#### Abstract

A common fixed point theorem for generalized -weak contraction in a metric space is established. As an application, some common fixed point results in normed linear spaces are obtained. We also study some results on best approximation via common fixed point theorems. Our result improves some results from the existing literature. Some illustrative examples to highlight the realized improvements are also furnished.

#### 1. Introduction

Let be a metric space. A mapping is said to be contraction if there exists such that for all If the metric space is complete, then the mapping satisfying (1) has a unique fixed point. This is known as the Banach contraction principle and is one of the significant results in nonlinear analysis. Inequality (1) also implies the continuity of .

Due to its importance and usefulness, generalizations of the above contraction principle have been a very active field of research for the last four decades (see, e.g., [1–27]).

In 1997, Alber and Guerre-Delabriere [1] introduced the concept of weakly contractive mappings and proved the existence of fixed points for weakly contractive mappings in Hilbert spaces. Thereafter, Rhoades [2] assumed -weakly contractive mappings which satisfies the condition where and is a continuous and nondecreasing function such that if and only if and shows that most of the results of Alber and Guerre-Delabriere [1] are still true in Banach spaces. If one takes , where , then (2) reduces to (1).

Recently, Zhang and Song [3] used the generalized -weak contraction and proved the following result.

Theorem 1 (see [3]). *Let be a complete metric space and let be a map such that for all **
where is a lower semicontinuous function with for and ,
**
Then there exists the unique point such that .*

Further, using the control function defined by Khan et al. [4], the above result has been generalized by many authors (see [5–7]). On the other hand, Berinde [8] introduced the notion of -weak contraction and proved that many well-known contractive conditions do imply -weak contraction. The concept of -weak contraction does not ask to be less than 1 as happens in condition (1). Afterward, many authors study this new class of weak contraction and obtained some significant result (see [8–12]).

*Definition 2. *A map is called generalized -weak contraction, if for each
where ,
and is a continuous monotone nondecreasing function with if and only if and is a lower semicontinuous function from right such that is positive on and .

If , then is said to be generalized -weak contraction. If , then is called generalized -weak contraction. If identity map, that is, coincides with , then is called generalized -weak contraction which is exactly the maps studied by Doric [5]. Again, if and , then is called generalized -weak contraction which is the same as generalized -weak contraction investigated by Akbar et al. (see [22]) and if for a constant with , then is called generalized -contraction which has been introduced by Song [21]. In Equation (5), if (identity map), then is called generalized ()-weak contraction.

*Remark 3. *It is obvious that the class of generalized -weak contraction contains the class of generalized -weak contraction and hence contains the class of generalized -weak contraction, but the converse is not true, as shown by Remark 30.

In this paper, we prove that there is a unique common fixed point for generalized -weak contractive mappings in a metric space. As an application, some common fixed point results in normed linear space are obtained. We apply these theorems to obtain some results on invariant approximation. Our results generalize and extend the corresponding results of [5, 11, 12, 16–23, 28] to the class of generalized -weak contractive maps.

#### 2. Preliminaries

We need the following known definitions and standard notations in the sequel.

Let be a nonempty subset of a normed space . The set is called the set of best approximants to out of , where . We denote and cl (resp., wcl) by the set of positive integers and the closure (resp., weak closure) of a set in . Let be mappings. The set of fixed point of is denoted by . A point is a coincidence point (common fixed point) of and if . The set of coincidence point of and is denoted by . The pair is called(1)commuting if for all ,(2)compatible [13] if whenever is a sequence in such that for some ,(3)weakly compatible [14] if for all ,(4)Banach operator pair [16] if the set is -invariant; that is, .Apparently, 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 [16]).

The set is called -star shaped with if the segment joining to is contained in for all . The map defined on is said to be (5) affine [18] if is convex and for all and and (6) -affine [18] if is -star shaped and for all and .

Suppose that is -star shaped with and is both - and -invariant. Then and are called(1)-commuting [18] if for all , where , where ,(2)-subweakly commuting on [15] if, for all , there exists a real number such that .

*Definition 4. *A map is said to be demiclosed at if whenever is a sequence in such that converges weakly to and converges strongly to 0; then .

*Definition 5. *A Banach space is said to satisfy Opial’s condition if whenever is a sequence in such that converges weakly to ; the inequality
holds for all . Every Hilbert space and the space satisfy Opial’s condition.

Lemma 6. *If and is a Cauchy sequence, then is also a Cauchy sequence.*

*Proof. *Since is a Cauchy sequence, we have that, for a given , there is a such that for every we have . Also, since we have that, given , there is a such that, for every , . Now, suppose ; then we have three possibilities.(a)If and are both even numbers, let , ; then .(b)If is even and is odd, let , ; then by the triangle inequality .(c)If are both odd numbers, let +1, ; then using the previous estimate .Therefore is Cauchy.

