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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 243-250

Published online September 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.3.243

© The Korean Institute of Intelligent Systems

Orthogonal F-Contraction Mapping on O-Complete Metric Space with Applications

Gunaseelan Mani1, Arul Joseph Gnanaprakasam2, Nasreen Kausar3, Mohammad Munir4, and Salahuddin5

1Department of Mathematics, Sri Sankara Arts and Science College (Autonomous), Madras University,Tamil Nadu, India
2Department of Mathematics, Faculty of Engineering and Technology, College of Engineering and Technology, SRM Institute of Science and Technology, Tamil Nadu, India
3Faculty of Arts and Science, Yildiz Technical University, ˙Istanbul, Turkey
4Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
5Department of Mathematics, Jazan University, Jazan, Saudi Arabia

Correspondence to :
Nasreen Kausar (kausar.nasreen57@gmail.com)

Received: December 25, 2020; Revised: May 2, 2021; Accepted: June 14, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce orthogonal concepts concerning F-contraction mappings and demonstrate some fixed-point theorems for self-mapping in a complete orthogonal metric space. Some well-known results in the literature are generalized and modified based on the demonstrated results. An example is provided to support our results, which are used in an application.

Keywords: Orthogonal set, Orthogonal complete metric space, Orthogonal continuous, Orthogonal preserving, Orthogonal F-contraction, Fixed point

One of the most important results of a mathematical analysis is the famous fixed-point result, called Banach contraction theory. In several branches of mathematics, this is the most commonly used fixed-point result, and has been generalized in many different directions. One natural way to reinforce the Banach contraction concept is to replace the metric space with other generalized metric spaces. The fixed-point result in the setting of complete metric spaces was established by Wardowski [1], which is a generalization of the Banach contraction principle in metric spaces. However, in many branches of mathematics, the notion of an orthogonal set has many applications and has several methods of orthogonality. Gordji et al. [2] introduced the current concept of orthogonality in metric spaces and demonstrated the fixed-point result for contraction mappings in metric spaces equipped with the new orthogonality. In a generalized orthogonal metric space, Gordji and Habibi [3] proved the theory of fixed points. Sawangsup et al. [4] introduced the new concept of orthogonal F-contraction mappings and fixed-point theorems on orthogonal-complete metric space were proven. Orthogonal contractive-type mappings have been studied by many authors, and important results were obtained in [59].

In Section 2, for the subsequent usage, we remember some essential notes and definitions from the current literature. In Section 3, in an orthogonal complete metric space, we introduce a new F-contraction mapping and prove some fixed-point theorems for this contraction mapping. We apply our results to demonstrate the presence of a unique solution to the ordinary differential equation in Section 4.

In this study, Ξ, ℝ+, and ℕ denote the non-empty set, positive real numbers, and positive integer sets, respectively.

First, we recall the notion of an auxiliary function, introduced in 2012 by Wardowski [1], as follows:

Definition 2.1[1]

Let (Ξ, d) be a metric space. A mapping Ω : ΞΞ is said to be an F-contraction if ι > 0 and F ∈ ϒ such that

ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ))

holds for any ϖ, ϱΞ with dϖ, Ωϱ) > 0, where ϒ is the set of all functions F : ℝ+ → ℝ satisfying the following properties:

  • (F1) F is strictly increasing;

  • (F2) for each sequence {ɛm} of positive numbers, we have

    limmɛm=0iff limmF(ɛm)=-;

    (F3) q(0,1)limɛ0+ɛqF(ɛ)=0.

In 2013, Secelean [10] demonstrated that, in Definition 2.1, the condition (F2) can be altered. The definition of an F-contraction of the Hardy-Rogers-type was introduced by Cosentino and Vetro [11]. In 2014, Piri and Kumam [12] demonstrated that in Definition 2.1, the condition (F3) can be altered. Later, in 2020, Popescu and Stan [13] proved some fixed point theorem concerning an F-contraction in a complete metric space as follows:

Theorem 2.1 [13]

Let Ω be a self-mapping of a complete metric space Ξ. Suppose that there exists ι > 0 such that ∀ ϖ, ϱΞ, dϖ, Ωϱ) > 0,

ι+F(d(Ωϖ,Ωϱ))F(θ1d(ϖ,ϱ)+θ2d(ϖ,Ωϖ)+θ3d(ϱ,Ωϱ)+θ4d(ϖ,Ωϱ)+θ5d(ϱ,Ωϖ)),

where F : ℝ+ → ℝ is an increasing mapping, θ1, θ2, θ3, θ4, θ5 ≥ 0, θ4<12, θ3 < 1, θ1 + θ2 + θ3 + 2θ4 = 1, 0 < θ1 + θ4 + θ5 ≤ 1. Then, Ω has a unique fixed point ϖ*Ξ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to ϖ*.

By contrast, the definition of an orthogonal set (or O set) and the properties of orthogonal sets were introduced by Gordji et al. [2] as follows:

Definition 2.2[2]

Let Ξφ and ⊥ ⊆ Ξ × Ξ be a binary relation. If ⊥ satisfies the following consecutive conditions,

ϖ0Ξ:(ϖΞ,ϖϖ0)         or         (ϖΞ,ϖ0ϖ),

then it is said to be an O set. We indicate this O set by (Ξ, ⊥).

Example 2.2[2]

Let Ξ = [0, ∞) and define ϖϱ if ϖϱ ∈ {ϖ, ϱ}. Then, by setting ϖ0 = 0 or ϖ0 = 1, (Ξ, ⊥) is an O-set.

Now, we present the concepts of an an orthogonal sequence (O-sequence),, an O-complete orthogonal metric space, and a ⊥-preserving mapping.

Definition 2.3 [2]

Let (Ξ, ⊥) be an O set. A sequence {ϖm} is said to be an O-sequence if

(m,ϖmϖm+1)         or         (m,ϖm+1ϖm).

Definition 2.4[2]

The triplet (Ξ, ⊥, d) is considered to be an orthogonal metric space if (Ξ, ⊥) is an O set, and (Ξ, d) is a metric space.

Definition 2.5 [2]

Let (Ξ, ⊥) be an O set. A mapping Ω : ΞΞ is called ⊥-preserving if Ωϖ⊥Ωϱ whenever ϖϱ.

