# Fixed Point of Orthogonal F-Suzuki Contraction Mapping on O-Complete b-Metric Spaces with Applications

Authors: Ismat Beg; Gunaseelan Mani; Arul Joseph Gnanaprakasam
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce the concept of generalized orthogonalF-Suzuki contraction mapping and prove some fixed point theorems on orthogonal b-metric spaces. Our results generalize and extend some of the well-known results in the existing literature. As an application of our results, we show the existence of a unique solution of the first-order ordinary differential equation.


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## Body

## 1. Introduction and Preliminaries

Banach contraction principle is one of the famous and useful results in mathematics. In last 100 years, it is extended in many different directions. The substitution of the metric space by other generalized metric spaces is one normal way to strengthen the Banach contraction principle. The concept of ab-metric space was introduced by Bakhtin [1] and Czerwik [2]. They also established the fixed point result in the setting of b-metric spaces which is a generalization of the Banach contraction principle. In 2015, Alsulami et al. [3] introduced the concepts of generalized F-Suzuki type contraction mappings and proved the fixed point theorems on complete b-metric spaces. On the other hand, Gordji et al. [4] introduced the new concept of an orthogonality in metric spaces and proved the fixed point result for contraction mappings in metric spaces endowed with this new type of orthogonality. Furthermore, they gave the application of this results for the existence and uniqueness of the solution of a first-order ordinary differential equation, while the Banach contraction mapping principle cannot be applied in this situation. This new concept of an orthogonal set has many applications, and there are also many types of the orthogonality. Afterward Eshaghi Gordji and Habibi [5] proved fixed point in generalized orthogonal metric spaces. Recently, Sawangsup et al. [6] introduced the new concept of an orthogonal F-contraction mappings and proved the fixed point theorems on orthogonal-complete metric spaces. Subsequently, many other researchers [7–11] studied the orthogonal contractive type mappings and obtained significant results. This paper is in continuation of these studies; first, we introduced the new concepts of generalized orthogonal F-Suzuki contraction mappings on an orthogonal b-metric space and then prove the fixed point theorems on orthogonal b-complete metric space with examples and applications to differential equations. Over results generalize/extend several results from the existing literature.In this paper, we denote byℕ, ℝ+, and W the set of positive integers, the set of positive real numbers, and the nonempty set, respectively. Next, we state the concept of a control function which was introduced by Wardowski [12].LetI denote the family of all functions F:ℝ+⟶ℝ satisfying the following properties:(F1) F is strictly increasing(F2) for each sequence αn of positive numbers, we have
(1)limn→∞αn=0⇐limn→∞Fαn=−∞;Bakhtin [1] and Czerwik [2] gave the concept of a b-metric space as follows.Definition 1 (see [2]).
LetW be a nonempty set and s≥1. Suppose that the mapping d:W×W⟶ℝ+ satisfies the following conditions for all u,v,w∈W:
(i)
du,v=0 if and only if u=v(ii)
du,v=dv,u(iii)
du,v≤sdu,w+dw,vThen,W,d is called a b-metric space with the coefficient s.Example 2 (see [2]).
Define a mappingd:W×W⟶ℝ+ by du,v=u−v2 for all u,v∈W. Then, W,d is a b-metric space with the coefficient s=2.The idea of generalizedF-Suzuki type contraction and F-Suzuki type contraction in complete b-metric spaces is due to Alsulami et al. [3]. Gordji et al. [4] introduced the notion of an orthogonal set (or O-set).Definition 3 (see [4]).
LetW≠ϕ and ⊥⊆W×W be a binary relation. If ⊥ satisfies the following condition:
(2)∃u0∈W:∀u∈W,u⊥u0or∀u∈W,u0⊥u,then it is called an orthogonal set (briefly O-set). We denote this O-set by W,⊥.Example 4 (see [4]).
LetW be the set all people in the world. Define the binary relation ⊥ on W by v⊥u if v can give blood to u. According to the Table 1, if u0 is a person such that his (her) blood type is O-, then we have u0⊥u for all u∈W. This means that W,⊥ is an O-set. In this O-set, u0 (in Definition 3) is not unique. Note that, in this example, u0 may be a person with blood type AB+. In this case, we have u⊥u0 for all u∈W.Table 1
Blood types.
TypeYou can give blood toYou can receive blood fromA+A+AB+A+A-O+O-O+O-A+B+AB+O+O-B+B+AB+B+B-O+O-AB+AB+EveryoneA-A+A-AB+AB-A-O-O-EveryoneO-B-B+B-AB+AB-B-O-AB-AB+AB-AB-B-O-A-Next, we give some basic definition for subsequent use.Definition 5 (see [4]).