#### 3. Main Results

The following result extends and improves Theorem 2.2 of [5], Theorem 2.4 of [11], Theorem 2.1 of [12], and Theorem 2.1 of [18].

Theorem 7. *Let be a nonempty subset of a metric space and , and are self-mappings of such that . Assume that is complete and is generalized -weak contraction. Then the pairs and have a unique point of coincidence in . Also, if the pairs and are weakly compatible, then is singleton.*

*Proof. *Let be an arbitrary point. Since , we can choose a point such that . Similarly a point can be chosen such that . Continuing this process, we obtain a sequence in such that and for every . If, for some , = , then turns out to be a constant sequence and hence it is Cauchy. Now suppose that .

Using the fact that is generalized -weak contraction, for each , we have
where
Since , therefore we have
This implies
As is a nondecreasing function, therefore for each we have
Now from the triangle inequality for we have
If , then > 0. It furthermore implies that
This is a contradiction; therefore we have
Similarly, it can be shown that
Therefore, for each , we have
Thus, the sequence is monotone nonincreasing and bounded. So there exist such that
After letting in (10), we obtain ≤ , which is a contradiction unless .

Hence
Because of (19) and Lemma 6, to show to be a Cauchy sequence in , it is sufficient to show that is a Cauchy in .

Suppose not, then there exists for which we can find subsequences and of with such that, for every , and . So we have
Now using (19), we have
Moreover, using the known relation , we obtain
Then by using (19) and (21), we get
Again from the relation
using (19) and (23), we get
Now from
using (19) and (25), we get
Again, using the fact that is generalized -weak contraction, we get
where
And
Letting in (28), we get
which is a contradiction with . Thus is a Cauchy sequence and hence is a Cauchy sequence; therefore by the completeness of there is some such that
Further, ; therefore there exist such that
Since is generalized -weak contraction, therefore
where
Now using (32) and (33), we can write
Therefore, letting in (34), we get
This is true only if ; that is, and is coincidence point of and .

Since is generalized -weak contraction, therefore
where
Therefore, (32) and (33) imply
Then, letting in (38), we obtain
This is true only if ; that is, and is coincidence point of and . Thus the pairs and have a common point of coincidence in .

If the pairs and are weakly compatible, then and and hence . Now, we have to show that .

Further, using the fact that is generalized -weak contraction, we have
where
Then from (42), we get
This is true only if which implies .

Moreover, it can be easily shown that this is unique and hence is singleton.

If we take = identity mapping in Theorem 7, then we have the following result.

Corollary 8. *Let be a nonempty subset of a metric space and let be a self-mapping of such that . Assume that is complete and is generalized -weak contraction. Then is singleton.*

Corollary 9 (see [5], Theorem 2.2). *Let be a complete metric space and let be a self-mapping of . If is generalized (-weak contractions, then there is a unique fixed point of .*

In Corollary 8, if and , then Theorem 1 can be obtained as a particular case of the following result.

Corollary 10. *Let be a subset of a metric space and let be a self-mapping of such that . Assume that is complete and is generalized -weak contractions. Then is singleton.*

*Remark 11. *(1) In Theorem 7, if and for a constant with , then we obtain Theorem 2.1 of Song [21] as a particular case of Theorem 7.

(2) In Theorem 7, if and for a constant with , then for = identity map (resp., ) we obtain Theorem 2.4 of Berinde [11] (resp., Theorem 2.1 of Abbas and Ilić [12]) as a particular case of Theorem 7.

As an application of Corollary 8, we obtain the following general common fixed point result.

Theorem 12. *Let be a nonempty subset of a metric space and , and are self-maps of . Assume that is complete, is generalized -weak contraction, is nonempty, and . Then is singleton.*

*Proof. * is complete by the completeness of . Also, for all , we have by generalized -weak contractiveness of :
where , .

Hence is generalized -weak contraction mapping on . As , thus, by Corollary 8, has a unique fixed point in and, consequently, is singleton.

Corollary 13. *Let be a nonempty subset of a metric space and and are Banach operator pairs on . Assume that is complete, is generalized -weak contraction, and is nonempty and closed. Then is singleton.*

*Proof. *Since and are Banach operator pairs on , therefore ; then by closedness of we have . By Theorem 12, has a unique fixed point in and, consequently, is singleton.

In Theorem 12, if , then we easily obtain the following results which properly contain Theorem 3.3 of Akbar et al. [22].

Corollary 14. *Let be a nonempty subset of a metric space and , and are self-maps of . Assume that is complete, is generalized -weak contraction, is nonempty, and . Then is singleton.*

In Corollary 14, if and for a constant with , then for we obtain the following result which improves Lemma 3.1 of Chen and Li [16] and Theorem 2.2 of Al-Thagafi and Shahzad [19].