In this study, we adapt the concepts of F-contraction mapping to orthogonal sets and prove some fixed-point theorems for an F-contraction mapping in the complete orthogonality of the metric space.

In this section, we introduce a new F-contraction mapping, inspired by the concepts of F-contraction mapping, described by Popescu and Stan [13], and demonstrate some fixed-point theorems in an orthogonal-complete metric space for this contraction mapping.

Theorem 3.1

Let (Ξ, ⊥, d) be an orthogonal-complete metric space, and an orthogonal element ϖ0. Let Ω be a self-mapping of Ξ. Suppose that ι > 0 exists such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ))F(θ1d(ϖ,ϱ))+θ2d(ϖ,Ωϖ)+θ3d(ϖ,Ωϱ)+θ4d(ϖ,Ωϱ)+θ5d(ϱ,Ωϖ))],

where F : ℝ+ → ℝ is an increasing mapping; θ1, θ2, θ3, θ4, θ5 are non-negative numbers; and θ4<12, θ3 < 1, θ1 + θ2 + θ3 +2θ4 = 1, 0 < θ1 + θ4 + θ5 ≤ 1, where Ω is ⊥-preserving. Then, Ω has a unique fixed-point tΞ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to t.

Proof

Because (Ξ, ⊥) is an O-set,

ϖ0Ξ:(ϖΞ,ϖϖ0)         or         (ϖΞ,ϖ0ϖ).

It follows that ϖ0⊥Ωϖ0 or Ωϖ0ϖ0. Let

ϖ1:=Ωϖ0,,ϖm+1:=Ωϖm=Ωm+1ϖ0,

m ∈ ℕ∪{0}. If ϖm = ϖm+1 for any m ∈ ℕ∪{0}, it is then evident that ϖm is a fixed-point of Ω. We assume that ϖmϖm+1m ∈ ℕ∪{0}. Thus, we derive dϖm, Ωϖm+1) > 0 ∀m ∈ ℕ ∪ {0}. Because Ω is ⊥-preserving, we obtain

ϖmϖm+1         or         ϖm+1ϖm,

m ∈ ℕ ∪ {0}. This means that {ϖm} is an O-sequence. Hence, we presume that

0<d(ϖm,Ωϖm)=d(Ωϖm-1,Ωϖm),m.

Now, let dm = d(ϖm, ϖm+1). Through the monotony of F, we derive ∀m ∈ ℕ

ι+F(dm)=ι+F(d(ϖm,ϖm+1))=ι+F(d(Ωϖm-1,Ωϖm))F(θ1d(ϖm-1,ϖm)+θ2d(ϖm-1,Ωϖm-1)+θ3d(ϖm,Ωϖm)+θ4d(ϖm-1,Ωϖm)+θ5d(ϖm,Ωϖm-1))=F(θ1d(ϖm-1,ϖm)+θ2d(ϖm-1,ϖm)+θ3d(ϖm,ϖm+1)+θ4d(ϖm-1,ϖm+1)+θ5d(ϖm,ϖm))=F(θ1dm-1+θ2dm-1+θ3dm+θ4d(ϖm-1,ϖm+1))F(θ1dm-1+θ2dm-1+θ3dm+θ4(dm-1+dm))=F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm).

It proceeds that

F(dm)F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm)-ι<F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm).

Thus, from the monotony of F, we derive

dm<(θ1+θ2+θ4)dm-1+(θ3+θ4)dm,

and hence

(1-θ3-θ4)dm<(θ1+θ2+θ4)dm-1,

m ∈ ℕ. Because θ3 ≠ 1 and θ1 + θ2 + θ3 + 2θ4 = 1, we conclude that 1 − θ3θ4 > 0, and thus

dm<θ1+θ2+θ41-θ3-θ4dm-1=dm-1,

m ∈ ℕ. Therefore, the sequence {dm}m∈ℕ is strictly decreasing, and thus, limm→∞dm = d. Suppose that d > 0. Because F is an increasing mapping, limϖd+F(ϖ) = F(d + 0). Because m → ∞ in Eq. (5) , we obtain F(d0- + 0) ≤ F(d + 0) − ι. Therefore,

limmdm=0.

Next, we prove that {ϖm}m∈ℕ is an O-Cauchy sequence. Suppose that ε > 0 and sequences {i(m)}m∈ℕ and {j(m)}m∈ℕ of natural numbers exist such that i(m) > j(m) > m,

d(ϖi(m),ϖj(m))>ɛ,         d(ϖi(m)-1,ϖj(m))ɛ,m.

We derive

ɛ<d(ϖi(m),ϖj(m))d(ϖi(m),ϖi(m)-1)+d(ϖi(m)-1,ϖj(m))d(ϖi(m)-1,ϖj(m))+ɛ.

From Eq. (6), we obtain

limmd(ϖi(m),ϖj(m))=ɛ.

Because d(ϖi(m), ϖj(m)) > ε > 0 and the monotony of F, we obtain

ι+F(d(ϖi(m),ϖj(m)))F(θ1d(ϖi(m)-1,ϖj(m)-1)+θ2d(ϖi(m)-1,Ωϖi(m)-1)+θ3d(ϖj(m)-1,Ωϖj(m)-1)+θ4d(ϖi(m)-1,Ωϖj(m)-1)+θ5d(Ωϖi(m)-1,ϖj(m)-1))=F(θ1d(ϖi(m)-1,ϖj(m)-1)+θ2d(ϖi(m)-1,ϖi(m))+θ3d(ϖj(m)-1,ϖj(m))+θ4d(ϖi(m)-1,ϖj(m))+θ5d(ϖi(m),ϖj(m)-1))F(θ1[d(ϖi(m),ϖj(m))+di(m)-1+dj(m)-1]+θ2di(m)-1+θ3dj(m)-1+θ4[d(ϖi(m),ϖj(m))+di(m)-1]+θ5[d(ϖi(m),ϖj(m))+dj(m)-1])=F((θ1+θ4+θ5)d(ϖi(m),ϖj(m))+(θ1+θ2+θ4)di(m)-1+(θ1+θ3+θ5)dj(m)-1)F(d(ϖi(m),ϖj(m))+di(m)-1+(θ1+θ3+θ5)dj(m)-1).

