## Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications

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# Some Integral Type Fixed Point Theorems for Non-Self-Mappings Satisfying Generalized -Weak Contractive Conditions in Symmetric Spaces

**Academic Editor:**Wei-Shih Du

#### Abstract

The aim of this paper is to obtain some new integral type fixed point theorems for nonself weakly compatible mappings in symmetric spaces satisfying generalized -contractive conditions employing the common limit range property. We furnish some interesting examples which support our main theorems. Our results generalize and extend some recent results contained in Imdad et al. (2013) to symmetric spaces. Consequently, a host of metrical common fixed theorems are generalized and improved. In the process, we also derive a fixed point theorem for four finite families of mappings which can be utilized to derive common fixed point theorems involving any number of finite mappings.

#### 1. Introduction

The celebrated Banach contraction principle is indeed the most fundamental result of metrical fixed point theory, which states that* a contraction mapping of a complete metric space into itself has a unique fixed point*. This theorem is very effectively utilized to establish the existence of solutions of nonlinear Volterra integral equations, Fredholm integral equations, and nonlinear integrodifferential equations in Banach spaces besides supporting the convergence of algorithms in computational mathematics. In [1], Hicks and Rhoades proved some common fixed point theorems in symmetric spaces and showed that a general probabilistic structure admits a compatible symmetric or semimetric.

The study of common fixed points for noncompatible mappings is equally interesting due to Pant [2]. Jungck [3] generalized the idea of weakly commuting pair of mappings due to Sessa [4] by introducing the notion of compatible mappings and showed that compatible pair of mappings commutes on the set of coincidence points of the involved mappings. In 1998, Jungck and Rhoades [5] introduced the notion of weakly compatible mappings in nonmetric spaces. For more details on systematic comparisons and illustrations of these described notions, we refer to Singh and Tomar [6] and Murthy [7]. Afterwards, Al-Thagafi and Shahzad [8] introduced an even weaker notion which they called occasionally weak compatibility. Many authors (see, e.g., [9–12]) used this notion to obtain common fixed point results. Recently, Đorić et al. [13] showed that, for single-valued mappings, the condition of occasionally weak compatibility reduces to weak compatibility in the case of a unique point of coincidence (and a unique common fixed point) of the given mappings. However, for hybrid pairs of maps, this is not the case.

On the other hand, in 2002, Aamri and El Moutawakil [14] introduced the notion of property (E.A) which is a special case of tangential property due to Sastry and Krishna Murthy [15]. Later on, Liu et al. [16] initiated the notion of common property (E.A) for hybrid pairs of mappings which contained property (E.A). In this continuation, Imdad et al. [17] and Soliman et al. [18] extended the results of Sastry and Krishna Murthy [15] and Pant [19] to symmetric spaces by utilizing the weak compatible property with common property (E.A). Since the notions of property (E.A) and common property (E.A) always require the completeness (or closedness) of underlying subspaces for the existence of common fixed point, hence Sintunavarat and Kumam [20] coined the idea of “common limit range property” which relaxes the requirement of completeness (or closedness) of the underlying subspace. Afterward, Imdad et al. [21] extended the notion of common limit range property to two pairs of self-mappings and proved some fixed point theorems in Menger and metric spaces. Most recently, Karapınar et al. [22] utilized the notion of common limit range property and showed that the new notion buys certain typical conditions utilized by Pant [19] up to a pair of mappings on the cast of a relatively more natural absorbing property due to Gopal et al. [23].

The concept of weak contraction was introduced by Alber and Guerre-Delabriere [24] in 1997 wherein authors introduced the following notion for mappings defined on a Hilbert space .

Consider the following set of real functions: A mapping is called a -weak contraction if there exists a function such that

Alber and Guerre-Delabriere [24] also showed that each -weak contraction on a Hilbert space has a unique fixed point. Thereafter, Rhoades [25] showed that the results contained in [24] are also valid for any Banach space. In particular, he generalized the Banach contraction principle which follows in case one chooses .

