# Discussion on Geraghty Type Hybrid Contractions

Authors: Sara Salem Alzaid; Andreea Fulga; Fehaid Salem Alshammari
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this manuscript, we define the notion of Geraghty type hybrid contractions in the setting ofb-metric spaces. We prove the existence of a fixed point for such mappings whenever b-metric space is complete. Our observed results not only unify several existing results but also extend some known results.


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## Body

## 1. Introduction and Preliminaries

The distance notion is one of the ancient and most basic concepts in the history of mathematics. In modern mathematics history, this notion was formally formulated by Frechét as “L-space.” Later, it was redefined as “metric space” by Hausdorff. After then, this concept has been extended and generalized in several ways. From all these generalizations of metric notions, the b-metric is the most interesting.In order to introduce the subject clearly, we first fix the basic concepts and notations. A functionδ, defined from X×X (where X is a nonempty set) to nonnegative reals, is said to be a distance function, if it is symmetrical, that is δu,ν=δν,u, for every u,ν∈X and δu,ν=0 if and only if u=ν.Moreover, a distance functionδ is a (standard) metric in case that
(1)δu,v≤δu,ν+δν,v,forallu,v,ν∈X.As we mentioned above, the distance notion, as well as the notion of the metric, has been extended and generalized in several directions. One of the outstanding generalizations of metric notion is namedb-metric. Indeed, the concept of b-metric was considered by distinct authors, in various periods of the time, involving Bakhtin [1] and Czerwik [2]. Later, many researchers were interested in this topic, and thus, a series of interesting results were obtained, see, e.g., [3–19].Definition 1.
A distance functionb on X is said to be a b-metric over constant s≥1 if the inequality (weighted triangle inequality)
(2)bu,v≤sbu,ν+bν,v,holds forallu,v,ν∈X.In what follows, we consider thatX,b,s denotes a b-metric space.An immediate observation is that the notion ofb-metric is more general than the concept of metric; for instance, when s=1, we recover the notion of metric space. Moreover, we mention that a b-metric is not necessarily continuous, see, e.g., [20, 21].Example 2.
The functionb on X=0,∞, where bu,ν=u−νq,q>1, is a b-metric over s=2q, but not a metric.Definition 3.
On ab-metric space X,b,s, let un be a sequence in X.
(a)
The sequenceun is convergent in X,b,s to u, if for every e>0, there exists n0∈ℕ such that bun,u<e for all n>n0. (We denote by un→u as n→∞ or limn→∞un=u.)(b)
The sequenceun is Cauchy, if for every e>0, there exists n0∈ℕ such that bun,un+l<e for all n>n0,l>0(c)
If every Cauchy sequence inX converges to a point u∈X, then the triplet X,b,s is said to be completeIn short,X∗,b,s denotes a complete b-metric space over s.Recently, Mitrovic et al. [22] introduced the following type of contractions.Definition 4 (see [22]).
LetX,b,s and T:X→X be a self-mapping. We say that T is a r,a-weight type contraction, if there exists κ∈0,1 such that
(3)bTu,Tν≤κ·MrT,u,ν,a,where r≥0, a=a1,a2,a3,ai≥0,i=1,2,3 such that a1+a2+a3=1 and
(4)MrT,u,ν,a=a1bu,νr+a2bu,Tur+a3bν,Tνr1/r,r>0bu,νa1bu,Tua2bν,Tνa3,r=0,for all u,ν∈X\FixT, where FixT=ϖ∈X,Tϖ=ϖ.In 1973, Geraghty [23] introduced a class of auxiliary functions to refine the Banach contraction principle. Let G be the set defined as
(5)G=βb:0,∞→0,1∣limn→∞βbtn=1implieslimn→∞tn=0.Theorem 5 (see Geraghty [23]).
On a complete metric spaceX,d, a mapping T:X→X admits a unique fixed point provided that there exists a function β∈G such that
(6)dTu,Tν≤βdu,νdu,ν,foranyu,ν∈X.


