On Some Interpolative Contractions of Suzuki Type Mappings

The goal of this study is to propose a new interpolative contraction mapping by using an interpolative approach in the setting of complete metric spaces. We present some fixed point theorems for interpolative contraction operators using 
 
 w
 
 -admissible maps which satisfy Suzuki type mappings. In addition, some results are given. These results generalize several new results present in the literature. Moreover, examples are provided to show the suitability of our given results.


Introduction
In 1922, Banach [1] proved his famous remarkable fixedpoint theorem; the result is known as the Banach contraction principle, which states that "Let ðK, dÞ be a complete metric space and S : K ⟶ K be a contraction, then S has a unique fixed point." The Banach contraction principle is one of the essential and most valuable theorems of analysis and is accepted as the main results of metric fixed-point theory. In the last century, the fixed point and its applications have been the subject of research by many authors in the literature, since it provides useful tools to solve many complex problems that have applications in different sciences like computer science, engineering, data science, physics, economics, game theory, and biosciences [2][3][4][5][6][7]. Due to several applications of "fixed point theory," researchers were motivated to further generalize it in different directions, by generalizing the contractive conditions underlying the space concept of completeness.
The background literature on the famous Banach contraction principle has been extended in various comprehensive directions by many researchers. One of the exciting generalizations was given by Kannan [8], which characterize the completeness of underlying metric spaces. Kannan introduced the following theorem. Theorem 1. [8] Let ðK, dÞ be a complete metric space. A mapping S : K ⟶ K is said to be a Kannan contraction if there exists λ ∈ ½0, 1/2Þ such that for all v, t ∈ K \ FixðSÞ. Then, S posses a unique fixed point.
The Kannan theorem has been generalized in different aspects by many authors; one of the crucial generalizations was given by Karapinar in [9]. Karapinar [9] introduced the notion of an interpolative Kannan contraction mapping and proved the following: A mapping S : K ⟶ K on ðK, dÞ a complete metric space such that where κ ∈ ½0, 1Þ and α ∈ ð0, 1Þ, for each v, t ∈ K \ FixðSÞ.
Then, S has a unique fixed point in K. Subsequently, Karapinar et al. [10] introduced the following notion of interpolative Ciric-Reich-Rus contractions.
The concept of w-orbital admissible mappings was introduced by Popescu as a clarification of the concept of α -admissible mappings of Samet et al. [18].
Definition 3 (see [19]). Let S be a self-map defined on K and w : K × K ⟶ ½0,∞Þ be a function. S is said to be an w -orbital admissible if for all v ∈ K , we have In our appointed theorems, if the continuity of the involved contractive mappings is removed, to handle this defect, it is necessary that ðK, dÞ be w-regular.
ðRÞ A space ðK, dÞ is defined as w-regular, if fv r g is a sequence in K such that wðv r , v r+1 Þ ≥ 1 for each r and v r ⟶ ω ∈ K as r ⟶ ∞, then wðv r , ωÞ ≥ 1 for all r.
Another most interesting Banach contraction principle generalization was given by Suzuki [25,26]. He introduced a weaker notion of contraction and discussed the existence of some new fixed point theorems. Besides the famous theorem, Suzuki generalized also the results of Nemytzki [27] and Edelstein [28] for compact metric spaces. One of the recently popular topics in fixed point theory is addressing the existence of fixed points of Suzuki type mappings. As with many generalizations of the famous Banach theorems, Suzuki type generalization can be said to have many applications, such as in computer science [29], game theory [30], and biosciences [31] and in other areas of mathematical sciences such as in dynamic programming, integral equations, data dependence, and homotopy [32,33]. Subsequently, Popescu [34] has modified the nonexpansiveness situation with the weaker C-condition presented by Suzuki. Accordingly, the existence of fixed points of maps satisfying the C-condition has been extensively studied (see [35][36][37][38]). Karapınar et al. [39] introduced the definition of a nonexpansive mapping satisfying the C-condition: Definition 4. A mapping S on a metric space ðK, dÞ satisfies the C -condition if for each v, t ∈ K:

