# Fixed Point Results forα,⊥F-Contractions in Orthogonal F-Metric Spaces with Applications

Authors: Jamshaid Ahmad; Ahmed Saleh Al-Rawashdeh; Abdullah Eqal Al-Mazrooei
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main objective of this article is to introduce the notion ofα,⊥F-contraction in the setting of orthogonal F-metric space and present some new fixed point results in such newly introduced space. We also furnish an example to manifest the originality of the obtained results. As application of our foremost result, we look into the solution of a nonlinear fractional differential equation.


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## Body

## 1. Introduction and Preliminaries

One of the most important part in the theory of fixed points is the underlying space as well as the contractive mapping and contractive condition. In 1905, a French mathematician Maurice Fréchet gave the study of metric space which plays a virtual role in the pioneer result in this theory. In the last few decadess, many researchers have presented interesting generalizations of metric spaces. Most of the generalizations are made by making some changes in the triangle inequality of the original definition. Some of these well-known generalizations of metric space areb-metric space due to Czerwik [1], rectangular metric space due to Branciari [2], and JS-metric space due to Jleli and Samet [3]. After all such generalizations, Jleli and Samet [4] gave a compulsive extension of a metric space which is known as F-metric space in this way.SupposeF be the set of functions ξ:0,+∞⟶ℝ satisfyingF10<ℏ1<ℏ2⟹ξℏ1≤ξℏ2F2 for all ℏn⊆ℝ+, limn⟶∞ℏn=0 if and only if limn⟶∞ξℏn=−∞Definition 1 (see [4]).
LetR be nonempty set, and let τ:R×R⟶0,+∞. Suppose that there exists ξ,α∈F×0,+∞ such that
(D1) ℏ,ς∈R×R, τℏ,ς=0 if and lnly if ℏ=ς
(D2) τℏ,ς=τς,ℏ, for all ℏ,ς∈R
(D3) for all ℏ,ς∈R×R, and ℏii=1N⊂R, with ℏ1,ℏN=ℏ,ς, we have
(1)τℏ,ς>0⟹ξτℏ,ς≤ξ∑i=1N−1τℏi,ℏi+1+α,for all N≥2. Then, R,τ is said to be an F-metric space (F-MS).Example 1 (see [4]).
LetR=ℝ. Then, τ:R×R⟶0,+∞ defined by
(2)τℏ,ς=ℏ−ς2ifℏ,ς∈0,3×0,3,ℏ−ςifℏ,ς∈0,3×0,3,with ξt=lnt and α=ln3 is an F-metric.Definition 2 (see [4]).
LetR,τ be an F-MS.
Letℏn be a sequence in R. Then, ℏn is said to be an F-convergent to ℏ∈R if ℏn is convergent to ℏ with regard to an F-metric τ.
(i)
A sequenceℏn is F-Cauchy, if(3)limn,m⟶∞τℏn,ℏm=0.
For more particulars in this way, we mention the researchers to [5–9].
On the other hand, Gordji et al. [10] initiated the conception of the orthogonal set (in short, O-set).Definition 3.
LetR be a non empty and ⊥⊆R×R. If R with the binary relation ⊥ satisfies the condition:
(4)Thereexistsℏ0forallς∈R,ς⊥ℏ0orforallς∈R,ℏ0⊥ς,then, it is said to be an orthogonal set. Moreover, ℏ0 is said to be an orthogonal point. We represent this O-set by R,⊥.Example 2 (see [10]).
LetR=ℤ. Define ⊥ on U by i⊥j if there exists l∈ℤ such that i=lj. It is very simple to note that 0⊥j, for all j∈ℤ. Thus, (R,⊥) is an O-set.Definition 4 (see [10]).
LetR,⊥ be an O-set. A sequence {ℏn} is called orthogonal sequence (in essence O-sequence) if
(5)foralln∈ℕ,ℏn⊥ℏn+1orforalln∈ℕ,ℏn+1⊥ℏn.
In this direction, these researchers [11–14] utilized O-sets in different ways to obtain their results.
Very recently, Kanwal et al. [15] combined both concepts and introduced the notion of orthogonal F-metric space in this way.Definition 5 (see. [15]).
