Some Valid Generalizations of Boyd and Wong Inequality and -Weak Contraction in Partially Ordered Metric Spaces
In this manuscript, we use -weak contraction to generalize coincidence point results which are established in the context of partially ordered b-metric spaces. The presented work explicitly generalized some recent results from the existing literature. Examples are also provided to show the authenticity of the established work.
Banach contraction principle  is one of the most important result in analysis. It is widely considered as the source of fixed point theory. Banach principle has been modified by many researchers by changing either the contractive condition or the underlying spaces. Boyd and Wong  modified the contraction condition in metric spaces with control function
The -weak contraction condition was initiated by Alber and Guerre-Delabriere in the Hilbert spaces, see . Rhoudes, in , showed that every -weak contractive mapping in metric spaces has a unique fixed point. The generalized -weak contractive condition in complete metric spaces was introduced by Zhang and Song in . In , Dutta and Choudhury studied unique fixed point results with the help of -weak contraction in metric spaces. Doric, in , studied the result of  with generalized -weak contraction in metric spaces. The concept of fixed point results for two pairs of mappings was introduced by Jungck in , for commuted mappings. Jungck in [8, 9] states this new assumption, with compatibility and weak compatibility of mappings. Some further work for generalized -weak contraction in metric spaces can be found in [6, 10, 11]. Ran and Reurring, in , introduced partially ordered metric spaces. A b-metric space with partial ordering is called partially ordered b-metric space. Recently, -weak contraction in partially ordered b-metric spaces gain the attention of many researcher. In this direction, fixed point and coincidence point results are discussed by many authors. For more details of fixed point and coincidence point results and their applications, comparison of different contraction conditions, and related results in b-metric spaces, we refer the reader to [13–40] along with the references mentioned therein. To show that a sequence is Cauchy in metric-type spaces, Jovanovic et al.  proved Lemma 1. As b-metric spaces have discontinuous structure, therefore, in this manuscript, we give a generalization of Boyd and Wong  inequality and -weak contraction, to establish coincidence and fixed point results.
Definition 1. (see ). Let be a partially ordered set, and assume that the map satisfies the following conditions, For all and : if and only if Then, is called partially ordered -metric space.
Lemma 1. (see ). “Let be b-metric space and . When a sequence satisfies the following condition,for some, and .
Then, is -Cauchy sequence in .
Definition 2. (see ). The pair of mappings is compatible in the metric space if and only ifwhenever in sequence such that
Definition 3. (see ). In the metric space , the pair of mappings , is weakly compatible if is commutative at the point of coincidence (i.e., whenever ).
Definition 4. (see ). The pair of mappings , defined on a partially ordered set is called(a)Weakly increasing if and , for all (b)Partially weakly increasing if ,
Definition 5. (see [44, 45]). Let , , be three mappings on partially ordered set such that and . Then, pair is said to be(i)Weakly increasing with respect to if and only if for all , , for all and , and for all (ii)Partially weakly increasing w.r.t if and only if for all .
Theorem 1. (see ). Let self-mappings be continuous mappings in partially ordered complete metric space with . Assume that the pair is weakly compatible, whereas is partially weakly increasing with respect to such that and are comparable elements and satisfying the following condition:where are altering distance functions. Then, mappings , have coincidence point.
Theorem 2. (see ). Let , be mappings on a partially ordered complete -metric space . Assume that , is partially weakly increasing and satisfying the following condition:where are altering distance and lower semicontinuous functions, respectively, with andwhere . Furthermore, assume that either or is continuous mapping or space is regular. Then, have a common fixed point.
Theorem 3. (see ). “Let be partially ordered complete -metric space and , , , and be continuous mappings such that and . Suppose for comparable elements and , the following condition holdwith altering distance functions, and . Let the pairs and be compatible and the pairs and be partially weakly increasing w.r.t and , respectively. Then, , and have a coincidence point.”
Theorem 4. (see ). “Let , , , be continuous mappings on a partially ordered complete -metric space with and . Assume that compatible pairs and and comparable elements and satisfy the following condition:where , while and the pairs of mappings and are partially weakly increasing w.r.t and , respectively. When and are comparable, then .”
Theorem 5. (see ). Let the mappings , be defined on a partially ordered complete -metric space and be -weakly isotone increasing. Let be a function such that for all and satisfy the condition:where . If the mappings or are continuous in . Then, has a common fixed point.
3. Main Results
We begin with the following result.
Theorem 6. Let , , , be continuous mappings on a partially ordered complete -metric space, with and . Assume that compatible pairs and and comparable elements and satisfy the following contractive condition:where and satisfy for all , , andwhere and the pairs and are partially weakly increasing w.r.t , respectively. Then, and have a coincidence point. Furthermore, if is a coincidence point for comparable elements and , then .
