for each , we have . Consider
This implies that for each . Thus for each , is a Banach operator pair on M. Similarly, for each , is a Banach operator on M. Now the result follows from Theorem 4.1. □
Corollary 4.6 Let X be a normed (respectively, Banach) space and let T, f, and g be self-maps of X. If and , is nonempty, has a contractive jointly continuous (respectively, jointly weakly continuous) family of functions , (respectively, ), is compact (respectively, is weakly compact), T is continuous on D (respectively, T is weakly continuous) and (4.1) holds for all , then .
Corollary 4.7 Let X be a normed (respectively, Banach) space and T, f, and g be self-maps of X. If and , is nonempty, has a contractive jointly continuous (respectively, jointly weakly continuous) family of , (respectively, ), is compact (respectively, is weakly compact), T is continuous on D (respectively, T is weakly continuous) and (4.1) holds for all , then .
Remark 4.8 (1) Theorems 4.1 and 4.2 of Chen and Li [10], Theorems 3 and 4 of Chandok and Narang [24] are particular cases of Corollaries 4.6 and 4.7.
(2) By Proposition 2.2 of Chen and Li [10], it can be concluded that Corollary 4.5 extends and generalizes Corollary 2.1 of Shahzad [28].
Now we present two examples in support of Theorem 4.1 and Theorem 3.5, respectively.
Example 4.9 Let be the set of real numbers with the usual norm and . Assume , for every x in M and define by
Then , and is compact. Suppose that is a family of functions from into , defined by
We observe that the family Γ is contractive jointly continuous for , . Thus all the conditions of Theorem 4.1 are satisfied. Consequently T, f, and g have a common fixed point. Here it is seen that is the common fixed point of T, f, and g.
Remark 4.10 (1) Theorem 2.2(i) of Chandok and Narang [27] cannot apply to Example 4.9, since f is not continuous.
(2) It is interesting to note that the results of Akbar et al. [22] cannot apply to Example 4.9, since is not q-starshaped.
Example 4.11 Let and let be given as
Then is a metric space. Let is defined, respectively, as follows:
and
Clearly and . Further T is a generalized almost -contraction for and . Hence, all the conditions of Theorem 3.5 are satisfied. Consequently T, f, and g have a unique common fixed point. Here it is seen that is the unique common fixed point of T, f, and g.
Remark 4.12 (1) In Example 4.11, , and , therefore is not contained in . Hence Theorem 2.1 of Song [19] cannot apply to Example 4.11.
(2) In Example 4.11, if we take then T and f does not satisfy the contractive condition of Lemma 3.1 of [10] and Theorem 2.2 of [16] at , . Hence Lemma 3.1 of [10] and Theorem 2.2 of [16] cannot apply to Example 4.11.
Remark 4.13 (1) Example 3.3 satisfies all the conditions of Theorem 3.5 except the condition . Note that .
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Rathee, S., Kumar, A. Some common fixed-point and invariant approximation results with generalized almost contractions. Fixed Point Theory Appl 2014, 23 (2014). https://doi.org/10.1186/1687-1812-2014-23
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DOI: https://doi.org/10.1186/1687-1812-2014-23