Let (R,⊥) be an O-set and τ be an F-metric on R. Then, triplet R,⊥,τ is claimed to be an orthogonal F-metric space (OF-MS).Example 3 (see [15]).
LetR=0,1. Define F-metric τ by
(6)τℏ,ς=eℏ−ς,ifℏ≠ς,0,ifℏ=ς,for all ℏ,ς∈R,ξt=−1/t,t>0 and α=1. Define ℏ⊥ς if ℏς≤ℏ or ℏς≤ς. Then, for all ℏ∈R, 0⊥ς, so (R,⊥) is an O-set. Then, R,⊥,τ is an OF-MS.
From now onwards, we represent (R,⊥) as an O-set and R,⊥,τ an OF-MS.Definition 6 (see [15]).
A mappingV:R,⊥,ρ⟶R,⊥,ρ is professed to be orthogonality continuous (in short ⊥-continuous) at ℏ∈R if for each O-sequence {ℏn} in R if ℏn⟶ℏ, then Vℏn⟶Vℏ. Also V is ⊥-continuous on R if V is ⊥-continuous in each ℏ∈R.Definition 7 (see [15]).
A setR of R,⊥,τ is professed to be orthogonally F-complete (in short O-F-complete) if every Cauchy O-sequence is F-convergent.
In 2012, Samet et al. [16] gave the conception of α-admissible in this way.Definition 8.
A mappingV is called α-admissible whenever αℏ,ς≥1 implies αVℏ,Vς≥1.Later on, Ramezani [17] extended the above concept to orthogonal set as follows.Definition 9.
A mappingV is called orthogonally α-admissible whenever ℏ⊥ς and αℏ,ς≥1 implies αVℏ,Vς≥1.
We give the following property: (OH)αℏ,ς≥1 for any ℏ,ς∈ℏ∗∈R~ℏ∗=Vℏ∗.
On the other side, Wardowski [18] introduced the following new notion of F-contraction in 2012.Definition 10.
LetR,τ be a metric space. A mapping V:R⟶R is said to be a F-contraction if there exists ⅁>0 such that for ℏ,ς∈R:
(7)τVℏ,Vς>0⟹⅁+IτVℏ,Vς≤Iτℏ,ς,where I:ℝ+⟶ℝ is a mapping satisfying (F1), (F2), and (F3), that is,
(F3) there exists 0<r<1 such that limℏ⟶0+ℏrIℏ=0.We representΨ, the set of the functions I:ℝ+⟶ℝ satisfying (F1)-(F3). Later on, Secelean [19] and Piri and Kumam [20] replaced the conditions (F2) and (F3) with some weaker symmetrical conditions repectively. Popescu and Stan [21] obtained two new fixed point results for F-contractions with these weaker symmetrical conditions. Many authors [22–29] used the concept of F-contraction introduced by Wardowski in order to define and prove new results on fixed points in complete metric spaces.In this article, we introduce the notion ofα,⊥F-contraction in the background of orthogonal F-metric space (OF-MS) and obtain some new results in such newly introduced space. We also furnish an example to manifest the originality of the obtained results. As application of our foremost result, we look into the solution of a nonlinear fractional differential equation.


## 2. Main Results

Definition 11.
LetR,⊥,τ be an OF-MS. A mapping V:R⟶R is said to be α,⊥F-contraction if there exists I∈Ψ,α:R×R⟶1,∞ and ⅁>0 such that
(8)forallℏ,ς∈R,ℏ⊥ς,τVℏ,Vς≠0⟹⅁+αℏ,ςIτVℏ,Vς≤Iτℏ,ς.Theorem 12.
LetR,⊥,τ be an O-complete OF-MS and V:R⟶R is an α,⊥F-contraction such that following assertions hold:
(i)
V is ⊥-preserving, that is, ℏ⊥ς implies Vℏ⊥Vς(ii)
V is orthogonally α-admissible(iii)
there existsℏ0∈R such that ℏ0⊥Vℏ0 and αℏ0,Vℏ0≥1(iv)
V is ⊥-continuous
Then, there existsℏ∗∈R such that Vℏ∗=ℏ∗. Furthermore, if R has the property (OH), then this fixed point is unique.Proof.