## 2. Main Results

Let the setGb=βb:0,∞→0,1/s∣limsupn→∞βbtn=1/simplieslimn→∞tn=0..Definition 6.
OnX,b,s, a mapping T:X→X is called Geraghty type hybrid contraction, if there exists βb∈Gb such that
(7)bTu,Tν≤βbMrT,u,ν,aMrT,u,ν,a,where r≥0, a=a1,a2,a3∈0,∞×0,∞×0,∞, with a1<1, a1+a2+a3=1 and
(8)MrT,u,v,a=a1bu,vr+a2bu,Tur+a3bv,Tvr1/r,r>0,u,v∈Xbu,va1bu,Tua2bv,Tva3,r=0,u,v∈X\FixT.Theorem 7.
OnX∗,b,s, a Geraghty type hybrid contraction T:X→X admits a unique fixed point ϖ∈X if one of the following hypotheses is satisfied:
(i)
T is continuous at ϖ(ii)
a2<1(iii)
a3<1
Moreover, for anyu0∈X the sequence Tnu0 converges to ϖ.Proof.
We take an arbitrary pointu0∈X. Starting from this initial point, we shall construct a recursive sequence un with the following formula:
(9)un+1=Tunforalln≥0.
It is evident that if there existsk0 such that uk0=uk0+1, then uk0 becomes a fixed point of T. Therefore, from now, we assume that
(10)un≠un+1foralln≥0.We shall discuss all possible situations.Case 8.
Suppose thatr>0. From (7), we have that
(11)bTun,Tun−1≤βbMrT,un,un−1,aMrT,un,un−1,a,where
(12)MrT,un,un−1,a=a1bun,un−1r+a2bun,un+1r+a3bun−1,unr1/r.It yields(13)bun+1,un≤βba1+a3bun,un−1r+a2bun,un+1r1/r·a1+a3bun,un−1r+a2bun,un+1r1/rwhich is equivalent to
(14)bun+1,una1+a3bun,un−1r+a2bun,un+1r1/r≤βba1+a3bun,un−1r+a2bun,un+1r1/r<1s.Therefore, we have(15)bun+1,un<a1+a3sr−a21/rbun,un−1≤bun,un−1.It yieldsbun+1,un is nonincreasing sequence bounded by 0. Thus, the sequence bun+1,un converges to a nonnegative real number, say L. We assert that L is 0. On the one hand, by taking the limsup of all sides of (13), we deduce that
(16)L≤limsupβbn→∞a1+a3bun,un−1r+a2bun,un+1r1/rL<1sL.Suppose on the contrary, thatL>0, we obtain
(17)1s≤1≤limsupn→∞βba1+a3bun,un−1r+a2bun,un+1r1/r<1s.Thus, the limitlimn→∞a1+a3bun,un−1r+a2bun,un+1r1/r=0. Consequently, L=0.We assert that the sequenceun is b-Cauchy.On contrary, supposing that the sequenceun is not b-Cauchy, we can find e>0 and two sequences of positive integers qi and pi, pi>qi≥i such that
(18)buqi,upi≥eandbuqi,upi−1<e,(19)e≤liminfi→∞buqi,upi≤limsupi→∞buqi,upi≤sees≤liminfi→∞buqi+1,upi≤limsupi→∞buqi+1,upi≤s2ees≤liminfi→∞buqi,upi+1≤limsupi→∞buqi,upi+1≤s2ees2≤liminfi→∞buqi+1,upi+1≤limsupi→∞buqi+1,upi+1≤s3e.On the other hand,(20)es≤buqi+1,upi=bTuqi,Tupi−1≤βbMrT,uqi,upi−1,aMrT,uqi,upi−1,a<1sMrT,uqi,upi−1,awhere
(21)MrT,uqi,upi−1,a=a1buqi,upi−1r+a2buqi,Tuqir+a3bupi−1,Tupi−1r1/r=a1buqi,upi−1r+a2buqi,uqi+1r+a3bupi−1,upir1/rTakinglimsup of (21), we find
(22)limsupi→∞MrT,uqi,upi−1,a≤a11/re<e.If we combine the observed inequalities above, in particular, (20) and (22), we have
(23)es≤limsupi→∞buqi+1,upi≤limsupi→∞βbMrT,uqi,upi−1,aMrT,uqi,upi−1,a<elimsupi→∞βbMrT,uqi,upi−1,a<es,since a1<1. It implies that