Corollary 15. *Let be a nonempty subset of a metric space and and are self-maps of . Assume that is complete, is generalized -contraction, is nonempty, and . Then is singleton.*

The following theorem properly contains Theorem 3.8 and Corollary 3.9 of Akbar et al. [22].

Theorem 16. *Let be a nonempty subset of a normed [resp., Banach] space and , and are self-maps of . Suppose that is -star shaped, resp., , and , and satisfy
**
for all and , where*(a)* is a continuous monotone nondecreasing function with if and only if ,*(b)* is a lower semicontinuous function from right such that is positive on and ,*(c)*, ,*(d)*.**Then provided that is compact [resp., is weakly compact] and is continuous [resp., is demiclosed at 0, where stands for identity map].*

*Proof. *For each , we define by for all and a fixed sequence of real numbers converging to 1. Since is -star shaped and [resp., , we have [resp., for each . Let and . Then by using (46), for all , we have
where
Clearly, is a lower semicontinuous function from right such that is positive on and and . Thus, for each is generalized -weak contraction. As is compact [resp., is weakly compact], therefore, for each , is compact [resp., is weakly compact]. Thus, [resp., is complete for each . By Theorem 12, for each , there exists in such that .

Again the compactness of implies that there exists a subsequence of such that . Since is a sequence in and , therefore . Moreover
As is continuous on , we have . Thus .

Next, weak compactness of implies that there exists a subsequence of such that converges weakly to. Since is a sequence in and , therefore . Also we have → 0 as . Further, demiclosedness of at 0 implies . Thus .

Corollary 17. *Let be a nonempty subset of a normed [resp., Banach] space and , and are self-maps of . Suppose that is -star shaped and closed [resp., weakly closed] and and are Banach operator pairs and satisfy (46). Then provided that is compact [resp., is weakly compact] and is continuous [resp., is demiclosed at 0, where stands for identity map].*

In Theorem 16, if and for a constant with , then we easily obtain the following result.

Corollary 18. *Let be a nonempty subset of a normed [resp., Banach] space and , and are self-maps of . Suppose that is -star shaped, [resp., , and , , and satisfy
**
where
**
Then provided that is compact [resp., wcl is weakly compact] and is continuous [resp., is demiclosed at 0, where stands for identity map].*

*Remark 19. *(1) By comparing Theorem 2.3(i) of Abbas and Ilić [12] with the first case of Corollary 18 (when ), their assumptions “ is -star shaped, , and and are weakly compatible on ” are replaced with “ is -star shaped and ."

(2) By comparing Theorem 2.3(ii) of Abbas and Ilić [12] with the second case of Corollary 18 (when ), their assumptions “ is -star shaped, , and are weakly compatible on , is weakly continuous, and is demiclosed at 0” are replaced with “ is -star shaped, , and is weakly continuous.”

(3) By comparing Theorem 2.4 of Song [21] with the first case of Corollary 18 (when ), his assumptions “ is -star shaped, , the pairs and are -commuting, and and are -affine and continuous on ” are replaced with is -star shaped and .”

(4) By comparing Theorem 2.2(i) of Hussain and Jungck [20] with the first case of Corollary 18 ( when ), their assumptions “ is complete and -star shaped, and are continuous and affine on , , , and and are -subweakly commuting pair on ” are replaced with “ is -star shaped and .”

(5) By comparing Theorem 2.2(ii) of Hussain and Jungck [20] with the second case of Corollary 18 ( when ), their assumptions “ is weakly compact and -star shaped, and are affine and continuous on , , , and are -subweakly commuting pair on , and is demiclosed at 0” are replaced with “ is weakly compact, is -star shaped, , and is demiclosed at 0.”

Corollary 20. *Let be a normed space [resp., Banach] and , , and are self-maps of . If and , is -star shaped, [resp., wcl], is compact [resp., wcl is weakly compact], is continuous on [resp., is demiclosed at 0, where stands for identity map], and (46) holds for all , then .*

*Remark 21. *Corollary 20 improves Theorem 2.8 of Hussain and Jungck [20], Theorems 3.1–3.4 of Song [21], and Corollary 2.5 of Al-Thagafi and Shahzad [19]. It is also noted that Corollary 3.13 of Akbar et al. [22] is a special case of Corollary 20.

We denote by the class of closed convex subsets of containing 0. For , we define . Clearly .

Theorem 22. *Let be a normed [resp., Banach] space and . If and such that , is compact [resp., wcl is weakly compact], and for all , then is nonempty, closed, and convex with . If, in addition, is a subset of , is -star shaped, [resp., wcl, is continuous on [resp., is demiclosed at 0, where stands for identity map], and (46) holds for all , then .*

*Proof. *We may assume that . If , then and so
Thus = . Assume that is compact; then by the continuity of norm, there exist such that .

If we assume that is weakly compact, then by using Lemma 5.5 of [24, p. 192] we can show the existence of such that . Thus, in both cases, we have