By assertion (iii), there existsℏ0∈R such that ℏ0⊥Vℏ0 and αℏ0,Vℏ0≥1. Let the sequence {ℏn} is defined as
(9)ℏ1=Vℏ0,⋯,ℏn+1=Vℏn=Vn+1ℏ0,for all n≥0. As V is ⊥-preserving, so {ℏn} is an O-sequence in R. As V is orthogonally α-admissible, so it follows that αVℏn,Vℏn+1>1, for all n≥0. If ℏn=ℏn+1, for any n∈ℕ∪0, it is then evident that ℏn is a fixed point of V. So we suppose that ℏn=ℏn+1, for all n∈ℕ∪0. Hence we have τVℏn−1,Vℏn=τℏn,ℏn+1>0, for all n∈ℕ∪0. As V is ⊥-preserving, we get
(10)ℏn⊥ℏn+1orℏn+1⊥ℏn,for all n∈ℕ∪0. This means that {ℏn} is an O-sequence. Hence, we presume that
(11)0<τℏn,Vℏn=τVℏn−1,Vℏn,for all n∈ℕ∪0. From (8) and (11), we get
(12)⅁+Iτℏn,ℏn+1=⅁+IτVℏn−1,Vℏn≤⅁+αℏn−1,ℏnIτVℏn−1,Vℏn≤Iτℏn−1,ℏn,hence,
(13)⅁+Iτℏn,ℏn+1≤Iτℏn−1,ℏn,for all n∈ℕ∪0. Consequently, we have
(14)Iτℏn,ℏn+1≤Iτℏn−1,ℏn−⅁≤Iτℏn−2,ℏn−1−2⅁≤⋮≤Iτℏ0,ℏ1−n⅁,for all n∈ℕ∪0. Now by applying n⟶∞ and by (F2), we have
(15)limn⟶∞Iτℏn,ℏn+1=−∞⇔limn⟶∞τℏn,ℏn+1=0.
From the condition (F3), there exists 0<r<1 such that
(16)limn⟶∞τℏn,ℏn+1rIτℏn,ℏn+1=0.
From (14) and (16), we have
(17)τℏn,ℏn+1rIτℏn,ℏn+1−τℏn,ℏn+1rIτℏ0,ℏ1≤τℏn,ℏn+1rIτℏ0,ℏ1−nτ−τℏn,ℏn+1rIτℏ0,ℏ1≤−nτsndbℏn,ℏn+1r≤0.
Takingn⟶∞, we have
(18)limn⟶∞nτℏn,ℏn+1r=0.
Hence, there existsn1∈ℕ such that
(19)τℏn,ℏn+1≤1n1/r,for all n>n1. This yields
(20)∑i=nm−1τℏi,ℏi+1≤∑i=nm−11i1/rfor m>n. Since ∑i=n∞1/i1/r is convergent, so there exists N∈ℕ such that
(21)0<∑i=nm−11i1/r<∑i=n∞1i1/r<δ,for n>N. Hence, by (21) and (F1), we get
(22)ξ∑i=nm−1τℏi,ℏi+1≤ξ∑i=n∞1i1/r<ξε−a,
m>n≥N. Using (D3) and (22), we get
(23)τℏn,ℏm>0,m>n≥N⟹ξτℏn,ℏm≤ξ∑i=nm−1τℏi,ℏi+1+a<ξε,which, from (F1), gives that
(24)τℏn,ℏm<ε,m>n≥N.
Hence,ℏn is a Cauchy O-sequence in R,⊥,τ. The O-completeness of R,⊥,τ guarantees that there exists ℏ∗∈R such that limn⟶∞ℏn⟶ℏ∗. Now, we prove that ℏ∗ is fixed point of V. By ⊥-continuity of V gives Vℏn⟶Vℏ∗ as n⟶∞. Thus,
(25)Vℏ∗=limn⟶∞Vℏn=limn⟶∞ℏn+1=ℏ∗.
Thus,ℏ∗ is a fixed point of V. Lastly, we assume that ℏ/=Vℏ/ is once more fixed point of V such that ℏ/≠ℏ∗. From (OH), we have ℏ∗⊥ℏ/ or ℏ/⊥ℏ∗ and αℏ∗,ℏ/≥1. Thus, from (8), we have
(26)⅁+Iτℏ∗,ℏ/=⅁+IτVℏ∗,Vℏ/≤⅁+αℏ∗,ℏ/IτVℏ∗,Vℏ/≤Iτℏ/,ℏ∗,which implies that
(27)Iτℏ∗,ℏ/≤Iτℏ/,ℏ∗−⅁<Iτℏ/,ℏ∗,that is a contradiction. Thus, ℏ/=ℏ∗. Hence, the fixed point is unique.Corollary 13.