(24)1s≤limsupi→∞βbMrT,uqi,upi−1,a≤1s.Sinceβb∈Gb, we conclude limi→∞MrT,uqi,upi−1,a=0. Attendantly,
(25)limi→∞nuqi,upi−1=0.Under these observations, by employing the weighted triangle axiom together with (18), we get
(26)e≤buqi,upi≤sbuqi,upi−1+bupi−1,upi→0asi→∞.Therefore,un is a b-Cauchy sequence in X∗,b,s, so we can find a point ϖ∈X such that
(27)limn→∞un=ϖ.We assert now that this point,ϖ is a fixed point of T.
(i)
Assuming that the mappingT is continuous at ϖ∈X, since limn→∞bϖ,un+1=0, we have(28)limn→∞bTun,Tϖ=bTϖ,Tϖ=0and from (49), we get bϖ,Tϖ=0, i.e., Tϖ=ϖ.For the other cases, we consider the inequality,(29)bϖ,Tϖ≤sbϖ,un+1+bTun,Tϖ,for any n∈ℕ.
(ii)
Suppose thata2<1. If Tϖ≠ϖ, we have(30)0<bTϖ,ϖ≤sbTϖ,un+1+bun+1,ϖ=sbTϖ,Tun+bun+1,ϖ≤sβbMrT,ϖ,un,aMrT,ϖ,un,a+bun+1,ϖ<MrT,ϖ,un,a+sbun+1,ϖ=a1bϖ,unr+a2bϖ,Tϖr+a3bun,Tunr1/r+sbun+1,ϖand when n→∞, we get
(31)0<bTϖ,ϖ≤a21/rbTϖ,ϖ.Sincea2<1, we get a contradiction, that is, Tϖ=ϖ.
(iii)
Suppose thata3<1. Assuming that bTϖ,ϖ>0, we have(32)0<bϖ,Tϖ≤sbϖ,un+1+bun+1,Tϖ=sbϖ,un+1+bTun,Tϖ≤sbϖ,un+1+βbMrT,un,ϖ,aMrT,un,ϖ,a<sbϖ,un+1+MrT,un,ϖ,a≤sbϖ,un+1+a1bun,ϖr+a2bun,un+1r+a3bϖ,Tϖr1/r.Takingn→∞, we have
(33)0<bTϖ,ϖ≤a31/rbTϖ,ϖ,which is a contradiction, since a3<1. Therefore, Tϖ=ϖ.Case 9.
r=0. Here, (7) and (8) become
(34)bTu,Tν≤βbIT,u,ν,aIT,u,ν,a,for every u,ν∈X, where
(35)IT,u,ν,a≔bu,νa1bu,Tua2bν,Tν1−a1−a2κ∈0,1 and a1,a2∈0,1.As in the proof of the caser>0, we shall consider a recursive sequence un=Tun−1, starting with an arbitrary point u∈X where u0=u. By using the same argument of this part of the proof, we presume that
(36)un≠un+1foralln≥0.Employingu=un−1 and ν=un in (34), we find that
(37)bun,un+1=bTun−1,Tun≤βbbun−1,una1bun−1,Tun−1a2bun,Tun1−a1−a2·bun−1,una1bun−1,Tun−1a2bun,Tun1−a1−a2=βbbun−1,una1+a2bun,un+11−a1−a2bun−1,una1+a2bun,un+11−a1−a2<1sbun−1,una1+a2bun,Tun1−a1−a2≤bun−1,una1+a2bun,Tun1−a1−a2It yields that(38)bun,un+1a1+a2≤bun−1,una1+a2iffbun,un+1≤bun−1,un,for each n∈ℕ. Attendantly, we deduce that the sequence of nonnegative numbers bun−1,un is a nonincreasing sequence. Ergo, there is a real number L≥0 such that limn→∞bun−1,un=L.As in the previous case, we assert thatL=0. Supposing on the contrary, that L>0, by taking limsup in (37), we derive that
(39)L≤limsupn→∞βbbun−1,una1bun−1,Tun−1a2bun,Tun1−a1−a2L.SinceL>0, we obtain
(40)1s≤1≤limsupn→∞βbbun−1,una1bun−1,una2bun,un+11−a1−a21s.Thus,limn→∞bun−1,una1+a2bun,un+11−a1−a2=0 and consequently, L=0.We claim thatun is a b-Cauchy sequence. On contrary, if we suppose that un is not a b-Cauchy sequence, that is, we can find e>0 and the sequences qi, pi of positive integers with pi>qi≥i such that