LetW,⊥ be an O-set. A sequence un is called an orthogonal sequence (briefly, O-sequence) if
(3)∀n∈ℕ,un⊥un+1or∀n∈ℕ,un+1⊥un.Definition 6 (see [4]).
The tripletW,⊥,d is called an orthogonal metric space if W,⊥ is an O-set and W,d is a metric space.Definition 7 (see [4]).
LetW,⊥,d be an orthogonal metric space. Then, a mapping G:W⟶W is said to be orthogonally continuous (or ⊥-continuous) in u∈W if for each O-sequence un in W with un⟶u as n⟶∞, we have Gun⟶Gu as n⟶∞. Also, G is said to be ⊥-continuous on W if G is ⊥-continuous in each u∈W.Remark 8 (see [4]).
Every continuous mapping is⊥-continuous and the converse is not true.Definition 9 (see [4]).
LetW,⊥,d be an orthogonal metric space. Then, W is said to be orthogonally complete (briefly, O-complete) if every Cauchy O-sequence is convergent.Remark 10 (see [4]).
Every complete metric space isO-complete and the converse is not true.Definition 11 (see [4]).
LetW,⊥ be an O-set. A mapping G:W⟶W is said to be ⊥-preserving if Gu⊥Gv when ever u⊥v.
An orthogonal sequenceun is said to satisfy the condition T if
(4)∀n,m∈ℕ,un⊥umor∀n,m∈ℕ,um⊥un.
Throughout the paper, we assume that an orthogonal sequenceun satisfies the condition T.


## 2. Main Results

In this section, inspired by Alsulami et al. [3], the notion of a generalized F-Suzuki type contraction mapping, we introduce a new generalized F-Suzuki contraction mapping and obtain fixed point results for this contraction mapping on an orthogonal b-metric space.Definition 12.
LetW,⊥,dbe an orthogonalb-metric space with constants≥1. A mapG:W⟶Wis said to be a generalized orthogonalF-Suzuki contraction mapping (briefly, generalizedG⊥-Suzuki contraction) onW,⊥,dif there areF∈Iandτ>0such that the following condition holds:(5)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟹τ+FdGu,Gv≤aFdu,v+bFdu,Gu+cFdv,Gv,where c∈0,1 and a,b∈0,1 are real numbers with a+b+c=1.In the above Definition12, if we take a=1 and b=c=0; we obtain the following Definition.Definition 13.
LetW,⊥,dbe an orthogonalb-metric space with constants≥1. A mapG:W⟶Wis said to be an orthogonalF-Suzuki contraction mapping (briefly,G⊥-Suzuki contraction) onW,⊥,dif there areF∈Iandτ>0such that the following condition holds:(6)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟹τ+FdGu,Gv≤Fdu,v.Theorem 14.
LetW,⊥,dbe anO-complete orthogonalb-metric space with constants≥1and an orthogonal elementu0and a mappingG:W⟶W. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving(ii)
G is generalized G⊥-Suzuki contraction mapping
Then,G has a fixed point t∈W, and for every u∈W, the sequence Gnu converges to t.Proof.
SinceW,⊥ is an O-set,
(7)∃u0∈W:∀u∈W,u⊥u0or∀u∈W,u0⊥u.
It follows thatu0⊥Gu0 or Gu0⊥u0. Let
(8)u1≔Gu0,u2≔Gu1=G2u0⋯⋯,un+1≔Gun=Gn+1u0,for all n∈ℕ∪0. If un=un+1 for any n∈ℕ∪0, then it is clear that un is a fixed point of G. Assume that un≠un+1 for all n∈ℕ∪0. Thus, we have dGun,Gun+1>0 for all n∈ℕ∪0. Since G is ⊥-preserving, we have
(9)un⊥un+1orun+1⊥un,for all n∈ℕ∪0. This implies that un is an O-sequence. Since G is generalized G⊥-Suzuki contraction mapping, we have
(10)12sdun,Gun<dun,Gun,∀n∈ℕ.
Thus, by hypothesis of theorem, we have(11)τ+FdGun,G2un≤aFdun,Gun+bFdun,Gun+cFdGun,G2un,which is equivalent to
(12)τ+1−cFdGun,G2un≤a+bFdun,Gun.
Sincea+b+c=1, the inequality (12) turns into
(13)FdGun,G2un≤Fdun,Gun−τa+b<Fdun,Gun.
FromF1, we conclude that
(14)dun+1,Gun+1=dGun,G2un<dun,Gun,∀n∈ℕ.
Therefore,dun,Gunn=1∞ is a decreasing sequence of real numbers which is bounded below. Therefore, dun,Gunn=1∞ converges and
(15)limn→∞dun,Gun=δ=infdun,Gun:∀n∈ℕ.