LetR,⊥,τ be an O-complete OF-MS. If V:R⟶R is ⊥-preserving and ⊥-continuous satisfying
(28)forallℏ,ς∈R,ℏ⊥ς,τVℏ,Vς≠0⟹⅁+IτVℏ,Vς≤Iτℏ,ς,then, there exists ℏ∗∈R such that Vℏ∗=ℏ∗ which is unique.Example 4.
Define the sequenceμn as follows:
(29)μ1=ln1,μ2=ln1+4,⋮μn=ln1+4+7+⋯+3n−2=lnn3n−12,for all n∈ℕ. Let R≪μn:n∈ℕ equipped with the F metric defined by
(30)τℏ,ς=eℏ−ς,ifℏ=ς,0,ifℏ=ς,
withξt=−1/t and a=1. For all μn,μm∈R, define μn⊥μm if and only if m≥2∧n=1. Hence, R,⊥,τ is an O-complete OF-MS. Define V:R⟶R by
(31)Vμn=μ1,ifn=1,μn−1,ifn>1,and α:R×R⟶1,+∞ by
(32)αμn,μm=1,ifμn≠μm,0,ifμn=μm.
Clearly,(33)limn⟶∞τVμn,Vμ1τμn,μ1=1,then V is not a contraction in the sense of [18].
It is easy to check thatV is ⊥-continuous and V is ⊥-preserving. Let the mapping I:0,∞⟶ℝ defined by
(34)It=lnt+t,t>0.
It is easy to show thatI∈Ψ. Now, to prove V is an ⊥F-contraction, that is
(35)τVμn,Vμm≠0⟹⅁+lnτVμn,Vμm+τVμn,Vμm≤lnτμn,μm+τμn,μm,for ⅁>0. The above condition is equivalent to
(36)τVμn,Vμm≠0⟹e⅁+lnτVμn,Vμm+τVμn,Vμm≤elnτμn,μm+τμn,μm.
So, we have to check that(37)τVμn,Vμm≠0⟹τVμn,Vμmτμn,μmeτVμn,Vμm−τμn,μm≤e−⅁.
For everym∈ℕ,m≥2, we have
(38)τVμm,Vμ1≠0⟹τVμm,Vμ1τμm,μ1eτVμm,Vμ1−τμm,μ1≤e−⅁τμm−1,μ1τμm,μ1eτμm−1,μ1−τμm,μ1=eμm−1−μ1eμm−μ1eeμm−1−μ1−eμm−μ1=m−13m−4m3m−1e−6m+4<e−1.
Thus, the inequality (8) is satisfied with ⅁=1>0. Hence, V is an (α,⊥F)-contraction. Thus, Theorem 12 implies that μ=ln1 is a unique fixed point of V.Example 5.
Define the sequenceμn as follows:
(39)μ1=log21,μ2=log21+log22,⋮μn=log21+log22+⋯+log2n=log2n!,for all n∈ℕ. Let R≪μn:n∈ℕ equipped with the F metric defined by
(40)τℏ,ς=2ℏ−ς,ifℏ≠ς,0,ifℏ=ς,
withξt=−1/t and a=1. For all μn,μm∈R, define μn⊥μm if and only if m≥2∧n=1. Hence, R,⊥,τ is an O-complete OF-MS. Define V:R⟶R by
(41)Vμn=μ1,ifn=1,μn−1,ifn>1..
