# On Jaggi-Suzuki-Type Hybrid Contraction Mappings

Authors: Maha Noorwali; Seher Sultan Yeşilkaya
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of the paper is to introduce the concept of new hybrid contractions that combine Jaggi hybrid type contractions and Suzuki type contractions withw-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.


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## Body

## 1. 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 X,d, 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]).
LetE be a continuous self-map defined on complete metric space X,d. Further, let E satisfy the following condition:
(1)dEu,Ez≤αdu,Eudz,Ezdu,z+βdu,z,for all distinct points u,z∈X and where α,β∈0,1 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 well-known 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]).
LetX,d be a metric space. A self-mapping E:X⟶X is said to be an interpolative Kannan-type contraction if there are two constants λ∈0,1 and α∈0,1 such that
(2)dEu,Ez≤λdu,Euα·dz,Ez1−α,for all u,z∈X with u≠Eu.Theorem 3 (see [6]).
LetX,d be a complete metric space and E:X⟶X be an interpolative Kannan-type contraction. Then, E has a unique fixed point in X.Recall the notion ofα-admissibility introduced by Samet et al. [8].Definition 4.
A mappingE:X⟶X is called α-admissible if, for all u,z∈X, we have
(3)αu,z≥1⇒αEu,Ez≥1,where α:X×X⟶0,∞ is a given function.Afterward, as a modification in the concept ofα-admissible maps, Popescu [9] introduced w-orbital admissible maps.Definition 5.
Letw:X×X⟶0,∞ be a mapping and X≠∅. A self-mapping E:X⟶X is called w-orbital admissible if, for all u∈X, we have
(4)wu,Eu≥1⇒wEu,E2u≥1.The following condition has often been considered in order to avoid the continuity of the involved contractive mappings.RA space X,d is called w-regular, if whenever um is a sequence in X such that αum,um+1≥1 for all m and um⟶u∈X as m⟶∞, then there exists a subsequence umk of um such that wumk,u≥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 [11–15]). 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 mappingE on a metric space X,d satisfies the C-condition if
(5)12du,Eu≤du,z⇒dEu,Ez≤du,z,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 hybrid-type 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.


## 2. Main Results

We start by recalling the definition of the familyΨ as the set of all nondecreasing self-mappings ψ on 0,∞ such that ∑m=1∞ψmν<∞ for every ν>0. Notice that for ψ∈Ψ, we have ψ0=0 and ψν<ν for all ν>0 (see [18]).Here are our main definition and theorem.Definition 7.
LetX,d 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⟶0,∞ such that
(6)12du,Eu≤du,z⇒wu,zdEu,Ez≤ψμEsu,z,
for eachu,z∈X where s≥0 and ρi≥0, i=1,2, such that ρ1+ρ2=1 with ρ1<1/2,
(7)μEsu,z=ρ1du,Eudz,Ezdu,zs+ρ2du,zs1/s,fors>0,u≠z,du,Euρ1dz,Ezρ2,fors=0,u,z∈X\FixE.Theorem 8.
LetX,d 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 wu0,Eu0≥1 for some u0∈X. Then, E has a fixed point in X provided that at least one of the following conditions holds:
(j1) X,d is w-regular
(j2) E is continuous
(j3) E2 is continuous and wu,Eu≥1 where u∈FixE2Proof.
Let the sequenceum in X be constructed by Emu0=um for all positive integer m, where u0∈X such that wu0,Eu0≥1.