In 2002, Branciari [26] firstly studied the integral analogue of Banach’s contraction principle. Some interesting results can be easily seen in [27–32]. Most recently, Vetro et al. [33] proved some integral type fixed point results for mappings in metric spaces employing common limit range property. Zhang and Song [34] proved a common fixed point theorem for two mappings by using -weak contraction. This result was extended by Đorić [35] and Dutta and Choudhury [36] to a pair of -weak contractive mappings. However, the main fixed point theorem for a self-mapping satisfying -weak contractive condition contained in Dutta and Choudhury [36] is given below, but, before that, we consider the following set of real functions:

Theorem 1. *Let be a complete metric space and let be a self-mapping satisfying
**
for some and and all . Then has a unique fixed point in .*

The object of this paper is to prove some integral type common fixed point theorems for two pairs of nonself weakly compatible mappings satisfying generalized -contractive conditions by using the common limit range property in symmetric spaces. We give some illustrative examples to highlight the superiority of our results over several results existing in the literature. As an extension of our main result, we state some fixed point theorems for six mappings and four finite families of mappings in symmetric spaces by using the notion of the pairwise commuting mappings which is studied by Imdad et al. [37].

#### 2. Preliminaries

A common fixed point result generally involves conditions on commutativity, continuity, and contraction along with a suitable condition on the containment of range of one mapping into the range of the other. Hence, one is always required to improve one or more of these conditions in order to prove a new common fixed point theorem. It can be observed that in the case of two mappings , where is metric space (or symmetric space), one can consider the following classes of mappings for the existence and uniqueness of common fixed points: where is some function and is the maximum of one of the sets. Thus,

A further possible generalization is to consider four mappings instead of two and ascertain analogous common fixed point theorems. In the case of four mappings , where is metric space (or symmetric space), the corresponding sets take the form In this case (5) is usually replaced by where is the maximum of one of the -sets.

Similarly, we can define the -sets for six mappings , where is metric space (or symmetric space), as and the contractive condition is again in the form (8).

By using different arguments of control functions, Radenović et al. [38] proved some common fixed point results for two and three mappings by using -weak contractive conditions and improved several known metrical fixed point theorems. Motivated by these results, we prove some common fixed point theorems for two pairs of weakly compatible mappings with common limit range property satisfying generalized -weak contractive conditions. Many known fixed point results are improved, especially the ones proved in [38] and also contained in the references cited therein. We also obtain a fixed point theorem for four finite families of self-mappings. Some related results are also derived besides furnishing illustrative examples.

The following definitions and results will be needed in the sequel.

A symmetric on a set is a function satisfying the following conditions:(1) if and only if for ,(2) for all .

Let be a symmetric on a set . For and , let . A topology on is defined as follows: if and only if, for each , there exists an such that . A subset of is a neighbourhood of if there exists such that . A symmetric is a semimetric if, for each and each , is a neighbourhood of in the topology . A symmetric (resp., semimetric) space is a topological space whose topology on is induced by symmetric (resp., semimetric) . The difference of a symmetric and a metric comes from the triangle inequality. Since a symmetric space is not essentially Hausdorff, therefore in order to prove fixed point theorems some additional axioms are required. The following axioms, which are available in Wilson [39], Aliouche [40], and Imdad et al. [17], are relevant to this presentation.

From now on symmetric space will be denoted by whereas a nonempty arbitrary set will be denoted by .Given , , and in , and imply [39].Given , , and in , and imply [39].Given , , and in , and imply [40].A symmetric is said to be 1-continuous if implies , where is a sequence in and [41].A symmetric is said to be continuous if and imply , where and are sequences in and [41].

Here, it is observed that , , and but the converse implications are not true. In general, all other possible implications amongst , , and are not true. For detailed description, we refer an interesting note of Cho et al. [42] which contained some illustrative examples. However, implies all the remaining four conditions, namely, , , , and . Employing these axioms, several authors proved common fixed point theorems in the framework of symmetric spaces (see [22, 43–48]).

*Definition 2. *Let be a pair of self-mappings defined on a nonempty set equipped with a symmetric . Then the mappings and are said to be (1)commuting if for all ,(2)compatible [3] if for each sequence in such that ,(3)noncompatible [2] if there exists a sequence in such that but is either nonzero or nonexistent,(4)weakly compatible [5] if they commute at their coincidence points, that is, whenever , for some ,(5)satisfied the property (E.A) [14] if there exists a sequence in such that , for some .