Because m → ∞, we derive ι+F(ε+0) ≤ F(ε+0). Therefore, {ϖm}m∈ℕ is an O-Cauchy sequence. Because (Ξ, d) is an O-complete metric space, we find that {ϖm}m∈ℕ converges to a certain point t in Ξ. If there exists a sequence {i(m)}m∈ℕ of natural numbers such that ϖi(m)+1 = Ωϖi(m) = Ωt, then limm→∞ϖi(m)+1 = t, and Ωt = t. Otherwise, there exists M ∈ ℕ such that ϖm+1 = Ωϖm ≠ Ωt, ∀mM. We assume that Ωtt. From Eq. (2), we obtain the following:

ι+F(d(Ωϖm,Ωt))F(θ1d(ϖm,t)+θ2d(ϖm,Ωϖm)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(Ωϖm,t)),

and thus

ι+F(d(Ωϖm,Ωt))=F(θ1d(ϖm,t)+θ2d(ϖm,ϖm+1)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(ϖm+1,t)).

Because F is increasing, we derive

d(Ωϖm,Ωt)<θ1d(ϖm,t)+θ2d(ϖm,ϖm+1)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(ϖm+1,t).

Because m → ∞, we have

d(t,Ωt)θ3d(t,Ωt)+θ4d(t,Ωt)<d(t,Ωt).

Therefore, Ωt = t. Let t, rΞ be two distinct fixed-points of Ω. Thus, Ωt = tr = Ωr. Hence, dt, Ωr) = d(t, r) > 0. Because 0 < θ1 + θ4 + θ5 ≤ 1, we obtain

ι+F(d(t,r))=ι+F(d(Ωt,Ωr))F(θ1d(t,r)+θ2d(t,Ωt)+θ3d(r,Ωr)+θ4d(t,Ωr)+θ5d(r,Ωt))=F(θ1d(t,r)+θ4d(t,r)+θ5d(t,r))F((θ1+θ4+θ5)d(t,r)F(d(t,r)).

Therefore, Ω has a unique fixed point.

Corollary 3.2

Let (Ξ, ⊥, d) be an orthogonal-complete metric space and an orthogonal element ϖ0. Let Ω be a self-mapping of Ξ. Presume that F : ℝ+ → ℝ is an increasing mapping, and ι > 0 such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ)F(d(ϖ,ϱ)),

for all ϖ, ϱΞ, and Ωϖ ≠ Ωϱ, where Ω is ⊥-preserving. Then, Ω has a unique fixed-point in Ξ.

Example 3.3

Let Ξ = [0, 1]∪[2, 3] and d : Ξ × Ξ → [0, ∞) be a mapping defined by

d(ϖ,ϱ)=max{ϖ,ϱ},

for all ϖ, ϱΞ. We define the binary relation ⊥ on Ξ by ϖϱ if ϖϱ ≤ (ϖϱ), where ϖϱ = ϖ or ϱ. Then, (Ξ, d) is an O-complete metric space. The mapping Ω : ΞΞ is defined as

Ωϱ={1,if ϱ[0,1],1ϱ,if ϱ[2,3].

Clearly, Ω is ⊥-preserving. Now, let us consider mapping F defined by F(t) = ln t. Let ϖϱ and ι > 0. Without a loss of generality, we may assume that ϖϱϖ. Note that if dϖ, Ωϱ) > 0, then

ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ)),ϖ,ϱΞ

is equivalent to

d(Ωϖ,Ωϱ)e-ι(d(ϖ,ϱ)),ϖ,ϱΞ.

We are now considering the following cases:

Case 1

Letting ϖ = 0 and ϱ ∈ [0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Case 2

Letting ϖ = 0 and ϱ ∈ [2, 3], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=ϱ.

It is clear that Eq. (9) is satisfied.

Case 3

Letting ϖ ∈ [0, 1] and ϱ ∈ (0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Case 4

Letting ϖ ∈ [2, 3] and ϱ ∈ [0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Therefore, all conditions of Corollary (3.2) are satisfied. Hence, Ω has a unique fixed point in Ξ, that is, a point ϖ = 1.

Theorem 3.4

Let Ω be a self-mapping of an orthogonal-complete metric space Ξ. Suppose that ι > 0 exists such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ))],

where F : ℝ+ → ℝ is a mapping that satisfies the conditions (F2) and (F3), where (F3) F is continuous on (0, α), where α is a positive real number, and Ω is ⊥-preserving. Then, Ω has a unique fixed-point tΞ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to t.

Proof

Because (Ξ, ⊥) is an O-set,

ϖ0Ξ:(ϖΞ,ϖϖ0)   or   (ϖΞ,ϖ0ϖ).

It follows that ϖ0⊥Ωϖ0 or Ωϖ0ϖ0. Let

ϖ1:=Ωϖ0,,ϖm+1:=Ωϖm=Ωm+1ϖ0,

for all m ∈ ℕ ∪ {0}. If ϖm = ϖm+1 for any m ∈ ℕ ∪ {0}, then it is evident that ϖm is a fixed point of Ω. We assume that ϖmϖm+1 for all m ∈ ℕ ∪ {0}. Thus, we derive dϖm, Ωϖm+1) > 0 for all m ∈ ℕ ∪ {0}. Because Ω is ⊥-preserving, we have the following:

ϖmϖm+1   or   ϖm+1ϖm,

for all m ∈ ℕ ∪ {0}. This means that {ϖm} is an O-sequence. Hence, we presume that

0<d(ϖm,Ωϖm)=d(Ωϖm-1,Ωϖm),m

From Eq. (10), we obtain

ι+F(d(Ωϖm-1,Ωϖm))F(d(ϖm-1,ϖm)),m.

That is

ι+F(d(ωϖm-1,Ωϖm))F(d(ϖm-1,ϖm))-ι=F(d(Ωϖm-2,Ωϖm-1))-ιF(d(Ωϖm-2,ϖm-1))-2ι=F(d(Ωϖm-3,Ωϖm-2))-2ιF(d(Ωϖm-3,ϖm-2))-3ι=F(d(Ωϖm-4,Ωϖm-3))-3ιF(d(ϖ0,ϖ1))-mι.

This means that

limmF(d(ϖm,ϖm+1))=limmF(d(Ωϖm-1,Ωϖm))=-.

By (F2), we derive the following:

limmF(d(ϖm,ϖm+1))=0.

