Abstract
In this paper, our focus is to acquaint with the Suzuki-type mappings to establish some fixed point results using the new w-interpolative approach. We present some results for interpolative contraction operators via the w-admissible maps which satisfy the Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions in quasi-partial b-metric space. Further, the outcomes so obtained are affirmed with relevant examples.
1. Introduction
In the early century, Fréchet [1], a French mathematician, initiated the concept of metric space, and due to its efficiency and practicable implementations, the idea has been upgraded, improved, and generalized by many authors. In 1922, Banach [2] discovered a remarkable result, that is, Banach contraction principle, which holds a significant position in the field of nonlinear analysis. Later, Karapinar [3] introduced quasi-partial metric spaces which were followed by the discovery of metric spaces in 1993, by Czerwik [4]. Gupta and Gautam [5] generalized quasi-partial metric to quasi-partial metric space and proved some fixed point results for such spaces. After all these classical results, Suzuki [6] introduced a new type of mappings which generalized the well-known Banach contraction principle.
In 2014, the notion of w-orbital admissible maps was introduced by Popescu [7] which is a refinement of the concept of -admissible maps of Samet et al. [8].
Suppose is a self-map defined on and is a mapping where is nonempty. The mapping is said to be w-orbital admissible if for all , we have
If the continuity of the involved contractive mappings is removed, we necessarily need (, ) to be w-regular, i.e., if is a sequence in (, ) such that as and 1 for each , then we have .
We show that the condition of w-regularity holds in quasi-partial b-metric space by using w-admissibility condition. In our earlier work [9], we have shown that w-admissibility holds in quasi-partial b-metric space, i.e., ; then, as , we get , and hence we get the condition for w-regularity.
Throughout the paper, , , and stand for the set of positive real numbers, natural numbers, and an empty set, respectively. Let be the set of all nondecreasing self-mappings on such that for every . Notice that for , we have and for all (see [10]).
2. Preliminaries
Definition 1 (see [5]). A function is said to be a quasi-partial b-metric on a nonempty set if it satisfies the properties(1) implies ;(2);(3);(4),where is called the coefficient of such that for all .
Definition 2 (see [5]). Suppose is a quasi-partial b-metric space. Then,(1)A sequence is called a Cauchy sequence if and only if and exist finitely.(2)A sequence converges to if and only if = .(3) is said to be complete if every Cauchy sequence converges with respect to to a point that holds(4)A mapping is said to be continuous at if for every , there exists such that .
Definition 3 (see [9]). Suppose is a quasi-partial b-metric space. A self-map is known as a w-interpolative Ćirić–Reich–Rus contraction if there exist and a map with real numbers satisfying , that holdsfor all , .
Definition 4 (see [11]). Suppose is a quasi-partial b-metric. Define a self-mapping and a map where that holdsfor all , and real numbers , , that satisfy the condition . Such a mapping is known as w-interpolative Hardy–Rogers-type contraction.
Definition 5 (see [10]). A mapping is said to satisfy C-condition on (, ), if it satisfiesfor all , .
Throughout the paper, and denote the quasi-partial b-metric space and complete quasi-partial b-metric space, respectively. One can see for more related point results in [12–15] and the references therein.
3. Main Results
We now define the main results for Suzuki-type mappings using the notion of w-interpolation (see 16, 17) and the fact that the condition of w-regularity holds in quasi-partial metric space (see 18–20]).
Definition 6. Let be a quasi-partial b-metric space and there exists a self-map [0, ) with a real number [0, 1). A self-map is said to be a --interpolative Kannan contraction of Suzuki type if there exist that satisfiesfor all , .
Theorem 1. Suppose is a and is a --interpolative Kannan contraction of Suzuki type. Let be a w-orbital admissible map and 1 for some . Then, possesses a fixed point in if any of the following conditions hold:(1)(G,) is w-regular.(2)S is continuous.(3) is continuous and 1 when .
Proof. Let with the condition 1 and be the sequence such that = for each positive integer . For some , we have . Hence, we get = , so is a fixed point of . Hence, the proof is complete.