Any pair of compatible as well as noncompatible self-mappings satisfies the property (E.A) but a pair of mappings satisfying the property (E.A) needs not be noncompatible.

*Definition 3 (see [16]). *Let be an arbitrary set and let be a nonempty set equipped with symmetric . Then the pairs and of mappings from into are said to share the common property (E.A), if there exist two sequences and in such that
for some .

*Definition 4 (see [20]). *Let be an arbitrary set and let be a nonempty set equipped with symmetric . Then the pair of mappings from into is said to have the common limit range property with respect to the mapping (denoted by ) if there exists a sequence in such that
where .

*Definition 5 (see [21]). *Let be an arbitrary set and let be a nonempty set equipped with symmetric . Then the pairs and of mappings from into are said to have the common limit range property with respect to mappings and , denoted by , if there exist two sequences and in such that
where .

*Remark 6. *It is clear that property implies the common property (E.A) but the converse is not true (see [49, Example 1]).

*Definition 7 (see [37]). *Two families of self-mappings and are said to be pairwise commuting if (1) for all ,(2) for all ,(3) for all and .

#### 3. Results

Now, we state and prove our main results for four mappings employing the common limit range property in symmetric spaces. Firstly, we prove the following lemma.

Lemma 8. *Let be a symmetric space wherein satisfies the conditions whereas is an arbitrary nonempty set with , , and . Suppose that *(1)*the pair (or ) satisfies the (or ) property,*(2)* (or ),*(3)* (or ) is a closed subset of ,*(4)* converges for every sequence in whenever converges (or converges for every sequence in whenever converges),*(5)*there exist and such that for all , we have
**where
**
and is a Lebesgue-integrable mapping which is summable and nonnegative such that
**
for all .**Then the pairs and satisfy the property.*

*Proof. *First, we show that the conclusion of this theorem holds for first case. Since the pair enjoys the property; therefore there exists a sequence in such that
where . Since , hence for each sequence there exists a sequence in such that . Therefore, by closedness of ,
for and in all . Thus, in all, we have , and as . Since by (4), converges, in all we need to show that as . Assume this contrary, we get as . Now, using inequality (13) with , we have
where
Taking limit as and using property in inequality (18), we get
that is,
where
Hence inequality (21) implies
that is, and so and, by the property of the function , we have or equivalently , which contradicts the hypothesis . Hence both the pairs and satisfy the property.

In the second case, it similar to the first case. So, in order to avoid repetition, the details of the proof are omitted. This completes the proof.

Theorem 9. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with , which satisfy the inequalities (13) and (15) of Lemma 8. Suppose that the pairs and satisfy the property. Then and have a coincidence point each. Moreover, if , then , , and have a unique common fixed point provided both the pairs and are weakly compatible.*

*Proof. *Since the pairs and enjoy the property, there exist two sequences and in such that
where . It follows from that there exists a point such that . We assert that . If not, then, using inequality (13) with and , one obtains
where
Taking limit as and using properties and in inequality (25), one obtains
that is,
where
From inequality (28), one gets
so that ; that is, . Hence which shows that is a coincidence point of the pair .

Also , there exists a point such that . We assert that . If not, then, using inequality (13) with , , we have
where
Hence inequality (31) implies
so that ; that is, . Therefore which shows that is a coincidence point of the pair .

Consider . Since the pair is weakly compatible and , hence . Now we assert that is a common fixed point of the pair . To accomplish this, using inequality (13) with and , one gets
where
From inequality (34), we have
so that ; that is, . Hence we have which shows that is a common fixed point of the pair .

Also the pair is weakly compatible and ; hence . Using inequality (13) with and , we have
where
Hence inequality (37) implies
so that ; that is, . Therefore, which shows that is a common fixed point of the pair . Hence is a common fixed point of the pairs and . Uniqueness of common fixed point is an easy consequence of the inequality (13). This concludes the proof.

Now, we furnish an illustrative example which demonstrates the validity of the hypotheses and degree of generality of Theorem 9.

*Example 10. *Consider equipped with the symmetric for all , which also satisfies and . Define the mappings , , and by
Then and . Now, consider the sequences and . Then
that is, both the pairs and satisfy the property.