Next, we prove that {ϖm}m∈ℕ is an O-Cauchy sequence. Suppose that 0 < ε < α and sequences {i(m)}m∈ℕ and {j(m)}m∈ℕ of natural numbers exist such that

i(m)>j(m)>m,d(ϖi(m),ϖj(m))ɛ,d(ϖi(m)-1,ϖj(m))<ɛ,m.

Based on the proof of Theorem 3.1, we have

limmd(ϖi(m),ϖj(m))=limmd(ϖi(m)-1,ϖj(m)-1)=ɛ.

From (10), we obtain

ι+F(d(Ωϖi(m)-1,Ωϖj(m)0-1))F(d(ϖi(m)-1,ϖj(m)-1)),m.

This means

ι+F(d(Ωϖi(m),Ωϖj(m)))F(d(ϖi(m)-1,ϖj(m)-1)),m.

Because m → ∞, using (F3), we obtain ι + F(ε) ≤ F(ε). Therefore, {ϖm}m∈ℕ is an O-Cauchy sequence. Because (Ξ, d) is an orthogonal-complete metric space, {ϖm}m∈ℕ converges to a certain point t in Ξ. Next, we prove that t is a fixed-point of Ω. Otherwise, Ωtt. If a sequence {i(m)}m∈ℕ of natural numbers exists such that ϖi(m)+1 = Ωϖi(m) = Ωt, then limm→∞ϖi(m)+1 = t and Ωt = t. Therefore, there exists M ∈ ℕ such that ϖm+1 = Ωϖm ≠ Ωt, ∀mM. Then, from Eq. (10), we obtain

ι+F(d(ϖm+1,Ωt))F(d(ϖm,t)),m.

Because m → ∞, using (F3), we obtain limm→∞F(d(ϖm+1, Ωt)) = −∞. Consequently, by (F2), we obtain limm→∞d(ϖm+1, Ωt) = 0, which allows d(t, Ωt) = 0. Therefore, t is a fixed point of Ω, and the uniqueness yields the same results as in the proof of Theorem 3.1.

Recall that, for any 1 ≤ p < ∞, the space Lp(Ξ, F, λ) (or Lp(Ξ)) consists of all values of complex-valued measurable β on the underlying space Ξ satisfying

Ξβ(w)pdλ(w),

where F is the σ-algebra of the measurable sets, and λ is the measure. When p = 1, the space L1(Ξ) consists of all integrable functions β on Ξ, and we define the L1-norm of β as

β1=Ξβ(w)dλ(w).

In this section, using Corollary 3.2, we show the existence of a solution to the following differential equation:

{j(r)=H(r,j(r)),a.e.         rI:=[0,T];j(0)=c,c1,

where G : I × ℝ → ℝ is an integrable function that satisfies the following conditions:

  • (i) G(s, y) ≥ 0 for all y ≥ 0 and sI;

  • (ii) for each p, qL1(I) with p(s)q(s) ≥ l(s) or p(s)q(s) ≥ q(s) for all sI, there exist γL1(I) and ι > 0 such that

    G(s,p(s))-G(s,p(s))γ(s)1+ιγ(s)p(s)-q(s),

    and

    p(s)-q(s)γ(s)eU(s),sI,

    where U(s):=0sγ(l)dl.

Theorem 4.1

Consider the differential Eq. (16). If (i) and (ii) are satisfied, then the differential Eq. (16) has a unique positive solution.

Proof

Let Ξ = {wC(I, ℝ) : w(r) > 0 ∀rI}. The orthogonality relation ⊥ on Ξ is defined as

pqp(r)q(r)p(r)   or   p(r)q(r)q(r),rI.

Because U(r):=0rγ(s)ds, we have U′(r) = |γ(r)| for almost all rI. We define a mapping d : Ξ × Ξ → [0, ∞) as

d(p,q)=p-qU=suprIe-U(r)p(r)-q(r),p,qΞ.

Thus, (Ξ, d) is a metric space and a complete metric space. We define a mapping Q : ΞΞ by

(Qp)(r)=c+orG(s,p(s))ds.

Now, we show that Q is ⊥-preserving. For each p, qW with pq and rI, we have

(Qp)(r)=c+orG(s,p(s))ds1.

Accordingly, [(Qp)(r)][(Qq)(r)] ≥ (Qq)(r), and thus (Qp)(r)⊥(Qq)(r). Then, Q is ⊥-preserving. Let p, qΞ, where pq. Suppose that Q(p) ≠ Q(q). For each sI, we have

p(s)-q(s)γ(s)eU(s),

and thus

d(p,q)=supsIe-U(s)p(s)-q(s)γ(s).

Accordingly,

1+ιd(p,q)1+ιγ(s).

From (ii), for each rI, we derive

(Qp)(r)-(Qq)(r)0rG(s,p(s))-G(s,q(s))ds0rγ(s)1+ιγ(s)p(s)-q(s)ds0rγ(s)1+ιd(p,q)p(s)-q(s)e-U(s)eU(s)dsd(p,q)1+ιd(p,q)0rγ(s)eU(s)dsd(p,q)1+τd(p,q)(eU(r)-1),

and thus

e-U(r)(Qp)(r)-(Qq)(r)e-U(r)(eU(r)-1)d(p,q)1+ιd(p,q)=(1-e-U(r))d(p,q)1+ιd(p,q)(1-e-β1)d(p,q)1+ιd(p,q).

Therefore, we obtain

d(Qp,Qq)d(p,q)1+ιd(p,q),1+ιd(p,q)d(p,q)1d(Qp,Qq),ι+1d(p,q)1d(Qp,Qq),τ-1d(Qp,Qq)-1d(p,q).

Thus, F : ℝ+ → ℝ by F(b)=-1b for all b > 0. By Corollary 3.2, Q has a unique fixed point; hence, the differential Eq. (16) has a unique positive solution.

In this article, we demonstrated fixed-point theorems for orthogonal F-contraction mappings on an orthogonal-complete metric space.

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Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 243-250

Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.243

Copyright © The Korean Institute of Intelligent Systems.