On the contrary, take for every positive integer . As is w-orbital admissible, we have the condition (, S ) = (, ) 1 which implies that (, S) = (, ) 1. Proceeding in a similar way, we getHence, choosing = and = S in (6) giveswhich impliesHence, we havewhich equivalently can be written asThus, we get that is a nonincreasing sequence of positive terms, so there exists 0 such that = . On the other hand, from the above equations and the nondecreasing nature of function , we obtainBy triangular inequality, for all , we getwhere . But, the series is convergent, so there exists a positive real number such that = . Letting and in the above inequality, we getHence, is a Cauchy sequence and using the completeness property of space, it shows that there exists such thatAlso, we claim that possesses a fixed point as .
In the case when assumption (1) holds true, we have 1 and we claim thatfor every . On the contrary, if the above condition is not true, then by triangular inequality in space, we havewhich is a contradiction, and hence our claim is proved. If the first condition holds, we obtainIf the second condition holds, we getTherefore, letting , we get , that is, .
In the case when assumption (2) holds, we have that the mapping is continuous, so we getIn the case when assumption (3) holds, we have = = and we prove that . On the contrary, take ; then,By (6), we getHence, it is a contradiction. Thus, , that is, is a fixed point of the mapping .
Example 1. Let = [0, /4] and define [0, ) such thatDefine a self-mapping asAlso, define [0, ) such thatChoose and define the function as . The only case we need to verify is when = /9 and = /5; as for the remaining cases, we have , which clearly implies that inequality (6) holds. So, when = /9 and = /5, we getwhich impliesHence, all the assumptions of Theorem 1 are satisfied, which follows that the mapping owns a fixed point, that is, , as shown in Figure 1.
Corollary 1. Let be a C and be a self-map on , satisfyingfor all , , where [0, 1). Then, owns a fixed point in .
Proof. For the proof, take = 1 in Theorem 1.
Corollary 2. Let be a C and be a self-map on , satisfyingfor all , , where [0, 1). Then, S owns a fixed point in .
Proof. For the proof, take , with [0, 1) and 0 in Corollary 1.
Definition 7. Suppose is a quasi-partial b-metric space. Define a self-mapping such that there exist satisfyingfor all , and real numbers , 0 that satisfy + 1. Such a mapping is called --interpolative Ćirić–Reich–Rus contraction of Suzuki type (see 21–23).
Theorem 2. Suppose is a C and is a --interpolative Ćirić–Reich–Rus contraction of Suzuki type. Let be a w-orbital admissible map and (, ) 1 for some . Then, possesses a fixed point in if any of the conditions hold:(1)(G, ) is w-regular.(2)S is continuous.(3) is continuous and 1 when .
Proof. Let with the condition 1 and be the sequence such that = for each positive integer . Assume that for some , we have the condition . Hence, we get = S, which implies is the fixed point of . So, the proof is complete.
On the contrary, take for each positive integer . Since is w-orbital admissible, we have (, S ) = (, ) 1, which implies that (, S ) = (, ) 1. Proceeding similarly,Hence, choosing = and = S in (30), we getwhich impliesThen, using for every 0, we getwhich equivalently can be written asSo, we getHence, it shows that is a decreasing sequence. Eventually, we haveIn a similar way, we getSince is a fundamental sequence, applying triangular inequality, we getTaking , we deduce that is a fundamental sequence in and by the completeness property of , there exists satisfyingIf assumption (1) holds, then we have 1 and we claim thatfor every . On the contrary, suppose the above inequality does not hold; then, by triangular inequality in , we haveHence, the contradiction occurs. Therefore, for every N, our claim holds. If the first condition holds, we obtainIf the second condition holds true, clearly is the fixed point of in a similar manner. Furthermore, if the w-regular condition is removed and is a continuous map, we get a fixed point in becauseFinally, if the last condition holds, i.e., is continuous, we easily obtain . Suppose on the contrary that , since 1 for any Fix andWe havewhich is a contradiction. So, , that is, the mapping owns a fixed point .
Example 2. Suppose = and define [0, ) such thatLet the transformation maps as follows:Also, define [0, ) such thatChoose and define the function as . We need to check when . So, the following cases occur.
Case 1. When = (0, 1),which implies
Case 2. When = (0, 1/3),which implies
Case 3. When = (1, 0),which impliesHence, all the assumptions of Theorem 2 are satisfied, and it follows that the mapping owns a fixed point, that is, , as shown in Figure 2.