We will show thatδ=0. On the contrary, suppose that δ>0. In other words, for every ε>0 there exits k∈ℕ, such that
(16)duk,Guk<δ+ε.
FromF1, we find that
(17)Fduk,Guk<Fδ+ε.
On the other hand, we have(18)12sduk,Guk<duk,Guk,due to (10). Since G is generalized G⊥-Suzuki contraction, we derive
(19)τ+FdGuk,G2uk≤aFduk,Guk+bFduk,Guk+cFdGuk,G2uk,which yields that
(20)τ+1−cFdGuk,G2uk≤a+bFduk,Guk,taking the fact that a+b+c=1 into account, we obtain
(21)FdGuk,G2uk≤Fduk,Guk−τa+b.
Analogously, again by (10), we have 1/2sdGuk,G2uk<dGuk,G2uk. Owing to the fact that G is generalized G⊥-Suzuki contraction, we conclude that
(22)τ+FdG2uk,G3uk≤aFdGuk,G2uk+bFdGuk,G2uk+cFdG2uk,G3uk.
It implies that(23)FdG2uk,G3uk≤FdGuk,G2uk−τa+b,since a+b+c=1. Furthermore, by combining (21) and (23), we get
(24)FdG2uk,G3uk≤FdGuk,G2uk−τa+b≤Fduk,Guk−2τa+b.
Iteratively, we obtain that(25)FdGnuk,Gn+1uk≤FdGnuk,Gn−1uk−τa+b≤FdGn−1uk,Gn−2uk−2τa+b⋮≤FdGuk,uk−nτa+b≤Fδ+ε−nτa+b.
By lettingn⟶∞, we find that
(26)limn→∞FdGnuk,Gn+1uk=−∞.
Consequently, fromF2, we derive that limn→∞dGnuk,Gn+1uk=0. Thus, there exists N1∈ℕ such that
(27)dGnuk,Gn+1uk<δ,∀n≥N1,and from (8), we get
(28)duk+n,Guk+n<δ,∀n≥N1.
This is a contradiction with the definition ofδ. Hence, we have
(29)limn→∞dun,Gun=0.
It what follows, we will prove that(30)limn,k→∞dun,uk=0.
Suppose, on the contrary, that there existsε>0 and sequences inn=1∞ and jnn=1∞ of natural numbers such that
(31)in>jn>n,duin,ujn≥ε,duin−1,ujn<ε,∀n∈ℕ.
From the triangle inequality, we have(32)duin,ujn≤sduin,uin−1+duin−1,ujn≤sduin,uin−1+sε=sduin−1,Guin−1+sε,∀n∈ℕ.
Owing to (29), there exists and N2∈ℕ such that
(33)duin,Guin<ε,∀n>N2.
Taking (33) into account, (32) yields that
(34)duin,ujn<2sε,∀n>N2.
So fromF2, we obtain
(35)Fduin,ujn<F2sε,∀n>N2.
On the other hand, we can easily get that(36)12sduin,Guin<ε2s<ε≤duin,ujn,∀n>N2.
Next, we claim that(37)dGuin,Gujn=duin+1,ujn+1>0,∀n∈ℕ.
Arguing by contradiction, there existsk≥ℕ such that
(38)duik+1,ujk+1=0.
It follows from (31), (36), and (38) that
(39)ε≤duik,ujk≤duik,uik+1+duik+1,ujk≤duik,uik+1+duik+1,ujk+1+dujk+1,ujk=duik,Guik+duik+1,ujk+1+dujk,Gujk<ε2s+0+ε2s=εs.
This contradiction establishes the relation (37). Since G is ⊥-preserving, we have
(40)uin⊥ujnorujn⊥uin.
SinceG is generalized G⊥-Suzuki contraction, ∀n>N2, we find that
(41)τ+FdGuin,Gujn≤aFduin,ujn+bFduin,Guin+cFdujn,Gujn.
Taking (35) into account, (41) turns into
(42)τ+FdGuin,Gujn<aF2sε+bFduin,Guin+cFdujn,Gujn,∀n>N2.
Regarding (29) and F2, we obtain that
(43)limn→∞FdGuin,Gujn=−∞.
FromF2, we get that
(44)limn→∞FdGuin,Gujn=0⇔limn→∞Fduin+1,ujn+1=0.
This is a contradiction with relations in (31). Hence, limk,n→∞dun,uk=0; that is unn=1∞ is a Cauchy sequence in W. On the account of completeness of W,d, there exists t∈W such that
(45)limn→∞dun,t=0.
We claim that, for everyn∈ℕ,
(46)12sdun,Gun<dun,tor12sdGun,G2un<dGun,t,∀n∈ℕ.