(41)buqi,upi≥eandbuqi,upi−1<e.By the weighted triangle inequality, we have(42)e≤buqi,upi≤sbuqi,uqi+1+buqi+1,upi.Since(43)buqi+1,upi=bTuqi,Tupi−1≤βbIT,uqi,upi−1,aIT,uqi,upi−1,a<1sIT,uqi,upi−1,awhere
(44)IT,uqi,upi−1,a=buqi,upi−1a1buqi,Tuqia2bupi−1,Tupi−11−a1−a2=buqi,upi−1a1buqi,uqi+1a2bupi−1,upi1−a1−a2,taking limsup of (43), we find
(45)limsupi→∞IT,uqi,upi−1,a=0andhencelimsupi→∞buqi+1,upi=0.If we combine the observed inequalities above, in particular, (42) and (45), we get that we get
(46)e≤buqi,upi≤sbuqi,uqi+1+buqi+1,upi→0,asi→∞.Therefore, the sequenceun is b-Cauchy in X∗,b,s, so it is convergent at a point ϖ∈X, that is
(47)limn→∞un=ϖ.Now, we assert thatϖ is a fixed point of T.If the assumption (i) holds, sincelimn→∞bϖ,un+1=0 we get
(48)limn→∞bTun,Tϖ=bTϖ,Tϖ=0and from the inequality
(49)bϖ,Tϖ≤sbϖ,un+1+bTun,Tϖ,for any n∈ℕ, we obtain bϖ,Tϖ=0, i.e., Tϖ=ϖ.Suppose thata2<1 or a3<1. Assuming that Tϖ≠ϖ, we have
(50)0<bTϖ,ϖ≤sbTϖ,un+1+bun+1,ϖ=sbTϖ,Tun+bun+1,ϖ≤sβbIT,ϖ,un,aIT,ϖ,un,a+bun+1,ϖ<IT,ϖ,un,a+sbun+1,ϖ=bϖ,una1bϖ,Tϖa2bun,Tun1−a1−a2+sbun+1,ϖ.At the limit asn→∞, we have bTϖ,ϖ=0. So, Tϖ=ϖ.Example 10.
We shall derive several distinct contractions from Definition6. Some examples are given below. Let T be a self-mapping on X.
(1)
Ifr=2,a=1/3,1/3,1/3, we obtain the following condition(51)bTu,Tν≤βb13b2u,ν+b2u,Tu+b2ν,Tν1/213b2u,ν+b2u,Tu+b2ν,Tν1/2(2)
Ifr=1,a=1/3,1/3,1/3, we obtain the following condition(52)bTu,Tν≤βb13bu,ν+bu,Tu+bν,Tν13bu,ν+bu,Tu+bν,Tν(3)
Ifr=0,a=0,a1,1−a1 with a1∈0,1, we obtain(53)bTu,Tν≤βbbu,Tua1bν,Tν1−a1bu,Tua1bν,Tν1−a1which means that T is an interpolative Kannan type Geraghty-contraction;
(4)
Ifr=0,a=a1,a2,1−a1−a2 with a1,a2∈0,1, we have(54)bTu,Tν≤βbbu,νa1bu,Tua2bν,Tν1−a1−a2bu,νa1bu,Tua2bν,Tν1−a1−a2that is T is an interpolative Reich-Rus-C′iric′ type Geraghty-contraction.Related to these examples, we can establish some consequences, by choosing proper values forr,a1,a2,a3 in Theorem 15.Corollary 11.
LetX∗,b,s and a self-mapping T on X. If there exists a function βb∈Gb such that
(55)bTu,Tν≤βbb2u,ν+b2u,Tu+b2ν,Tν1/23b2u,ν+b2u,Tu+b2ν,Tν1/23,for all u,ν∈X then T admits a unique fixed point ϖ∈X.Corollary 12.
LetX∗,b,s and a self-mapping T on X. If there exists a function βb∈Gb such that
(56)bTu,Tν≤βbbu,ν+bu,Tu+bν,Tν3bu,ν+bu,Tu+bν,Tν3,for all u,ν∈X then T admits a unique fixed point ϖ∈X.Corollary 13.
LetX∗,b,s be a complete b-metric space, a self-mapping T on X and a1∈0,1. If there exists a function βb∈Gb such that
(57)bTu,Tν≤βbbu,Tua1bν,Tν1−a1bu,Tua1bν,Tν1−a1,for all u,ν∈XFixT, then T admits a fixed point ϖ∈X.Corollary 14.
LetX∗,b,s be a complete b-metric space, a self-mapping T on X and a1,a2∈0,1. If there exists a function βb∈Gb such that
(58)bTu,Tν≤βbbu,νa1bu,Tua2bν,Tν1−a1−a2bu,νa1bu,Tua2bν,Tν1−a1−a2,for all u,ν∈X\FixT, then T admits a fixed point ϖ∈X.