Orthogonal F-Contraction Mapping on O-Complete Metric Space with Applications

Gunaseelan Mani1, Arul Joseph Gnanaprakasam2, Nasreen Kausar3, Mohammad Munir4, and Salahuddin5

1Department of Mathematics, Sri Sankara Arts and Science College (Autonomous), Madras University,Tamil Nadu, India
2Department of Mathematics, Faculty of Engineering and Technology, College of Engineering and Technology, SRM Institute of Science and Technology, Tamil Nadu, India
3Faculty of Arts and Science, Yildiz Technical University, ˙Istanbul, Turkey
4Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
5Department of Mathematics, Jazan University, Jazan, Saudi Arabia

Correspondence to:Nasreen Kausar (kausar.nasreen57@gmail.com)

Received: December 25, 2020; Revised: May 2, 2021; Accepted: June 14, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we introduce orthogonal concepts concerning F-contraction mappings and demonstrate some fixed-point theorems for self-mapping in a complete orthogonal metric space. Some well-known results in the literature are generalized and modified based on the demonstrated results. An example is provided to support our results, which are used in an application.

Keywords: Orthogonal set, Orthogonal complete metric space, Orthogonal continuous, Orthogonal preserving, Orthogonal F-contraction, Fixed point

1. Introduction

One of the most important results of a mathematical analysis is the famous fixed-point result, called Banach contraction theory. In several branches of mathematics, this is the most commonly used fixed-point result, and has been generalized in many different directions. One natural way to reinforce the Banach contraction concept is to replace the metric space with other generalized metric spaces. The fixed-point result in the setting of complete metric spaces was established by Wardowski [1], which is a generalization of the Banach contraction principle in metric spaces. However, in many branches of mathematics, the notion of an orthogonal set has many applications and has several methods of orthogonality. Gordji et al. [2] introduced the current concept of orthogonality in metric spaces and demonstrated the fixed-point result for contraction mappings in metric spaces equipped with the new orthogonality. In a generalized orthogonal metric space, Gordji and Habibi [3] proved the theory of fixed points. Sawangsup et al. [4] introduced the new concept of orthogonal F-contraction mappings and fixed-point theorems on orthogonal-complete metric space were proven. Orthogonal contractive-type mappings have been studied by many authors, and important results were obtained in [59].

In Section 2, for the subsequent usage, we remember some essential notes and definitions from the current literature. In Section 3, in an orthogonal complete metric space, we introduce a new F-contraction mapping and prove some fixed-point theorems for this contraction mapping. We apply our results to demonstrate the presence of a unique solution to the ordinary differential equation in Section 4.

2. Preliminaries

In this study, Ξ, ℝ+, and ℕ denote the non-empty set, positive real numbers, and positive integer sets, respectively.

First, we recall the notion of an auxiliary function, introduced in 2012 by Wardowski [1], as follows:

Definition 2.1[1]

Let (Ξ, d) be a metric space. A mapping Ω : ΞΞ is said to be an F-contraction if ι > 0 and F ∈ ϒ such that

ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ))

holds for any ϖ, ϱΞ with dϖ, Ωϱ) > 0, where ϒ is the set of all functions F : ℝ+ → ℝ satisfying the following properties:

  • (F1) F is strictly increasing;

  • (F2) for each sequence {ɛm} of positive numbers, we have

    limmɛm=0iff limmF(ɛm)=-;

    (F3) q(0,1)limɛ0+ɛqF(ɛ)=0.

In 2013, Secelean [10] demonstrated that, in Definition 2.1, the condition (F2) can be altered. The definition of an F-contraction of the Hardy-Rogers-type was introduced by Cosentino and Vetro [11]. In 2014, Piri and Kumam [12] demonstrated that in Definition 2.1, the condition (F3) can be altered. Later, in 2020, Popescu and Stan [13] proved some fixed point theorem concerning an F-contraction in a complete metric space as follows:

Theorem 2.1 [13]

Let Ω be a self-mapping of a complete metric space Ξ. Suppose that there exists ι > 0 such that ∀ ϖ, ϱΞ, dϖ, Ωϱ) > 0,

ι+F(d(Ωϖ,Ωϱ))F(θ1d(ϖ,ϱ)+θ2d(ϖ,Ωϖ)+θ3d(ϱ,Ωϱ)+θ4d(ϖ,Ωϱ)+θ5d(ϱ,Ωϖ)),

where F : ℝ+ → ℝ is an increasing mapping, θ1, θ2, θ3, θ4, θ5 ≥ 0, θ4<12, θ3 < 1, θ1 + θ2 + θ3 + 2θ4 = 1, 0 < θ1 + θ4 + θ5 ≤ 1. Then, Ω has a unique fixed point ϖ*Ξ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to ϖ*.

By contrast, the definition of an orthogonal set (or O set) and the properties of orthogonal sets were introduced by Gordji et al. [2] as follows:

Definition 2.2[2]

Let Ξφ and ⊥ ⊆ Ξ × Ξ be a binary relation. If ⊥ satisfies the following consecutive conditions,

ϖ0Ξ:(ϖΞ,ϖϖ0)         or         (ϖΞ,ϖ0ϖ),

then it is said to be an O set. We indicate this O set by (Ξ, ⊥).

Example 2.2[2]

Let Ξ = [0, ∞) and define ϖϱ if ϖϱ ∈ {ϖ, ϱ}. Then, by setting ϖ0 = 0 or ϖ0 = 1, (Ξ, ⊥) is an O-set.

Now, we present the concepts of an an orthogonal sequence (O-sequence),, an O-complete orthogonal metric space, and a ⊥-preserving mapping.

Definition 2.3 [2]

Let (Ξ, ⊥) be an O set. A sequence {ϖm} is said to be an O-sequence if

(m,ϖmϖm+1)         or         (m,ϖm+1ϖm).

Definition 2.4[2]

The triplet (Ξ, ⊥, d) is considered to be an orthogonal metric space if (Ξ, ⊥) is an O set, and (Ξ, d) is a metric space.

Definition 2.5 [2]

Let (Ξ, ⊥) be an O set. A mapping Ω : ΞΞ is called ⊥-preserving if Ωϖ⊥Ωϱ whenever ϖϱ.

In this study, we adapt the concepts of F-contraction mapping to orthogonal sets and prove some fixed-point theorems for an F-contraction mapping in the complete orthogonality of the metric space.

3. Main Results

In this section, we introduce a new F-contraction mapping, inspired by the concepts of F-contraction mapping, described by Popescu and Stan [13], and demonstrate some fixed-point theorems in an orthogonal-complete metric space for this contraction mapping.