Proof. Let . Since and , then there exist such that and . The sequence is constructed as follows:The pairs of mappings and are partially weakly increasing w.r.t and , respectively. Therefore,By repeating this process, we deduce thatAssume thatWe discuss three steps: Step I: at this step, we have to prove that . Since and are comparable, therefore, from (11), we have with To show that , suppose that Here, we discuss three possible cases of . Case (1): if , from (17), we have Since for , therefore which contradict assumption (19). Case (2): if , since for , therefore, from (17), one can write which is held unless , which implies that , but it contradicts (16). Therefore, Case (3): if . Using triangle inequality and assumption (19), we have Since for all , thus, from (17), one has This is again contradiction. Hence, for all cases, we conclude that Step II: in this step, we have to prove that the sequence is -Cauchy by using Lemma 1, for all three cases. Case (1): if . From (17), one can write As for , so Taking , we have Case (2): if , from (26), we have . From (17), one has Since , for , therefore Thus, Let . Then, Case (3): if . Using triangle inequality and (26), it is reduced to case (2). Therefore, (2) holds for three cases. Similarly, if and , one can easily show that Hence, Define . Suppose , for some , then . In case , then . We have If Then, from (17), Since for all and , therefore, This is only possible if . Hence, . Thus, . Similarly, one can show for the remaining two cases. By the same process, if , then , which gives . Therefore, is a constant sequence with , and (2) is also valid for constant sequence. Hence, is -Cauchy sequence. The completeness of space implies that there exist such that . Consequently, the subsequences will also converge to : Step III: now, we show that is a coincidence point of , sinceAs from the compatibility of , we haveMoreover, and .
As and are continuous mappings, thereforeUsing triangle inequality twice, one can writeTaking limit and using (41) and (42) in the above inequality, we obtainThus, . Thus, . Therefore, is coincidence at point of .
From comparability of element and (11), one can writewhereIf , thenThis is possible if . Hence, .
On the same process, from the other two cases, we can show that .
In the following, the result condition of continuity and compatibility is relaxed for mappings.
Theorem 7. Let , , , be four mappings on regular partially ordered complete -metric spaces such that , , and , are -closed subsets of . Suppose the pairs and are partially weakly increasing with respect to , , respectively, whereas pairs are weak compatible and satisfy the following condition:where and with for , where , andwith . Then, , and have a coincidence point. Furthermore, if are comparable, then .
Proof. Following the lines of the proof of Theorem 6, one can easily show that there exists a sequence which converges to some . Therefore,If and but and are -closed subsets of , then there must be some , from which and . Hence, from the construction of the sequence given in Theorem 6, we haveNow, we prove that .
From regularity of , we have and .
Thus, from (49), one can writewhereSince for all and and using (54) in (53), we obtainThis is only possible if . It implies that . Hence, . Now, from weak compatibility of and , we can write . Thus, . Similarly, we can show that . Furthermore, following the last lines of the proof of Theorem 6, we have .
By putting (identity mapping) in Theorems (6 and 7), then the following corollary is obtained with
Corollary 1. Let , be two mappings on . Let be partially weakly increasing, whereas with for and , satisfy the following condition with comparable elements
Also,(i) or is continuous Or(ii) is regular
Then, b is a common fixed point of , .
Example 1. Let . Define , for all , then clearly is a partially ordered -metric on with and partially ordering is defined byAssume that are continuous mappings defined as follows:Since . Therefore, . Therefore, . Thus, the pair is partially weakly increasing. Now, we show thatDefine , where and take . Here, we discuss the following cases. Case (1): if , then Using mean value theorem for , where , we obtain Case (2): if , then . Therefore, . Hence, condition of Corollary 1 holds, and 0 is coincidence point.
Remark 2. As are partially weakly increasing, therefore is -weakly isotone increasing. Thus, by substitution , in Corollary 1, we get Theorem 2.1 of . Moreover, if and in Corollary 1, then Corollary 2.4 of  is obtained.
Remark 3. Clearly, the conditions of our Corollary 1 holds for Example 2.8 of  and the corresponding conclusion holds. By substituting in Example 2, then condition of Theorem 2.1 of  does not hold but our condition (57) of Corollary 1 holds.
In the remaining part of this manuscript, we discuss coincidence point of two compatible pairs of mappings with generalized -weak contractive condition.
Throughout the rest of this paper, we consider the following, , and for all , we defineand are altering distance functions and is partially ordered complete -metric space.
Theorem 8. Let , , , be continuous self-mappings on , where and . Suppose that the pairs and are compatible, whereas the pairs and are partially weakly increasing with respect to and , respectively, and satisfy the following condition:
Then , and have a coincidence point. Moreover, if are comparable then .
Proof. Let . Since and , so there exist such that and . Construct the sequence as follows:Since the pairs and are partially weakly increasing with respect to and , therefore,Repeating the above process, we can writeAssume thatWe discuss the proof in three steps. Step I: first, we prove that . Since and are comparable, therefore (64) implies that where To show that , suppose that Here, we discuss three possible cases of . Case (1): if , then As is nondecreasing, therefore which contradicts assumption (69). Thus, Case (2): if , then This is only possible if . However, if . Hence, , which implies that which contradicts (68). Hence, Case (3): if . Using triangle inequality and (69), we have Then, which implies that , Therefore, . So, contradicts (68). Therefore, in all three cases, we concluded that Step II: in this step, we will show that the sequence is -Cauchy sequence by using Lemma 1 for all three cases. Case (1): if . From (69), one can write Since for , therefore Since is nondecreasing, therefore