Definition 8. Let be a quasi-partial b-metric space. Define a self-mapping with satisfyingfor all , and , 0 with the condition + 1. Such a mapping is called a -interpolative Ćirić–Reich–Rus contraction of Suzuki type.
Corollary 3. Suppose (G, ) is a C and S is a -interpolative Ćirić–Reich–Rus contraction of Suzuki type. Then, S owns a fixed point in G.
Proof. For the proof, take = 1 in Theorem 2.
Corollary 4. Suppose (G, ) is a C and S is an interpolative Ćirić–Reich–Rus contraction of Suzuki type if there exist [0, 1) and positive reals , 0, with + 1 such thatfor all , G. Then, S possesses a fixed point in G.
Proof. In Theorem 2, it is sufficient to put , for all 0 and , for the proof.
Definition 9. Let be a space and define a map [0, ). A mapping is --interpolative Hardy–Rogers contraction of Suzuki type if there exists with real numbers , , 0, holding + + 1 such thatfor all , and (see [24]).
Theorem 3. Suppose (G, ) is a C and is a -interpolative Hardy–Rogers contraction of Suzuki type. Let S be a w-orbital admissible mapping and (, S ) 1 for some G. Then, S possesses a fixed point in G if any of the conditions hold:(1)(G, ) is w-regular.(2)S is continuous.(3) is continuous and (S, ) 1 when Fix .
Proof. Let with the condition (, S) 1 and be the sequence such that = for each positive integer . Assume that for some , we have the condition . Hence, we get = S which implies is a unique fixed point of . Hence, the proof is complete.
Now, consider for each positive integer . As S is w-orbital admissible, we have the condition (, S) = (, ) 1 which implies that (, S) = (, ) 1. Proceeding in a similar way, we getChoosing = and = S in (58), we getwhich impliesThen, using for every 0, we getwhich is equivalent toHence, is a decreasing sequence. Eventually, we haveIn a similar way, we getSince is a fundamental sequence, applying triangular inequality,Taking , we deduce that is a fundamental sequence in space and by the completeness property of , there exists satisfyingNow we show that is the fixed point of . If assumption (1) holds true, then we have 1 and we claim thatfor every . Suppose the above condition does not hold; then, by triangular inequality in , we havewhich is a contradiction. Therefore, for every N, eitherholds. If the first condition holds, we obtainIf assumption (2) holds, we get that is the fixed point of in a similar manner. Furthermore, if the w-regular condition is removed and is continuous, then we get that owns a unique fixed point in becauseFinally, if the last condition holds, i.e., is continuous, we easily obtain . Suppose on the contrary that , since 1 for any Fix andWe havewhich is a contradiction. So, , which implies that is a fixed point of the map .
Definition 10. Let be a quasi-partial b-metric space. A mapping is said to be a -interpolative Hardy–Rogers contraction of Suzuki type if there exist and , , 0, with the condition + + 1 such thatfor all , .
Corollary 5. Let (G, ) be a C and S be a -interpolative Hardy–Rogers contraction of Suzuki type. Then, the mapping S possesses a fixed point in G.
Proof. For the proof, take = 1 in Theorem 3.
Example 3. Let = [0, 3] and define [0, ) such thatLet the mapping be defined asAlso, define [0, ) such thatChoose and define the function as . For , [0, 2), we have , which clearly implies that inequality (58) holds. As per the definition of function , the only case left is when we have = 0 and = 3 as , sowhich implieswhere we assume . Hence, the assumptions of Theorem 3 are satisfied, so the mapping owns a fixed point, that is, , as shown in Figure 3.
4. Conclusion and Future Aspects
The paper propounds the idea of using interpolation in noteworthy Suzuki-type mappings in the quasi-partial b-metric space. The incentive behind the paper was to introduce new concepts on completeness of --interpolative Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions of Suzuki-type mappings in quasi-partial b-metric space. Further, some fixed point results are obtained and are validated by illustrative examples. Interpolation is a noble concept which can be utilized to obtain different interpolative contraction of Suzuki-type mappings in other well-known spaces in the future. We are certain that the paper is a significant improvement of the known results in the existing fixed point literature.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.