On Jaggi-Suzuki-Type Hybrid Contraction Mappings

The aim of the paper is to introduce the concept of new hybrid contractions that combine Jaggi hybrid type contractions and Suzuki type contractions with w-orbital admissible. We investigate the existence and uniqueness of such new hybrid contractions in theorems and results. Further, an illustrated example is given. With the results of this study, we generalize several well-known results in the recent fixed point literature.


Introduction and Preliminaries
In the last three to four decades, metric fixed point theory was one of the crucial and useful topics of functional analysis. It has attracted the attention of researchers from several distinct disciplines. Almost a century ago, Banach [1] initiated the metric fixed point theory with a magnificently simple but enormously useful result, known as the "Banach contraction principle." It is one of the useful theorems for solving differential integral equations to guarantee both the existence and the uniqueness of the solution. A wide variety of problems arising in different areas of pure and applied mathematics, such as differential equations, discrete and continuous dynamical systems, and nonlinear analysis, can be modeled as fixed point equations of the form u = Eu. Soon afterward, the Banach contraction principle was extended by Caccioppoli [2] to the setting of a complete metric space, which states that, in a complete metric space ðX, dÞ, any Banach contraction E : X ⟶ X is a Picard operator; we refer to it as the "Banach-Caccioppoli theorem." These main theorems have been generalized in different directions by many researchers. Among all, the authors observed a very general result in a general setting by using a simulation function. For more interesting results, see, e.g., [3,4].
Jaggi [5] demonstrated the following theorem satisfying a contractive condition of a rational type.
Theorem 1 (see [5]). Let E be a continuous self-map defined on complete metric space ðX, dÞ. Further, let E satisfy the following condition: for all distinct points u, z ∈ X and where α, β ∈ ½0, 1Þ with α + β < 1. Then, E has a unique fixed point in X.
Very recently in 2018, Karapinar [6] obtained a new interesting type of contraction generalized from the wellknown Kannan contraction by adopting an interpolative approach. In [7], a common fixed point of the interpolative Kannan contraction was considered.
Karapinar's definition and theorem are listed as follows.
Definition 2 (see [6]). Let ðX, dÞ be a metric space. A selfmapping E : X ⟶ X is said to be an interpolative Kannan-type contraction if there are two constants λ ∈ ½0, 1Þ and α ∈ ð0, 1Þ such that for all u, z ∈ X with u ≠ Eu.
Theorem 3 (see [6]). Let ðX, dÞ be a complete metric space and E : X ⟶ X be an interpolative Kannan-type contraction. Then, E has a unique fixed point in X.
Afterward, as a modification in the concept of α-admissible maps, Popescu [9] introduced w-orbital admissible maps.
Definition 5. Let w : X × X ⟶ ½0,∞Þ be a mapping and The following condition has often been considered in order to avoid the continuity of the involved contractive mappings.
ðRÞA space ðX, dÞ is called w-regular, if whenever fu m g is a sequence in X such that αðu m , u m+1 Þ ≥ 1 for all m and u m ⟶ u ∈ X as m ⟶ ∞, then there exists a subsequence fu mðkÞ g of fu m g such that wðu mðkÞ , uÞ ≥ 1 for all k.
In 2008, Suzuki [10] published one of the most comprehensive generalizations of Banach's and Edelstein's basic results. Suzuki contraction is when the contractive condition required to satisfy is not for all points of the domain. The existence and uniqueness of fixed points of maps satisfying a Suzuki type contraction has been extensively studied (see [11][12][13][14][15]). Later, Popescu [9] modified the nonexpansiveness situation with the weaker C-condition presented by Suzuki. The C-condition is defined as follows.