We will prove the claim above by the method of reductio ad absurdum. Suppose, on the contrary, that there existsk∈ℕ such that
(47)12sduk,Guk≥duk,t,(48)12sdGuk,G2uk≥dGuk,t.
From (14) and F1, we have
(49)dGuk,G2uk<duk,Guk.
It follows from (48) and (49) that
(50)duk,Guk≤sduk,t+sdt,Guk≤12duk,Guk+12dGuk,G2uk<12duk,Guk+12duk,Guk=duk,Guk.
This is a contradiction. Hence, (46) holds. Since G is generalized G⊥-Suzuki contraction, (46) yields that, for every n∈ℕ, either
(51)τ+FdGun,Gt≤aFdun,t+bFdun,Gun+cFdt,Gt,(52)τ+FdG2un,Gt≤aFdGun,t+bFdGun,G2un+cFdt,Gt,holds. On account of F2, the limits in (29) and (45) imply that
(53)limn→∞Fdun,t=−∞,limn→∞Fdun,Gun=−∞.
Thus, lettingn⟶∞ in (51), we conclude that
(54)limn→∞FdGun,Gt=−∞.
Again byF2, we observe that
(55)limn→∞FdGun,Gt=0.
By regarding the triangle inequality with (8), we derive that
(56)dt,Gt≤sdt,Gun+dGun,Gt=sdt,un+1+sdGun,Gt.
By lettingn⟶∞ in the inequality above together with the limits in (45) and (55), we conclude that dt,Gt=0. Thus, t is a fixed point of G; that is t=Gt. Let us analyze the second case (51). Regarding (12), we have
(57)FdG2un,Gt<τ+FdG2un,Gt≤aFdGun,t+bFdGun,G2un+cFdt,Gt=aFdun+1,t+bFdun+1,Gun+1+cFdt,Gt.
As it was discussed above, from (29), (45), and F2, we conclude that
(58)limn→∞FdG2un,Gt=−∞.
FromF2 equivalently, we get
(59)limn→∞FdG2un,Gt=0.
Again by the triangle inequality together with (8), we find that
(60)dt,Gt≤sdt,G2un+dG2un,Gt=sdt,un+2+sdG2un,Gt.
By lettingn⟶∞ in the inequality above together with the limits in (45) and (59), we obtain that dt,Gt=0. Thus, t is a fixed point of G. Let h,g∈W be two fixed points of G and suppose that Gnh=h≠g=Gng for all n∈ℕ. By choice of u0, we obtain
(61)u0⊥handu0⊥gorh⊥u0andg⊥u0.
SinceG is ⊥-preserving, we have
(62)Gnu0⊥GnhandGnu0⊥GngorGnh⊥Gnu0andGng⊥Gnu0,for all n∈ℕ. Now,
(63)dh,g=dGnh,Gng≤sdGnh,Gnu0+dGnu0,Gng.
Asn⟶∞, we obtain dh,g≤0. Thus, h=g. Hence, G has a unique fixed point in W.Example 15.
LetW=0,1∪2,4andd:W×W⟶0,∞be a mapping defined by(64)dh,g=h−g2,for all h,g∈W. Define the binary relation ⊥ on W by h⊥g if hg≤h∨g, where h∨g=horg. Then, W,d is an O-complete b-metric space. Define the mapping G:W⟶W by
(65)Gg=ln1+g2ifg∈0,1,g−1ifg∈2,4.
Leta=1/2, b=1/4, c=1/4 and τ>0. Let h=2, g=4, then, hg≤g. We have
(66)dGh,Gg=4,dh,g=4,dh,Gh=1,anddg,Gg=1.
Therefore,(67)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟹τ+dGh,Gg≥aFdh,g+bFdh,Gh+cFdg,Gg.
Hence,G is not generalized F-Suzuki contraction mapping, and so Theorem 9 in [3] is not applicable. Now, let us consider the mapping F defined by Ft=lnt, t∈0,∞. Then, G is G⊥-Suzuki contraction with τ=ln4, c∈0,1, and a,b∈0,1 are real numbers, a+b+c=1. First, we observe that
(68)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟸h,g∈0,1∨h∈0,1andg∈2,4∨h∈2,4andg∈0,1.
Note that if(69)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟹τ+FdGu,Gv≤aFdu,v+bFdu,Gu+cFdv,Gv,Clearly,G is ⊥-preserving. Let h⊥g. Without loss of generality, we may assume that hg≤g. Then, we consider the following cases:Case 16.
Leth,g∈0,1, then dGh,Gg=ln1+h/2−ln1+g/22, dh,g=h−g2, dh,Gh=h−ln1+h/22, and dg,Gg=g−ln1+g/22. It is clear that (69) is satisfied.Case 17.