Ifum0=um0+1 for some integer m0, then um0 is a fixed point of E, so we shall assume that um≠um+1 for all positive integer m. Since E is w-orbital admissible, so wu0,Eu0=wu0,u1≥1 implies that wu1,Eu1=wu1,u2≥1. Continuing this argument, we get
(8)wum,um+1≥1.☐Case 1.
s>0, by taking μEsu,z choosing u=um−1 and z=Eum−1=um in (6), we get
(9)12dum−1,Eum−1=12dum−1,um≤dum−1,um⇒wum−1,umdEum−1,Eum≤ψμEsum−1,um, where
(10)μEsum−1,Eum−1=ρ1dum−1,Eum−1dEum−1,E2um−1dum−1,Eum−1s+ρ2dum−1,Eum−1s1/s=ρ1dum−1,umdum,um+1dum−1,ums+ρ2dum−1,ums1/s=ρ1dum,um+1s+ρ2dum−1,ums1/s.Hereafter, we find that(11)dum,um+1≤wum−1,umdEum−1,Eum≤ψμEsum−1,um=ψρ1dum,um+1s+ρ2dum−1,ums1/s.Suppose thatdum,um+1≥dum−1,um, so, using (11), we write
(12)dum,um+1≤ψρ1dum,um+1s+ρ2dum−1,ums1s≤ψρ1dum,um+1s+ρ2dum,um+1s1s≤ψρ1+ρ2dum,um+1s1s=ψdum,um+1s1s=ψdum,um+1<dum,um+1,which is a contradiction. Then, we get
(13)dum,um+1<dum−1,um.Eventually, from (11), we have
(14)dum,um+1≤ψdum−1,um<dum−1,um, and by repeating this process, we find that
(15)dum,um+1≤ψmdu0,u1,for anym∈ℕ. We claim that um is a Cauchy sequence in X,d. Then, using the triangle inequality with (15), we can write
(16)dum,um+l≤dum,um+1+dum+1,um+2+⋯+dum+l−1,um+l≤ψmdu0,u1+ψm+1du0,u1+⋯+ψm+l−1du0,u1=∑k=mm+l−1ψkdu0,u1=Sm+l−1−Sm−1,whereSt=∑k=0tψkdu0,u1. But, ψ∈Ψ, so the series ∑k=0∞ψkdu0,u1 is convergent, then there exists a positive real number S such that limt⟶∞St=S. As a result, letting m,l⟶∞ in the above inequality, we obtain
(17)dum,um+l⟶0.Thus,um is a Cauccess of the space X,d; it follows that there exists v∈X such that
(18)limm⟶∞dum,v=0.We assert thatv is a fixed point of E.If the assumptionj1 holds, we get wum,v≥1, and we assert that
(19)either12dum,Eum≤dum,vor12dEum,EEum≤dEum,v,for allm∈ℕ. Since, if we suppose that
(20)12dum,Eum>dum,v,12dEum,EEum>dEum,v,then, taking into account the triangle inequality and the fact that the sequencedum,um+1 is decreasing, we obtain that
(21)dum,um+1=dum,Eum≤dum,v+dv,Eum<12dum,Eum+12dEum,EEum=12dum,um+1+12dum+1,um+2≤12dum,um+1+12dum,um+1=dum,um+1,a contradiction. Thus, for allm∈ℕ, either
(22)12dum,Eum≤dum,v,or
(23)12dEum,EEum≤dEum,v,holds. In case that (22) holds, we get
(24)dum+1,Ev≤wum,vdEum,Ev≤ψρ1dum,Eumdv,Evdum,vs+ρ2dum,vs1/s<ρ1dum,Eumdv,Evdum,vs+ρ2dum,vs1/s=ρ1dum,um+1dv,Evdum,vs+ρ2dum,vs1/s≤ρ12dum,vdv,Evdum,vs+ρ2dum,vs1/s.If the second condition, (23), holds, we can write
(25)dum+2,Ev≤wum+1,vdE2um,Ev≤ψρ1dEum,E2umdv,EvdEum,vs+ρ2dEum,vs1/s<ρ1dEum,E2umdv,EvdEum,vs+ρ2dEum,vs1/s=ρ1dum+1,um+2dv,Evdum+1,vs+ρ2dum+1,vs1/s≤ρ12dum+1,vdv,Evdum+1,vs+ρ2dum+1,vs1/s.Hence, takingm⟶∞ in (24) and (25),
(26)dv,Ev<2ρ11/sdv,Ev≤dv,Ev,which is a contraction. Therefore, we get that dv,Ev=0, that is, v=Ev.If the assumptionj2 is true, that is, the mapping E is continuous, we obtain
(27)Ev=limm⟶∞Eum=limm⟶∞um+1=v.In case that last assumptionj3 holds, from above, we get E2v=limm⟶∞E2um=limm⟶∞um+2=v and we aim to prove that also Ev=v. Assuming on the contrary that v≠Ev, from
(28)12dEv,E2v=12dEv,v≤dEv,v,using (6), we obtain that
(29)dv,Ev=dE2v,Ev≤wEv,vdE2v,Ev≤ψρ1dEv,E2vdv,EvdEv,vs+ρ2dEv,vs1/s<ρ1dEv,E2vdv,EvdEv,vs+ρ2dEv,vs1/s=dv,Ev,a contradiction. As a result,v=Ev, that is, v is a fixed point of the mapping E.Case 2.