Theorem 3.1

Let (Ξ, ⊥, d) be an orthogonal-complete metric space, and an orthogonal element ϖ0. Let Ω be a self-mapping of Ξ. Suppose that ι > 0 exists such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ))F(θ1d(ϖ,ϱ))+θ2d(ϖ,Ωϖ)+θ3d(ϖ,Ωϱ)+θ4d(ϖ,Ωϱ)+θ5d(ϱ,Ωϖ))],

where F : ℝ+ → ℝ is an increasing mapping; θ1, θ2, θ3, θ4, θ5 are non-negative numbers; and θ4<12, θ3 < 1, θ1 + θ2 + θ3 +2θ4 = 1, 0 < θ1 + θ4 + θ5 ≤ 1, where Ω is ⊥-preserving. Then, Ω has a unique fixed-point tΞ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to t.

Proof

Because (Ξ, ⊥) is an O-set,

ϖ0Ξ:(ϖΞ,ϖϖ0)         or         (ϖΞ,ϖ0ϖ).

It follows that ϖ0⊥Ωϖ0 or Ωϖ0ϖ0. Let

ϖ1:=Ωϖ0,,ϖm+1:=Ωϖm=Ωm+1ϖ0,

m ∈ ℕ∪{0}. If ϖm = ϖm+1 for any m ∈ ℕ∪{0}, it is then evident that ϖm is a fixed-point of Ω. We assume that ϖmϖm+1m ∈ ℕ∪{0}. Thus, we derive dϖm, Ωϖm+1) > 0 ∀m ∈ ℕ ∪ {0}. Because Ω is ⊥-preserving, we obtain

ϖmϖm+1         or         ϖm+1ϖm,

m ∈ ℕ ∪ {0}. This means that {ϖm} is an O-sequence. Hence, we presume that

0<d(ϖm,Ωϖm)=d(Ωϖm-1,Ωϖm),m.

Now, let dm = d(ϖm, ϖm+1). Through the monotony of F, we derive ∀m ∈ ℕ

ι+F(dm)=ι+F(d(ϖm,ϖm+1))=ι+F(d(Ωϖm-1,Ωϖm))F(θ1d(ϖm-1,ϖm)+θ2d(ϖm-1,Ωϖm-1)+θ3d(ϖm,Ωϖm)+θ4d(ϖm-1,Ωϖm)+θ5d(ϖm,Ωϖm-1))=F(θ1d(ϖm-1,ϖm)+θ2d(ϖm-1,ϖm)+θ3d(ϖm,ϖm+1)+θ4d(ϖm-1,ϖm+1)+θ5d(ϖm,ϖm))=F(θ1dm-1+θ2dm-1+θ3dm+θ4d(ϖm-1,ϖm+1))F(θ1dm-1+θ2dm-1+θ3dm+θ4(dm-1+dm))=F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm).

It proceeds that

F(dm)F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm)-ι<F((θ1+θ2+θ4)dm-1+(θ3+θ4)dm).

Thus, from the monotony of F, we derive

dm<(θ1+θ2+θ4)dm-1+(θ3+θ4)dm,

and hence

(1-θ3-θ4)dm<(θ1+θ2+θ4)dm-1,

m ∈ ℕ. Because θ3 ≠ 1 and θ1 + θ2 + θ3 + 2θ4 = 1, we conclude that 1 − θ3θ4 > 0, and thus

dm<θ1+θ2+θ41-θ3-θ4dm-1=dm-1,

m ∈ ℕ. Therefore, the sequence {dm}m∈ℕ is strictly decreasing, and thus, limm→∞dm = d. Suppose that d > 0. Because F is an increasing mapping, limϖd+F(ϖ) = F(d + 0). Because m → ∞ in Eq. (5) , we obtain F(d0- + 0) ≤ F(d + 0) − ι. Therefore,

limmdm=0.

Next, we prove that {ϖm}m∈ℕ is an O-Cauchy sequence. Suppose that ε > 0 and sequences {i(m)}m∈ℕ and {j(m)}m∈ℕ of natural numbers exist such that i(m) > j(m) > m,

d(ϖi(m),ϖj(m))>ɛ,         d(ϖi(m)-1,ϖj(m))ɛ,m.

We derive

ɛ<d(ϖi(m),ϖj(m))d(ϖi(m),ϖi(m)-1)+d(ϖi(m)-1,ϖj(m))d(ϖi(m)-1,ϖj(m))+ɛ.

From Eq. (6), we obtain

limmd(ϖi(m),ϖj(m))=ɛ.

Because d(ϖi(m), ϖj(m)) > ε > 0 and the monotony of F, we obtain

ι+F(d(ϖi(m),ϖj(m)))F(θ1d(ϖi(m)-1,ϖj(m)-1)+θ2d(ϖi(m)-1,Ωϖi(m)-1)+θ3d(ϖj(m)-1,Ωϖj(m)-1)+θ4d(ϖi(m)-1,Ωϖj(m)-1)+θ5d(Ωϖi(m)-1,ϖj(m)-1))=F(θ1d(ϖi(m)-1,ϖj(m)-1)+θ2d(ϖi(m)-1,ϖi(m))+θ3d(ϖj(m)-1,ϖj(m))+θ4d(ϖi(m)-1,ϖj(m))+θ5d(ϖi(m),ϖj(m)-1))F(θ1[d(ϖi(m),ϖj(m))+di(m)-1+dj(m)-1]+θ2di(m)-1+θ3dj(m)-1+θ4[d(ϖi(m),ϖj(m))+di(m)-1]+θ5[d(ϖi(m),ϖj(m))+dj(m)-1])=F((θ1+θ4+θ5)d(ϖi(m),ϖj(m))+(θ1+θ2+θ4)di(m)-1+(θ1+θ3+θ5)dj(m)-1)F(d(ϖi(m),ϖj(m))+di(m)-1+(θ1+θ3+θ5)dj(m)-1).