Leth∈0,1 and g∈2,4, then dGh,Gg=ln1+h/2−g+12, dh,g=h−g2, dh,Gh=h−ln1+h/22, and dg,Gg=1. It is clear that (69) is satisfied.Case 18.
Leth∈2,4 and g∈0,1, then dGh,Gg=h−1−ln1+g/22, dh,g=h−g2, dh,Gh=1, and dg,Gg=g−ln1+g/22. It is clear that (69) is satisfied. Hence, all the hypothesis of Theorem 14 are satisfied, note that G has a unique fixed point g=0.Theorem 19.
LetW,⊥,dbe anO-complete orthogonalb-metric space with constants≥1and an orthogonal elementu0and a mappingG:W⟶W. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving(ii)
G is a G⊥-Suzuki contraction mapping
Then,G has a unique fixed point t∈W, and for every u∈W, the sequence Gnu converges to t.Proof.
By takinga=1 and b=c=0 in Theorem 14, the proof is complete.Example 20.
LetW=0,1∪4,6andd:W×W⟶0,∞be a mapping defined by(70)dh,g=h−g2,for all h,g∈W. Define the binary relation ⊥ on W by h⊥g if hg≤h∨g, where h∨g=horg. Then, W,d is an O-complete b-metric space. Define the mapping G:W⟶W by
(71)Gg=ln1+g3ifg∈0,1,g−1ifg∈4,6.
Leth=2, g=4, then hg≤g. Since dG4,G6=4=d4,6, then for all F∈I and τ>0, we have
(72)τ+FdG4,G6>Fd4,6.
Therefore,G is not F-Suzuki contraction, and so Theorem 11 in [3] is not applicable. Now, let us consider the mapping F defined by Ft=lnt, t∈0,∞. Then, G is G⊥-Suzuki contraction with τ=ln4. First, we observe that
(73)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟸h,g∈0,1∨h∈0,1andg∈4,6∨h∈4,6andg∈0,1.
Note that if(74)∀u,v∈Wwithu⊥vdGu,Gv>0,12sdu,Gu<du,v⟹τ+FdGu,Gv≤Fdu,v.Clearly,G is ⊥-preserving. Let h⊥g. Without loss of generality, we may assume that hg≤g. Then, we consider the following cases:Case 21.
Leth,g∈0,1, then dGh,Gg=ln1+h/3−ln1+g/32, dh,g=h−g2, dh,Gh=h−ln1+h/32, and dg,Gg=g−ln1+g/32. It is clear that (74) is satisfied.Case 22.
Leth∈0,1 and g∈4,6, then dGh,Gg=ln1+h/3−g+12, dh,g=h−g2, dh,Gh=h−ln1+h/32, and dg,Gg=1. It is clear that (74) is satisfied.Case 23.
Leth∈4,6 and g∈0,1, then dGh,Gg=h−1−ln1+g/32, dh,g=h−g2, dh,Gh=1, and dg,Gg=g−ln1+g/32. It is clear that (74) is satisfied. Hence, all the hypothesis of Theorem 19 are satisfied, note that G has a unique fixed point g=0.Corollary 24.
LetGbe a self mapping on anO-complete metric spaceW,⊥,d. Assume that there existF∈Iandτ>0such that the following condition hold:(i)
G is ⊥-preserving(ii)
∀u,v∈Wwithu⊥vdGu,Gv>0,1/2du,Gu<du,v⟹τ+FdGu,Gv≤aFdu,v+bFdu,Gu+cFdv,Gvwhere c∈0,1 and a,b∈0,1 are real numbers with a+b+c=1.
Then,G has a unique fixed point t∈W.Proof.
Since any metric space is ab-metric space with constant s=1, so from Theorem 14 the proof is complete.Corollary 25.
LetGbe a self mapping on anO-complete metric spaceW,⊥,d. Assume that there existF∈Iandτ>0such that the following condition hold:(i)
G is ⊥-preserving(ii)
∀u,v∈Wwithu⊥vdGu,Gv>0,1/2du,Gu<du,v⟹τ+FdGu,Gv≤Fdu,v
Then,G has a unique fixed point t∈W.Proof.
Since any metric space is ab-metric space with constant s=1, so by taking a=1 and b=c=0 from Theorem 14, the proof is complete.Proof.
By takinga=1 and b=c=0 in Theorem 14, the proof is complete.
Next, we give the fixed point theorem for orthogonal continuous mapping on anO-complete b-metric space W,⊥,d.Theorem 26.
LetW,⊥,dbe anO-completeb-metric space with constants≥1and an orthogonal elementu0and a mappingG:W⟶W. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving(ii)
∀u,v∈Wwithu⊥vdGu,Gv>0⟹τ+FdGu,Gv<aFdu,v+bFdu,Gu+cFdv,Gvwherec∈0,1 and a,b∈0,1 are real numbers with a+b+c=1.(iii)
G is ⊥-continuous
Then,G has a unique fixed point t∈W.Proof.