s=0, by taking μEsu,z choosing u=um−1 and z=Eum−1=um in (6), we obtain
(30)12dum−1,Eum−1=12dum−1,um≤dum−1,um⇒(31)dum,um+1≤wum−1,umdEum−1,Eum≤ψμEsum−1,zm=ψdum−1,Eum−1ρ1·dum,Eumρ2<dum−1,umρ1·dum,um+1ρ2.Due to the above inequality, we obtain(32)dum,um+11−ρ2<dum−1,umρ1,and from ρ1+ρ2=1, we attain that dum,um+1<dum−1,um for every m∈ℕ. Using (31), we get
(33)dum,um+1≤ψdum−1,um,and using previous reasoning
(34)dum,um+1≤ψmdu0,u1.By using the same methods as in the case ofs>0, we obviously obtain that um forms a Cauchy sequence in complete metric space. Subsequently, there exists v∈X such that limm⟶∞dum,v=0, and we claim that this v is a fixed point of E. In case the space X,d is w-regular and um verifies (8), that is, wum,um+1≥1 for every m∈ℕ, we get wum,v≥1. On the other hand, we know (see the proof of case 1) that either
(35)12dum,Eum≤dum,v,or
(36)12dEum,EEum≤dEum,v,holds, for allm∈ℕ. If (35) is taken, we get
(37)dum+1,Ev≤wum,vdEum,Ev≤ψdum,Eumρ1·dv,Evρ2,=ψdum,um+1ρ1·dv,Evρ2,<dum,um+1ρ1·dv,Evρ2.If the second condition, (36), is taken, we get
(38)dum+2,Ev≤wum+1,vdE2um,Ev≤ψdEum,E2umρ1·dv,Evρ2,=ψdum+1,um+2ρ1·dv,Evρ2,<dum+1,um+2ρ1·dv,Evρ2.Hence, takingm⟶∞ in the last two inequalities, we find that dv,Ev=0, that is, v=Ev. If the mapping E is continuous, we obtain v=Ev that we know (see the proof of case 1).If the last assumption (j3) holds, as above, we get E2v=limm⟶∞E2um=limm⟶∞um+2=v, and we aim to prove that Ev=v. Assuming, on the contrary, that v≠Ev, from
(39)12dEv,E2v=12dEv,v≤dEv,v,using (6), we find that
(40)dv,Ev=dE2v,Ev≤wEv,vdE2v,Ev≤ψdEv,E2vρ1·dv,Evρ2<dEv,vρ1·dv,Evρ2=dEv,v,a contradiction. As a result,v=Ev, that is, v is a fixed point of the mapping E.Theorem 9.
Besides the hypothesis of Theorem8, if we suppose that wv,v∗≥1 for any v,v∗∈FixEX. Thereupon, E has a unique fixed point in X.Proof.
We shall show thatv 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
(41)12dv,Ev≤dv,v∗implies,dv,v∗≤wv,v∗dEv,Ev∗≤ψμEsv,v∗<μEsv,v∗=ρ1dv,Evdv∗,Ev∗dv,v∗s+ρ2dv,v∗s1/s.