Because m → ∞, we derive ι+F(ε+0) ≤ F(ε+0). Therefore, {ϖm}m∈ℕ is an O-Cauchy sequence. Because (Ξ, d) is an O-complete metric space, we find that {ϖm}m∈ℕ converges to a certain point t in Ξ. If there exists a sequence {i(m)}m∈ℕ of natural numbers such that ϖi(m)+1 = Ωϖi(m) = Ωt, then limm→∞ϖi(m)+1 = t, and Ωt = t. Otherwise, there exists M ∈ ℕ such that ϖm+1 = Ωϖm ≠ Ωt, ∀mM. We assume that Ωtt. From Eq. (2), we obtain the following:

ι+F(d(Ωϖm,Ωt))F(θ1d(ϖm,t)+θ2d(ϖm,Ωϖm)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(Ωϖm,t)),

and thus

ι+F(d(Ωϖm,Ωt))=F(θ1d(ϖm,t)+θ2d(ϖm,ϖm+1)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(ϖm+1,t)).

Because F is increasing, we derive

d(Ωϖm,Ωt)<θ1d(ϖm,t)+θ2d(ϖm,ϖm+1)+θ3d(t,Ωt)+θ4d(ϖm,Ωt)+θ5d(ϖm+1,t).

Because m → ∞, we have

d(t,Ωt)θ3d(t,Ωt)+θ4d(t,Ωt)<d(t,Ωt).

Therefore, Ωt = t. Let t, rΞ be two distinct fixed-points of Ω. Thus, Ωt = tr = Ωr. Hence, dt, Ωr) = d(t, r) > 0. Because 0 < θ1 + θ4 + θ5 ≤ 1, we obtain

ι+F(d(t,r))=ι+F(d(Ωt,Ωr))F(θ1d(t,r)+θ2d(t,Ωt)+θ3d(r,Ωr)+θ4d(t,Ωr)+θ5d(r,Ωt))=F(θ1d(t,r)+θ4d(t,r)+θ5d(t,r))F((θ1+θ4+θ5)d(t,r)F(d(t,r)).

Therefore, Ω has a unique fixed point.

Corollary 3.2

Let (Ξ, ⊥, d) be an orthogonal-complete metric space and an orthogonal element ϖ0. Let Ω be a self-mapping of Ξ. Presume that F : ℝ+ → ℝ is an increasing mapping, and ι > 0 such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ)F(d(ϖ,ϱ)),

for all ϖ, ϱΞ, and Ωϖ ≠ Ωϱ, where Ω is ⊥-preserving. Then, Ω has a unique fixed-point in Ξ.

Example 3.3

Let Ξ = [0, 1]∪[2, 3] and d : Ξ × Ξ → [0, ∞) be a mapping defined by

d(ϖ,ϱ)=max{ϖ,ϱ},

for all ϖ, ϱΞ. We define the binary relation ⊥ on Ξ by ϖϱ if ϖϱ ≤ (ϖϱ), where ϖϱ = ϖ or ϱ. Then, (Ξ, d) is an O-complete metric space. The mapping Ω : ΞΞ is defined as

Ωϱ={1,if ϱ[0,1],1ϱ,if ϱ[2,3].

Clearly, Ω is ⊥-preserving. Now, let us consider mapping F defined by F(t) = ln t. Let ϖϱ and ι > 0. Without a loss of generality, we may assume that ϖϱϖ. Note that if dϖ, Ωϱ) > 0, then

ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ)),ϖ,ϱΞ

is equivalent to

d(Ωϖ,Ωϱ)e-ι(d(ϖ,ϱ)),ϖ,ϱΞ.

We are now considering the following cases:

Case 1

Letting ϖ = 0 and ϱ ∈ [0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Case 2

Letting ϖ = 0 and ϱ ∈ [2, 3], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=ϱ.

It is clear that Eq. (9) is satisfied.

Case 3

Letting ϖ ∈ [0, 1] and ϱ ∈ (0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Case 4

Letting ϖ ∈ [2, 3] and ϱ ∈ [0, 1], then

d(Ωϖ,Ωϱ)=1and d(ϖ,ϱ)=1.

It is obvious that Eq. (9) is satisfied.

Therefore, all conditions of Corollary (3.2) are satisfied. Hence, Ω has a unique fixed point in Ξ, that is, a point ϖ = 1.

Theorem 3.4

Let Ω be a self-mapping of an orthogonal-complete metric space Ξ. Suppose that ι > 0 exists such that

ϖ,ϱΞwith ϖϱ[d(Ωϖ,Ωϱ)>0ι+F(d(Ωϖ,Ωϱ))F(d(ϖ,ϱ))],

where F : ℝ+ → ℝ is a mapping that satisfies the conditions (F2) and (F3), where (F3) F is continuous on (0, α), where α is a positive real number, and Ω is ⊥-preserving. Then, Ω has a unique fixed-point tΞ, and for every ϖΞ, the sequence {Ωmϖ}m∈ℕ converges to t.

Proof

Because (Ξ, ⊥) is an O-set,

ϖ0Ξ:(ϖΞ,ϖϖ0)   or   (ϖΞ,ϖ0ϖ).

It follows that ϖ0⊥Ωϖ0 or Ωϖ0ϖ0. Let

ϖ1:=Ωϖ0,,ϖm+1:=Ωϖm=Ωm+1ϖ0,

for all m ∈ ℕ ∪ {0}. If ϖm = ϖm+1 for any m ∈ ℕ ∪ {0}, then it is evident that ϖm is a fixed point of Ω. We assume that ϖmϖm+1 for all m ∈ ℕ ∪ {0}. Thus, we derive dϖm, Ωϖm+1) > 0 for all m ∈ ℕ ∪ {0}. Because Ω is ⊥-preserving, we have the following:

ϖmϖm+1   or   ϖm+1ϖm,

for all m ∈ ℕ ∪ {0}. This means that {ϖm} is an O-sequence. Hence, we presume that

0<d(ϖm,Ωϖm)=d(Ωϖm-1,Ωϖm),m

From Eq. (10), we obtain

ι+F(d(Ωϖm-1,Ωϖm))F(d(ϖm-1,ϖm)),m.

That is

ι+F(d(ωϖm-1,Ωϖm))F(d(ϖm-1,ϖm))-ι=F(d(Ωϖm-2,Ωϖm-1))-ιF(d(Ωϖm-2,ϖm-1))-2ι=F(d(Ωϖm-3,Ωϖm-2))-2ιF(d(Ωϖm-3,ϖm-2))-3ι=F(d(Ωϖm-4,Ωϖm-3))-3ιF(d(ϖ0,ϖ1))-mι.