SinceW,⊥ is an O-set,
(75)∃u0∈W:∀u∈W,u⊥u0or∀u∈W,u0⊥u.
It implies thatu0⊥Gu0 or Gu0⊥u0. Let
(76)u1≔Gu0,u2≔Gu1=G2u0⋯,un+1≔Gun=Gn+1u0,for all n∈ℕ∪0.
Ifun=un+1 for any n∈ℕ∪0, then it is clear that un is a fixed point of G. Assume that un≠un+1 for all n∈ℕ∪0. Therefore, we have dGun,Gun+1>0 for all n∈ℕ∪0. As G is ⊥-preserving, we have
(77)un⊥un+1orun+1⊥un
for alln∈ℕ∪0. It implies that un is an O-sequence. By condition (ii), we have
(78)0<dun,Gun=dGun−1,Gun.∀n∈ℕ.
So we have(79)τ+FdGun−1,Gun≤aFdun−1,un+bFdun−1,Gun−1+cFdun,Gun=aFdun−1,un+bFdun−1,un+cFdGun−1,Gun.
Thus,(80)τ+1−cFdGun−1,Gun≤a+bFdun−1,un.As a+b+c=1, we get
(81)FdGun−1,Gun≤Fdun−1,un−τa+b<Fdun−1,un.
FromF1, we conclude that
(82)dun,Gun=dGun−1,Gun<dun−1,Gun−1,∀n∈ℕ.
Thus,dun,Gunn=1∞ is a decreasing sequence of real numbers which is bounded below. Therefore, dun,Gunn=1∞ converges and
(83)limn→∞dun,Gun=δ=infdun,Gun:∀n∈ℕ.
Next, we show thatδ=0. Arguing by contradiction, we get δ>0. For every ε>0, there exits k∈ℕ, such that
(84)duk,Guk<δ+ε.
FromF1, we get
(85)Fduk,Guk<Fδ+ε.
On the other hand (78), we have
(86)0<duk,Guk=duk−1,Guk.
So we obtain(87)τ+FdGuk−1,Guk≤aFduk−1,uk+bFduk−1,Guk−1+cFduk,Guk=aFduk−1,uk+bFduk−1,uk+cFdGuk−1,Guk,and thus
(88)τ+1−cFdGuk−1,Guk<a+bFduk−1,uk.
Sincea+b+c=1, we obtain
(89)FdGuk−1,Guk≤Fduk−1,uk−τa+b.
Also from (78), we have 0<dGuk,Guk+1<dGuk,G2uk, and thus, by assumption of theorem, we have
(90)τ+FdGuk,Guk+1≤aFduk,uk+1+bFduk,Guk+cFduk+1,Guk+1=aFduk,uk+1+bFduk,uk+1+cFdGuk,Guk+1,and therefore,
(91)τ+1−cFdGuk,Guk+1≤a+bFduk,Guk+1.
Sincea+b+c=1, we get
(92)FdGuk,Guk+1≤Fduk,Guk+1−τa+b.
Now by using (85) and continuing similar method as used in (89) and (92), we obtain
(93)Fduk+n,uk+n+1=FdGuk+n−1,Guk+n≤Fduk+n−1,uk+n−τa+b=FdGuk+n−2,Guk+n−1−τa+b≤Fduk+n−2,uk+n−1−2τa+b=FdGuk+n−3,Guk+n−2−2τa+b≤Fduk+n−3,uk+n−2−3τa+b⋮≤Fduk+1,uk+2−n−1τa+b=FdGuk,Guk+1−n−1τa+b≤Fδ+ε−nτa+b.
Lettingn⟶∞, we find that
(94)limn→∞Fduk+n,uk+n+1=−∞.
Consequently, fromF2, we have limn→∞dduk+n,uk+n+1=0. Thus, there exists N1∈ℕ such that
(95)duk+n,uk+n+1<δ,∀n≥N1,and from (76), we get
(96)duk+n,Guk+n<δ,∀n≥N1.
This is a contradiction with the definition ofδ. So δ=0 and from (85) we have
(97)limn→∞dun,Gun=0.
Next, we claim that(98)limn,k→∞dun,uk=0.
Suppose, otherwise, that there existsε>0 and sequences inn=1∞ and jnn=1∞ of natural numbers such that
(99)in>jn>n,duin,ujn≥ε,duin−1,ujn<ε,∀n∈ℕ.
From the triangle inequality, we have(100)duin,ujn≤sduin,uin−1+duin−1,ujn≤sduin,uin−1+sε=sduin−1,Guin−1+sε,∀n∈ℕ.