Then,(42)dv,v∗<ρ21/sdv,v∗≤dv,v∗,which is a contradiction. In the case that s=0, thus, from (6), we get that
(43)0<dv,v∗<0,
a contradiction. Consequently,v=v∗, that is, E has a unique fixed point in X.☐Example 10.
LetX=0,2 and d:X×X⟶0,+∞ be the usual metric on ℝ and the mapping E:X⟶X be determined as
(44)Eu=13,ifu∈0,1,u7,ifu∈1,2.Make allowances for the functionψ∈Ψ with ψt=t/3 and w:X×X⟶0,∞, such that
(45)wu,z=3,ifu,z∈0,1,1,ifu=0,z=2,0,otherwise.The mappingE is not continuous, but E2 is a continuous mapping. We choose ρ1=1/5, ρ2=4/5, and s=2, then we have
(46)μE2u,z=12du,Eudz,Ezdu,z2+12du,z21/2.We obtain that the following cases.Case 1.
Foru,z∈0,1, we get dEu,Ez=0, then (6) holds.Case 2.
Foru=0 and z=2,
(47)12d0,E0=16<2=d0,2⇒,w0,2dE0,E2=0.04761904761≤0.59780399339=1315272+4522=ψρ1du,Eudz,Ezdu,z2+ρ2du,z2.Other cases are provided, sincewu,z=0. Eventually, the presumptions of Theorem 8, being satisfied, and u=1/3 is the fixed point of E.Corollary 11.
LetX,d be a complete metric space and E:X⟶X a continuous mapping. If there exist ψ∈Ψ such that
(48)12du,Eu≤du,zimpliesdEu,Ez≤ψμEsu,z,for eachu,z∈X where s≥0; ρi≥0,i=1,2; and ρ1+ρ2=1 with ρ1<1/2 and α∈0,1(49)μEsu,z=ρ1du,Eudz,Ezdu,zs+ρ2du,zs1/s,fors>0,u≠z,du,Euαdz,Ez1−α,fors=0,u,z∈X\FixE,therefore, E has a fixed point in X.Proof.
Case 1: fors>0, it follows from Theorem 8, letting wu,z=1 for u,z∈X.
Case 2: fors=0, it follows from Theorem 8, letting wu,z=1 for u,z∈X and ρ1=α, ρ2=1−α.☐Corollary 12.
LetX,d be a complete metric space and E:X⟶X a continuous mapping. If there exist λ∈0,1 such that
(50)12du,Eu≤du,zimpliesdEu,Ez≤λμEsu,z,for eachu,z∈X where s≥0; ρi≥0,i=1,2; and ρ1+ρ2=1 with ρ1<1/2 and α∈0,1(51)μEsu,z=ρ1du,Eudz,Ezdu,zs+ρ2du,zs1/s,fors>0,u≠z,du,Euαdz,Ez1−α,fors=0,u,z∈X\FixE,therefore, E has a fixed point in X.Proof.
Case 1: fors>0, it follows from Corollary 11, letting ψν=λν for any ν≥0.
Case 2: fors=0, it follows from Corollary 11, letting ψν=λν for any ν≥0 and ρ1=α, ρ2=1−α.☐Corollary 13.
LetX,d be a complete metric space and E:X⟶X a continuous mapping. If there exist λ∈0,1 such that
(52)12du,Eu≤du,zimpliesdEu,Ez≤λdu,Eudz,Ez,for eachu,z∈X\FixE; therefore, E has a fixed point in X.Proof.
Fors=0, it follows from Corollary 12, letting α=1/2.☐Corollary 14.
LetX,d be a complete metric space. A continuous mapping E:X⟶X such that
(53)12du,Eu≤du,z⇒dEu,Ez≤αdu,Eudz,Ezdu,z+βdu,z,u≠z,for each u,z∈X, where α,β∈0,1, α+β<1. Thus, E has a fixed point in X.Proof.
Fors=1, it follows from Corollary 12, letting α=λρ1 and β=λρ2 with λ∈0,1 for u,z∈X.



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