Main Results
We start the section with the following essential definition: and a real number β ∈ ½0, 1Þ , such that for each v, t ∈ K \ FixðSÞ.
Theorem 6. Let ðK, dÞ be a complete metric space and S : K ⟶ K be an wψ -interpolative Kannan contraction of the Suzuki type. Suppose that S is an w -orbital admissible mapping and wðv 0 , Sv 0 Þ ≥ 1 for some v 0 ∈ K . Then, S has a fixed point in K provided that at least one of the following conditions holds: Let v 0 ∈ K such that wðv 0 , Sv 0 Þ ≥ 1 and fv r g be the sequence constructed by S r ðv 0 Þ = v r for each positive integer r: Assuming that for some r 0 ∈ ℕ, v r 0 = v r 0+1 , we get v r 0 = Sv r 0 , so v r 0 is a fixed point of S: Then, v r ≠ v r+1 for each positive integer r: Similarly, continuing this process, we have Thereupon, choosing v = v r−1 and t = Sv r−1 in (6), we get Journal of Function Spaces whence it follows that or equivalent Thus, on the one hand, it follows that the sequence fdðv r−1 , v r Þg is a nonincreasing sequence with positive terms, so there exists l ≥ 0 such that lim r⟶∞ dðv r−1 , v r Þ = l. On the other hand, combining (8) and (10) and keeping in mind that the function ψ is nondecreasing, we obtain Now, applying the triangle inequality and using (11), for all j ≥ 1, we get where P k = ∑ k m=0 ψ m ðdðv 0 , v 1 ÞÞ: But, ψ ∈ Ψ, the series ∑ ∞ m=0 ψ m ðdðv 0 , v 1 ÞÞ is convergent, so there exists a positive real number P such that lim k⟶∞ P k = P. Consequently, letting r, j ⟶ ∞ in the above inequality, we get Therefore, fv r g is a Cauchy sequence, and taking into account the completeness of the space ðK, dÞ, it follows that there exists ω ∈ K such that and we claim that this ω is a fixed point of S. In case that the assumption (a) holds, we have wðv r , ωÞ ≥ 1, and we claim that or for every r ∈ ℕ. Supposing on the account of the triangle inequality, we have which is a contradiction. Thereupon, for every r ∈ ℕ, either or holds. In the case that (19) holds, we obtain If the second condition, (20), holds, we have Therefore, letting r ⟶ ∞ in (21) and (22), we get that dðω, SωÞ = 0, that is, ω = Sω: In the case that the assumption (b) is true, that is, the mapping S is continuous, If the last assumption, (c), holds, as above, we have S 2 ω = lim r⟶∞ S 2 v r = lim r⟶∞ v r+2 = ω and we want to show that also Sω = ω. Supposing on the contrary, that ω ≠ Sω, since 3 Journal of Function Spaces by (6), we get which is a contradiction. Consequently, ω = Sω, that is, ω is a fixed point of the mapping S. ☐ Example 7. Let K = ½0, 3 and d : K × K ⟶ ½0,+∞Þ be the usual distance on ℝ . Consider the mapping S : K ⟶ K be defined as Let also w : We remark that the space is not regular since, for example, considering the sequence fv r g, with v r = ðr + 8Þ/ð2r + 4Þ we have v r ⟶ 1/2 as r ⟶ ∞, wðv r , v r+1 Þ = v 2 r + v 2 r+1 ≤ 1, but w ðv r , 1/2Þ = 0. On the other hand, the mapping S is not continuous, but since S 2 = 4/5, we have that S 2 is a continuous mapping. Let the function ψ ∈ Ψ defined as ψðzÞ = z/3 and we choose β = 1/9. Thus, we have to check that (6) holds. We have to consider the following cases: (1) For v, t ∈ ½0, 1, respectively, v, t ∈ ð1, 2Þ, we have d ðSv, StÞ = 0, so (6) holds (2) For v = 0 and t = 3 (3) All other cases are not interesting because wðv, tÞ = 0 Consequently, the assumptions of Theorem 6 being satisfied, it follows that the mapping S has a fixed point, which is v = 4/5.