This means that

limmF(d(ϖm,ϖm+1))=limmF(d(Ωϖm-1,Ωϖm))=-.

By (F2), we derive the following:

limmF(d(ϖm,ϖm+1))=0.

Next, we prove that {ϖm}m∈ℕ is an O-Cauchy sequence. Suppose that 0 < ε < α and sequences {i(m)}m∈ℕ and {j(m)}m∈ℕ of natural numbers exist such that

i(m)>j(m)>m,d(ϖi(m),ϖj(m))ɛ,d(ϖi(m)-1,ϖj(m))<ɛ,m.

Based on the proof of Theorem 3.1, we have

limmd(ϖi(m),ϖj(m))=limmd(ϖi(m)-1,ϖj(m)-1)=ɛ.

From (10), we obtain

ι+F(d(Ωϖi(m)-1,Ωϖj(m)0-1))F(d(ϖi(m)-1,ϖj(m)-1)),m.

This means

ι+F(d(Ωϖi(m),Ωϖj(m)))F(d(ϖi(m)-1,ϖj(m)-1)),m.

Because m → ∞, using (F3), we obtain ι + F(ε) ≤ F(ε). Therefore, {ϖm}m∈ℕ is an O-Cauchy sequence. Because (Ξ, d) is an orthogonal-complete metric space, {ϖm}m∈ℕ converges to a certain point t in Ξ. Next, we prove that t is a fixed-point of Ω. Otherwise, Ωtt. If a sequence {i(m)}m∈ℕ of natural numbers exists such that ϖi(m)+1 = Ωϖi(m) = Ωt, then limm→∞ϖi(m)+1 = t and Ωt = t. Therefore, there exists M ∈ ℕ such that ϖm+1 = Ωϖm ≠ Ωt, ∀mM. Then, from Eq. (10), we obtain

ι+F(d(ϖm+1,Ωt))F(d(ϖm,t)),m.

Because m → ∞, using (F3), we obtain limm→∞F(d(ϖm+1, Ωt)) = −∞. Consequently, by (F2), we obtain limm→∞d(ϖm+1, Ωt) = 0, which allows d(t, Ωt) = 0. Therefore, t is a fixed point of Ω, and the uniqueness yields the same results as in the proof of Theorem 3.1.

4. Applications

Recall that, for any 1 ≤ p < ∞, the space Lp(Ξ, F, λ) (or Lp(Ξ)) consists of all values of complex-valued measurable β on the underlying space Ξ satisfying

Ξβ(w)pdλ(w),

where F is the σ-algebra of the measurable sets, and λ is the measure. When p = 1, the space L1(Ξ) consists of all integrable functions β on Ξ, and we define the L1-norm of β as

β1=Ξβ(w)dλ(w).

In this section, using Corollary 3.2, we show the existence of a solution to the following differential equation:

{j(r)=H(r,j(r)),a.e.         rI:=[0,T];j(0)=c,c1,

where G : I × ℝ → ℝ is an integrable function that satisfies the following conditions:

  • (i) G(s, y) ≥ 0 for all y ≥ 0 and sI;

  • (ii) for each p, qL1(I) with p(s)q(s) ≥ l(s) or p(s)q(s) ≥ q(s) for all sI, there exist γL1(I) and ι > 0 such that

    G(s,p(s))-G(s,p(s))γ(s)1+ιγ(s)p(s)-q(s),

    and

    p(s)-q(s)γ(s)eU(s),sI,

    where U(s):=0sγ(l)dl.

Theorem 4.1

Consider the differential Eq. (16). If (i) and (ii) are satisfied, then the differential Eq. (16) has a unique positive solution.

Proof

Let Ξ = {wC(I, ℝ) : w(r) > 0 ∀rI}. The orthogonality relation ⊥ on Ξ is defined as

pqp(r)q(r)p(r)   or   p(r)q(r)q(r),rI.

Because U(r):=0rγ(s)ds, we have U′(r) = |γ(r)| for almost all rI. We define a mapping d : Ξ × Ξ → [0, ∞) as

d(p,q)=p-qU=suprIe-U(r)p(r)-q(r),p,qΞ.

Thus, (Ξ, d) is a metric space and a complete metric space. We define a mapping Q : ΞΞ by

(Qp)(r)=c+orG(s,p(s))ds.

Now, we show that Q is ⊥-preserving. For each p, qW with pq and rI, we have

(Qp)(r)=c+orG(s,p(s))ds1.

Accordingly, [(Qp)(r)][(Qq)(r)] ≥ (Qq)(r), and thus (Qp)(r)⊥(Qq)(r). Then, Q is ⊥-preserving. Let p, qΞ, where pq. Suppose that Q(p) ≠ Q(q). For each sI, we have

p(s)-q(s)γ(s)eU(s),

and thus

d(p,q)=supsIe-U(s)p(s)-q(s)γ(s).

Accordingly,

1+ιd(p,q)1+ιγ(s).

From (ii), for each rI, we derive

(Qp)(r)-(Qq)(r)0rG(s,p(s))-G(s,q(s))ds0rγ(s)1+ιγ(s)p(s)-q(s)ds0rγ(s)1+ιd(p,q)p(s)-q(s)e-U(s)eU(s)dsd(p,q)1+ιd(p,q)0rγ(s)eU(s)dsd(p,q)1+τd(p,q)(eU(r)-1),

and thus

e-U(r)(Qp)(r)-(Qq)(r)e-U(r)(eU(r)-1)d(p,q)1+ιd(p,q)=(1-e-U(r))d(p,q)1+ιd(p,q)(1-e-β1)d(p,q)1+ιd(p,q).

Therefore, we obtain

d(Qp,Qq)d(p,q)1+ιd(p,q),1+ιd(p,q)d(p,q)1d(Qp,Qq),ι+1d(p,q)1d(Qp,Qq),τ-1d(Qp,Qq)-1d(p,q).

Thus, F : ℝ+ → ℝ by F(b)=-1b for all b > 0. By Corollary 3.2, Q has a unique fixed point; hence, the differential Eq. (16) has a unique positive solution.

5. Conclusion

In this article, we demonstrated fixed-point theorems for orthogonal F-contraction mappings on an orthogonal-complete metric space.

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