Due to (97), there exists an N2∈ℕ such that
(101)duin,Guin<ε,∀n>N2.
It implies from (100) and (101) that
(102)duin,ujn<2sε,∀n>N2.
So fromF2, we obtain
(103)Fduin,ujn<F2sε,∀n>N2.
Also from (99), we can easily get that
(104)ε≤duin+1,ujn+1=dGuin,Gujn,∀n>N2.
Therefore, from (103) and assumption of theorem, we get
(105)τ+FdGuin,Gujn≤aFduin,ujn+bFduin,Guin+cFdujn,Gujn<aF2sε+bFduin,Guin+cFdujn,Gujn,∀n∈ℕ.
Using (100) and F2, we obtain
(106)limn→∞FdGuin,Gujn=−∞.
Thus, from this andF2, we get
(107)limn→∞FdGuin,Gujn=0⇔limn→∞Fduin+1,ujn+1=0.
A contradiction with relations in (99). Hence, limk,n→∞dun,uk=0. On the account of completeness of W,d, there exists t∈W such that
(108)limn→∞dun,t=0.
SinceG is ⊥-continuous, we get
(109)limn→∞dGun,Gt=0.
Asdt,Gt≤sdt,un+dxn,Gt, thus, dt,Gt=0 and so t is a fixed point of G.
Leth,g∈W be two fixed points of G and suppose that Gnh=h≠g=Gng for all n∈ℕ. By choice of u0, we obtain
(110)u0⊥handu0⊥gorh⊥u0andg⊥u0.
SinceG is ⊥-preserving, we have
(111)Gnu0⊥GnhandGnu0⊥GngorGnh⊥Gnu0andGng⊥Gnu0for all n∈ℕ. Now,
(112)dh,g=dGnh,Gng≤sdGnh,Gnu0+dGnu0,Gng.
Asn⟶∞, we obtain dh,g≤0. Thus, h=g. Hence, G has a unique fixed point in W.Example 27.
LetW=ℝandd:W×W⟶0,∞be a mapping defined by(113)dh,g=maxh,g2for all h,g∈W. Define the binary relation ⊥ on W by h⊥g if h,g∈3,n+4 for some n∈ℕ or h=0. Then, W,d is an O-complete b-metric space. Define the mapping G:W⟶W by
(114)Gg=0ifg=0g+1ifg∈1,2,g+1ifg∈2,∞.Leth∈W and hn be an arbitrary ⊥-sequence in W such that converges to h, then the following cases are satisfied:Case 28.
Ifhn=0 for all n, then h=0 and Ghn=0=Gh.Case 29.
Ifhn≠0 for some n, then there exists m∈ℕ such that h∈3,m+4, and Ghn=Gh. Therefore, G is ⊥-continuous on W, but it is not continuous on W. Let a=1/2, b=1/4, and c=1/4. Since dG1,G2=9, d1,2=4, d1,G1=4, and d2,G2=9, then for all F∈I and τ>0 we have
(115)∀h,g∈Wwithh⊥gdGh,Gg>0⟹τ+dGh,Gg≥aFdh,g+bFdh,Gh+cFdg,Gg
Therefore, Theorem 16 in [3] is not applicable. Now, let us consider the mapping F defined by Ft=lnt, t∈0,∞. First, we observe that
(116)∀h,g∈Wwithh⊥gdGh,Gg>0⇐h=0andg∈2,∞.
Note that if(117)∀h,g∈Wwithh⊥gdGh,Gg>0,12sdh,Gh<dh,g⟹τ+FdGh,Gg≤aFdh,g+bFdh,Gh+cFdg,Gg,
Clearly,G is ⊥-preserving. Let h⊥g and h=0,g∈2,∞, τ=ln2, then dGh,Gg=g−12, dh,g=g2, dh,Gh=0, and dg,Gg=g2. It is clear that (117) is satisfied.Hence, all the hypothesis of Theorem26 are satisfied, note that G has a unique fixed point g=0.Theorem 29.
LetW,⊥,dbe anO-completeb-metric space with constants≥1and an orthogonal elementu0and a mappingG:W⟶W. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving ∀u,v∈Wwithu⊥vdGu,Gv>0⟹τ+FdGu,Gv<Fdu,v;(ii)
G is ⊥-continuous
Then,G has a unique fixed point t∈W.Proof.
By takinga=1 and b=c=0 in Theorem 26, the proof is complete.Theorem 30.
LetW,⊥,dbe anO-complete metric space. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving(ii)
∀u,v∈Wwithu⊥vdGu,Gv>0⟹τ+FdGu,Gv<aFdu,v+bFdu,Gu+cFdv,Gv,wherec∈0,1 and a,b∈0,1 are real numbers with a+b+c=1(iii)
G is ⊥-continuous
Then,G has a unique fixed point t∈W.Proof.