Corollary 8.
Let ðK, dÞ be a complete metric space and S be a self-mapping on K , such that, for each v, t ∈ K \ FixðSÞ, where ψ ∈ Ψ and β ∈ ½0, 1Þ. Then, S possesses a fixed point in K.
Proof. Theorem 6 is sufficient to get wðv, tÞ = 1 for the proof. ☐ Moreover, taking ψðzÞ = zκ, with κ ∈ ½0, 1Þ in Corollary (8), we obtain the following consequence. Corollary 9. Let ðK, dÞ be a complete metric space and S be a self-mapping on K , such that for each v, t ∈ K \ FixðSÞ, where β ∈ ½0, 1Þ. Then, the mapping S possesses a fixed point in K.
Theorem 11. Let ðK, dÞ be a complete metric space and the mapping S : K ⟶ K be an wψ -interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Suppose that S is w -orbital admissible and wðv 0 , Sv 0 Þ ≥ 1 for some v 0 ∈ K . If ðK, dÞ is w -regular or either (1) S is continuous or (2) S 2 is continuous and wðSω, ωÞ ≤ 1 for any v ∈ FixðS 2 Þ, then the mapping S has a fixed point in K Proof. Let v 0 ∈ K satisfy wðv 0 , Sv 0 Þ ≥ 1 and fv r g be the sequence defined by S r ðv 0 Þ = v r for each positive integer r: If v r 0 = v r 0+1 for some r 0 ∈ ℕ, we get v r 0 = Sv r 0 , that means v r 0 is a fixed point of S: Then, we can assume that v r ≠ v r+1 for each positive integer r: Moreover, due to the assumption that S is w-orbital admissible, as in the previous proof, we 4 Journal of Function Spaces By letting v = v r−1 and t = Sv r−1 = v r in (31), we obtain then, using ψðzÞ < z for every z > 0.
or equivalent So, for every r ∈ ℕ. Therefore, the positive sequence fdðv r−1 , v r Þg is decreasing. Eventually, by (33), we have and by repeating this process, we find that We assert that fv r g is a fundamental sequence in ðK, dÞ. Thus, using the triangle inequality with (38), we can write Taking r ⟶ ∞ in (39), we deduce that fv r g is a fundamental sequence in ðK, dÞ, and using the completeness ðK, dÞ, there exists ω ∈ K such that We claim that the point ω is a fixed point of S: In the case of the space ðK, dÞ being w-regular and fv r g verifies (32), that is, wðv r , v r+1 Þ ≥ 1 for every r ∈ ℕ, we get wðv r , ωÞ ≥ 1. On the other hand, we know (see the proof of Theorem 6) that either or holds, for every r ∈ ℕ. If (41) is holds, we obtain Letting r ⟶ ∞ in the above inequality, we get that dðω, SωÞ = 0, that is, ω = Sω: If the second condition (42) is true, we get that ω is a fixed point S by a similar argument.

Journal of Function Spaces
Taking into account the definition of the function w, the only interesting situations are for v = 0, y = 1/2, respectively, v = 0, y = 1. For the first case, we have For the second case, Definition 13. Let ðK, dÞ be a metric space. The mapping S : K ⟶ K is called an ψ -interpolative Ćirić-Reich-Rus contraction of Suzuki type if there exist ψ ∈ Ψ and the constants β, α > 0 , with β + α < 1, such that for each v, t ∈ K \ FixðSÞ.

Theorem 14.
Let ðK, dÞ be a complete metric space and the mapping S : K ⟶ K be an ψ -interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Then, the mapping S has a fixed point in K.
Theorem 16. Let ðK, dÞ be a complete metric space and S : K ⟶ K be an interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Therefore, S has a fixed point in K.

Conclusions
In this manuscript, we introduce new concepts on completeness of w-ψ-interpolative Kannan contraction of Suzuki type and w-ψ-interpolative Ćirić-Reich-Rus contraction of Suzuki type mappings in metric space. We prove the existence of some fixed point theorems for mappings these concepts. Further, we obtain some fixed point results and give examples to show that the new results are applicable. Interpolation contraction, which is generalized from the Kannan type contraction, is a new and interesting contraction in fixed point theory, and different interpolation contractions of Suzuki type studies can be obtained by combining it with a Suzuki type contraction in the future. Additionally, these proposed contractions can be generalized in other well-known spaces and can give new fixed point results.

Data Availability
No data were used to support this study.