Definition 6.
A mapping E on a metric space ðX, dÞ satisfies the C-condition if for each u, z ∈ X.
Recently, Mitrovic et al. [16] used interpolation contraction and Reich contraction together and combined these two contractions in b-metric spaces. The combination of these two types of contractions has been called the new hybrid type contraction. In the last years, inspired by the result in [6], Karapinar and Fulga [17] introduced a new hybridtype contraction that combines Jaggi type contractions and interpolative-type contractions in the framework of complete metric spaces. In this paper, we investigate the Jaggi-Suzuki type hybrid contraction, inspired by the new contraction in [17], which is a combination of Jaggi hybrid type contractions and Suzuki type contractions with w-orbital admissible in the framework of complete metric spaces. We introduce the existence and uniqueness of a fixed point for this contraction.
Here are our main definition and theorem.
Definition 7. Let ðX, dÞ be a metric space. We say that the mapping E : X ⟶ X is a Jaggi-Suzuki type hybrid contraction if there exist ψ ∈ Ψ and w : X × X ⟶ ½0,∞Þ such that for each u, z ∈ X where s ≥ 0 and ρ i ≥ 0, i = 1, 2, such that ρ 1 + ρ 2 = 1 with ρ 1 < 1/2, Theorem 8. Let ðX, dÞ be a complete metric space and E : X ⟶ X be a Jaggi-Suzuki-type hybrid contraction. Assume also that E is w-orbital admissible mapping and wðu 0 , Eu 0 Þ ≥ 1 for some u 0 ∈ X. Then, E has a fixed point in X provided that at least one of the following conditions holds: If u m 0 = u m 0+1 for some integer m 0 , then u m 0 is a fixed point of E, so we shall assume that u m ≠ u m+1 for all positive integer m: Since E is w-orbital admissible, so wðu 0 , Eu 0 Þ = wðu 0 , u 1 Þ ≥ 1 implies that wðu 1 , Eu 1 Þ = wðu 1 , u 2 Þ ≥ 1. Continuing this argument, we get  (6), we get where Hereafter, we find that Suppose that dðu m , u m+1 Þ ≥ dðu m−1 , u m Þ, so, using (11), we write which is a contradiction. Then, we get Eventually, from (11), we have and by repeating this process, we find that for any m ∈ ℕ. We claim that fu m g is a Cauchy sequence in ðX, dÞ. Then, using the triangle inequality with (15), we can write where S t = ∑ t k=0 ψ k ðdðu 0 , u 1 ÞÞ: But, ψ ∈ Ψ, so the series ∑ ∞ k=0 ψ k ðdðu 0 , u 1 ÞÞ is convergent, then there exists a positive real number S such that lim t⟶∞ S t = S. As a result, letting m, l ⟶ ∞ in the above inequality, we obtain Thus, fu m g is a Cauccess of the space ðX, dÞ; it follows that there exists v ∈ X such that We assert that v is a fixed point of E.
If the assumption ðj 1 Þ holds, we get wðu m , vÞ ≥ 1, and we assert that for all m ∈ ℕ. Since, if we suppose that then, taking into account the triangle inequality and the fact that the sequence fdðu m , u m+1 Þg is decreasing, we obtain that holds. In case that (22) holds, we get If the second condition, (23), holds, we can write Hence, taking m ⟶ ∞ in (24) and (25), which is a contraction. Therefore, we get that dðv, EvÞ = 0, that is, v = Ev: If the assumption ðj 2 Þ is true, that is, the mapping E is continuous, we obtain In case that last assumption ðj 3 Þ holds, from above, we get E 2 v = lim m⟶∞ E 2 u m = lim m⟶∞ u m+2 = v and we aim to prove that also Ev = v. Assuming on the contrary that v ≠ E v, from using (6), we obtain that a contradiction. As a result, v = Ev, that is, v is a fixed point of the mapping E.  (6), we obtain Due to the above inequality, we obtain and from ρ 1 + ρ 2 = 1, we attain that dðu m , u m+1 Þ < dðu m−1 , u m Þ for every m ∈ ℕ. Using (31), we get and using previous reasoning By using the same methods as in the case of s > 0, we obviously obtain that fu m g forms a Cauchy sequence in complete metric space. Subsequently, there exists v ∈ X such that lim m⟶∞ dðu m , vÞ = 0, and we claim that this v is a fixed point of E. In case the space ðX, dÞ is w-regular and fu m g verifies (8), that is, wðu m , u m+1 Þ ≥ 1 for every m ∈ ℕ, we get wðu m , vÞ ≥ 1. On the other hand, we know (see the proof of case 1) that either using (6), we find that a contradiction. As a result, v = Ev, that is, v is a fixed point of the mapping E. Theorem 9. Besides the hypothesis of Theorem 8, if we suppose that wðv, v * Þ ≥ 1 for any v, v * ∈ Fix E ðXÞ: Thereupon, E has a unique fixed point in X.
Proof. We shall show that v is a unique fixed point of E. Assume that another v * is fixed point of E, that is, Ev * = v * with v ≠ v * : In the case that s > 0, hence, from (6), we have Then, which is a contradiction. In the case that s = 0, thus, from (6), we get that a contradiction. Consequently, v = v * , that is, E has a unique fixed point in X.☐ Example 10. Let X = ½0, 2 and d : X × X ⟶ ½0,+∞Þ be the usual metric on ℝ and the mapping E : X ⟶ X be determined as Make allowances for the function ψ ∈ Ψ with ψðtÞ = t/3 and w : X × X ⟶ ½0,∞Þ, such that w u, z ð Þ= 3, if u, z ∈ 0, 1 ½ , 1, if u = 0, z = 2, 0, otherwise: The mapping E is not continuous, but E 2 is a continuous mapping. We choose ρ 1 = 1/5, ρ 2 = 4/5, and s = 2, then we have We obtain that the following cases.