## 3. Immediate Consequences

By lettingβbt=κ, we shall observe the Definition 4, [22].Theorem 15 (see [22]).
LetX∗,b,s. A r,a-weight type contraction mapping T:X→X admits a fixed point ϖ∈X if one of the following holds:
(i)
T is continuous at such point ϖ(ii)
sra2<1(iii)
sra3<1
Moreover, for anyu0∈X the sequence Tnu0 converges to ϖ.We list the following corollaries.Corollary 16.
On the completeb-metric space X∗,b,s let T:X→X be a mapping. If there exists κ∈0,1 such that
(59)bTu,Tν≤κ·ba1u,ν·ba2u,Tu·ba3ν,Tν,for all u,ν∈X\FixT, a1,a2,a3≥0 and ∑i=13ai=1, then T has a fixed point ϖ∈X. u0∈X the sequence Tnu0 converges to ϖ.Proof.
Put in Theorem15, r=0 and a=a1,a2,a3.Corollary 17.
On the completeb-metric let T:X→X be a mapping such that
(60)bTu,Tν≤κbu,νbu,Tubν,Tν3,for all u,ν∈X\FixT, where κ∈0,1. Then, T has a fixed point ϖ∈X.Proof.
Put in Theorem15, r=0 and a=1/3,1/3,1/3.Corollary 18.
LetX∗,b,s be a complete b-metric space and T:X→X be a mapping such that for every u,ν∈X\FixT(61)bTu,Tν≤κ3bu,ν+bu,Tu+bν,Tν,where κ∈0,1. The mapping T has a fixed point ϖ provided that one of the following hold:
(i)
T is continuous at ϖ∈X(ii)
s<3
Then,T has a fixed point ϖ. Moreover for any u0∈X, the sequence Tnu0 converges to ϖ.Proof.
Letr=1 and a=1/3,1/3,1/3 in Theorem 15.Corollary 19.
LetX∗,b,s be a complete b-metric space and T:X→X be a mapping such that
(62)bTu,Tν≤κ3b2u,ν+b2u,Tu+b2ν,Tν1/2,for all u,ν∈XFixT, where κ∈0,1. Assume that one of the following conditions hold:
(i)
T is continuous at ϖ∈X(ii)
s2<3
Then,T has a fixed point ϖ and for any u0∈X, the sequence Tnu0 converges to ϖ.Proof.
Taker=2 and a=1/3,1/3,1/3 in Theorem 15.Corollary 20 (see [24]).
LetX∗,b,s, T be a self-mapping on X and a1∈0,1. If there exists a function βb∈Gb such that
(63)bTu,Tν≤κ·ba1u,Tub1−a1ν,Tν,for all u,ν∈inX\FixT; then, T admits a unique fixed point ϖ∈X.Proof.
Chooseβbt=κ in Corollary 13.Corollary 21 (see [25]).
LetX∗,b,s, T be a self-mapping on X and a1,a2∈0,1. If there exists κ∈0,1 such that
(64)bTu,Tν≤κ·ba1u,νba2u,Tub1−a1−a2ν,Tν,for all u,ν∈inX\FixT; then, T admits a unique fixed point ϖ∈X.Proof.
Chooseβbt=κ in Corollary 14.


## 4. Conclusions

In this paper, we combine linear and nonlinear contractions to unify and extend the several existing results. This approach may bring new frames to the topic of metric fixed point theory. In particular, interpolative contraction may extend several results in the setting of Banach space.We also mention that, for the cases=1, we find a series of results known in the context of metric spaces, see, e.g., [25–36].



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