It is sufficient to takes=1 in Theorem 26 the proof is complete.Theorem 31.
LetW,⊥,dbe anO-complete metric space. Suppose that there existF∈Iandτ>0such that the following conditions hold:(i)
G is ⊥-preserving(ii)
∀u,v∈Wwithu⊥vdGu,Gv>0⟹τ+FdGu,Gv<Fdu,v(iii)
G is ⊥-continuous
Then,G has a unique fixed point t∈W.Proof.
Since every metric space is ab-metric space with constant s=1, from Theorem 29, G has a fixed point t∈W. Indeed, if there is another fixed point s∈W of G, such that ∀t,s∈Wwitht⊥sdGt,Gs>0. Since τ>0, from our assumption of theorem, we obtain
(118)Fdt,s=FdGt,Gs<τ+FdGt,Gs≤aFdt,s.
This is contradiction. SoG has a unique fixed point.


## 3. Application to Ordinary Differential Equations

For any1≤p<∞, the space LpW,F,λ (or LpW) consists of all complex-valued measurable β on the underlying space W satisfying
(119)∫Wβwpdλw.HereF is the σ-algebra of measurable sets and λ is the measure.Whenp=1, the space L1W consists of all integrable functions β on W. We define the L1-norm of β by
(120)∥β∥1=∫W∣βw∣dλw.Now using Theorem19, we prove the existence of a solution of the differential equation:
(121)j′t=Gt,jt,a.e.t∈I≔0,T,(122)j0=c,c≥1,where G:I×ℝ⟶ℝ is an integrable function satisfying the conditions:
(i)
Gs,y≥0 for all y≥0 and s∈I(ii)
for eachl,m∈L1I with lsms≥ls or lsms≥ms for all s∈I, there exist γ∈L1I and τ>0 such that
(123)∣Gs,ls−Gs,ms∣≤γs1+τγsls−ms2,ls−ms2≤γse2Bsfor all s∈I, where Bs≔∫0s∣γr∣dr.Theorem 32.
Consider the differential Eq. ((122)). If (i) and (ii) are satisfied, then the differential Eq. ((122)) has a unique positive solution.Proof.
LetW=w∈L1W:wt>0forallt∈I. Define the orthogonality relation ⊥ on V by
(124)l⊥m⇐ltmt≥ltorltmt≥mtforallt∈I.
AsBt=∫0t∣γs∣ds, we have B′t=∣γt∣ for almost everywhere t∈I. Now define a mapping d:W×W⟶0,∞ by
(125)dl,m=∥l−m∥B=supt∈Ie−2Btlt−mt2,for all l,m∈W. Therefore, W,d is a complete b-metric space. Define a mapping Q:W⟶W by
(126)Qlt=c+∫0tGs,lsds.
Next, we show thatQ is ⊥-preserving. For each l,m∈W with l⊥m and t∈I, we have
(127)Qlt=c+∫0tGs,lsds≥1.
It follows thatQltQmt≥Qmt and so Qlt⊥Qmt. Then, Q is ⊥-preserving.
Now, we claim thatQ is an G⊥-Suzuki contraction. Let l,m∈W with l⊥m. Suppose that Ql≠Qm. For each s∈I, we have
(128)ls−ms2≤γse2Bs.and so
(129)dl,m=sups∈Ie−2Bsls−ms2≤γs.
Therefore,(130)1+τdl,m≤1+τγs.
Fromii, for each t∈I, we get
(131)∣Qlt−Qmt∣≤∫0t∣Gs,ls−Gs,ms∣ds≤∫0tγs1+τγsls−ms2ds≤∫0tγs1+τdl,mls−ms2e−2Bse2Bsds≤dl,m1+τdl,m∫0t∣γs∣e2Bsds≤dl,m1+τdl,me2Bt−1,and so
(132)e−2Bt∣Qlt−Qmt∣≤e−2Bte2Bt−1dl,m1+τdl,m=1−e−2Btdl,m1+τdl,m≤1−e−∥β∥1dl,m1+τdl,m.
Thus,(133)dQl,Qm≤dl,m1+τdl,m1+τdl,mdl,m≤1dQl,Qmτ+1dl,m≤1dQl,Qmτ+1dl,m≤1dQl,Qmτ−1dQl,Qm≤−1dl,m.
Taking a functionF:ℝ+⟶ℝ by Fb=−1/b for all b>0, it implies that Q is an G⊥-Suzuki contraction with a=1, b=0 and c=0. Using Theorem 19, Q has a unique fixed point. Hence, the differential Eq. (122) has a unique positive solution.



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*Source: 6692112-2021-03-10.xml*