Let Lebesgue-integrable defined by . Take and given by and . In order to check the contractive condition (13), consider the following nine cases:(i),
(ii), ,(iii), ,(iv), ,(v),
(vi), ,(vii), ,(viii), ,(ix).

In the cases (i), (iii), (vii), and (ix), we get that and (13) is trivially satisfied.

In the cases (ii) and (viii), we have and . Now we have Therefore, we get the fact that (13) holds.

In the case (iv), we get and . Now we have This implies that (13) holds.

In the case (vi), we have and . Now we have This shows that (13) holds.

Finally, in the case (v), we obtain and . Now we have that is, the contractive condition (13) holds.

Hence, all the conditions of Theorem 9 are satisfied and 2 is a unique common fixed point of the pairs and which also remains a point of coincidence as well. Here, one may notice that all the involved mappings are discontinuous at their unique common fixed point .

Corollary 11. *Let be a symmetric space wherein satisfies the condition whereas is an arbitrary nonempty set with satisfying all the hypotheses of Lemma 8. Then and have a coincidence point each. Moreover, if , then , , and have a unique common fixed point provided both the pairs and are weakly compatible.*

*Proof. *Owing to Lemma 8, it follows that the pairs and enjoy the property. Hence, all the conditions of Theorem 9 are satisfied, and and have a unique common fixed point provided both the pairs and are weakly compatible.

*Remark 12. *The conclusions of Lemma 8, Theorem 9, and Corollary 11 remain true if we choose or .

Our next result shows the importance of common limit range property over common property (E.A).

Theorem 13. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with , which satisfy the inequalities (13) and (15) of Lemma 8. Suppose that *(1)*the pairs and satisfy the common property (E.A),*(2)* and are closed subsets of .**Then and have a coincidence point each. Moreover, if , then , , and have a unique common fixed point provided both the pairs and are weakly compatible.*

*Proof. *Since the pairs and satisfy the common property (E.A), there exist two sequences and in such that
for some . Since and are closed subsets of , therefore . Since , there exists a point such that . Also, since , there exists a point such that . The rest of the proof runs on the lines of the proof of Theorem 9.

*Remark 14. *The conclusions of Theorem 13 remain true if condition (2) of Theorem 13 is replaced by one of the following conditions: and , where and denote the closure of ranges of the mappings and , and are closed subsets of provided and .

By setting , , and suitably, we can deduce corollaries involving two and three self-mappings. As a sample, we can deduce the following corollary involving two self-mappings.

Corollary 15. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with . Suppose that *(1)*the pair satisfies the property,*(2)*there exist and such that, for all , we have
**where , , and is a Lebesgue-integrable mapping which is summable and nonnegative such that (15) holds.**Then has a coincidence point. Moreover, if , then and have a unique common fixed point in provided the pair is weakly compatible.*

As an application of Theorem 9, we have the following result involving four finite families of self-mappings.

Theorem 16. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with , , , satisfying the inequalities (13) and (15) of Lemma 8 where , , and . Suppose that the pairs and satisfy the property. Then and have a point of coincidence each.**Moreover, if , then , , , and have a unique common fixed point provided the families and commute pairwise where , , , and .*

Now, we indicate that Theorem 16 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample, we can derive a common fixed point theorem for six mappings by setting two families of two members while setting the rest two of single members.

Corollary 17. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with . Suppose that *(1)*the pairs and share the property,*(2)*there exist and such that, for all , we have
**where , , and is a Lebesgue-integrable mapping which is summable and nonnegative such that (15) holds.**Then and have a coincidence point each. Moreover, if , then , , , , and have a unique common fixed point provided , , , , , and .*

By setting , , , and in Theorem 16, one gets the following corollary.

Corollary 18. *Let be a symmetric space wherein satisfies the conditions and whereas is an arbitrary nonempty set with . Suppose that *(1)*the pairs and share the property, where , , , are fixed positive integers,*(2)*there exist and such that, for all , we have
**where , , and is a Lebesgue-integrable mapping which is summable and nonnegative such that (15) holds.**Moreover, if , then , , and have a unique common fixed point provided and .*

*Remark 19. *The above Corollary 18 is a slight but partial generalization of Theorem 9 as the commutativity requirements (i.e., and ) in this corollary are relatively stronger as compared to weak compatibility in Theorem 9.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

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