Clearly,(42)limn⟶∞τVμn,Vμ1τμn,μ1=1,then V is not a contraction in the sense of [18].It is easy to check thatV is ⊥-continuous and V is ⊥-preserving. Let the mapping I:0,∞⟶ℝ defined by
(43)It=lnt+t,t>0.It is easy to show thatI∈Ψ. Now, to prove V is an ⊥F-contraction, that is,
(44)τVμn,Vμm≠0⟹⅁+lnτVμn,Vμm+τVμn,Vμm≤lnτμn,μm+τμn,μm,for ⅁>0. The above condition is equivalent to
(45)τVμn,Vμm≠0⟹e⅁+lnτVμn,Vμm+τVμn,Vμm≤elnτμn,μm+τμn,μm.So, we have to check that(46)τVμn,Vμm≠0⟹τVμn,Vμmτμn,μmeτVμn,Vμm−τμn,μm≤e−⅁.For everym∈ℕ,m≥2, we have
(47)τVμm,Vμ1τμm,μ1eτVμm,Vμ1−τμm,μ1=τμm−1,μ1τμm,μ1eτμm−1,μ1−τμm,μ1=2μm−1−μ12μm−μ1e2μm−1−μ1−2μm−μ1=m−1!m!e−m−1m+1!<e−1.For everym∈ℕ,m>n>1, we have
(48)τVμn,Vμmτμn,μmeτVμn,Vμm−τμn,μm=τμn−1,μm−1τμn,μmeτμn−1,μm−1−τμn,μm=2μm−1−μn−12μm−μne2μm−1−μn−1−2μm−μn=m−1!n−1!×n!m!em−1!/n−1!−n!/m!=nmem−1!/n−1!n−m/n<e−1.Thus, the inequality (8) is satisfied with ⅁=1>0. Thus, Corollary 13 implies that μ=log21 is a unique fixed point of V.Theorem 14.
LetR,τ be an F-complete F-metric space, V:R⟶R be an α-admissible mapping and there exist I∈Ψ and α:R×R⟶1,∞ such that
(49)τVℏ,Vς≠0⟹⅁+αℏ,ςIτVℏ,Vς≤Iτℏ,ς,for all ℏ,ς∈R and ⅁>0. Assume that there exists ℏ0∈R such that αℏ0,Vℏ0≥1, then, there exists ℏ∗∈R such that Vℏ∗=ℏ∗. Furthermore, if R has the property (OH), then this fixed point is unique.Proof.
Suppose that(50)ℏ⊥ςifandonlyifτVℏ,Vς≠0.
Fixℏ0∈R. Since V satisfies the inequality (49), for all ς∈R,ℏ0⊥ς. This implies that R,⊥ is O-set. Now, it is obviously that R is O-complete and V is ⊥-continuous and ⊥-preserving. Thu, by applying Theorem 12, V has a unique fixed point in R.Corollary 15 (see [30]).
LetR,τ be an F-complete F-metric space, V:R⟶R be an α-admissible mapping and there exist I∈Ψ such that
(51)τVℏ,Vς≠0⟹⅁+IτVℏ,Vς≤Iτℏ,ς,for all ℏ,ς∈R and ⅁>0, then, there exists unique ℏ∗∈R such that Vℏ∗=ℏ∗.
If we takeξt=lnt, for t>0 and α=1 in Definition 5, then OF-MS reduces to O-metric space and we get the following result.Theorem 16.
LetR,⊥,τ be an O-complete metric space and V:R⟶R is an α,⊥F-contraction such that following assertions hold:
(i)
V is ⊥-preserving, that is, ℏ⊥ς implies Vℏ⊥Vς(ii)
V is orthogonally α-admissible(iii)
there existsℏ0∈R such that ℏ0⊥Vℏ0 and αℏ0,Vℏ0≥1(iv)
V is ⊥-continuous
Then, there existsℏ∗∈R such that Vℏ∗=ℏ∗. Furthermore, if R has the property (OH), then this fixed point is unique.
If we takeαℏ,ς=1, for all ℏ,ς∈R in Theorem 16, then we get the following result of Mani et al. [31].Corollary 17.
LetR,⊥,τ be an O-complete metric space, V:R⟶R and there exist I∈Ψ such that
(52)forallℏ,ς∈R,ℏ⊥ς,τVℏ,Vς≠0⟹⅁+αℏ,ςIτVℏ,Vς≤Iτℏ,ς.
Assume that the following conditions hold:(i)
V is ⊥-preserving, that is, ℏ⊥ς implies Vℏ⊥Vς(ii)
V is orthogonally α-admissible(iii)
there existsℏ0∈R such that ℏ0⊥Vℏ0 and αℏ0,Vℏ0≥1(iv)
V is ⊥-continuous
Then, there existsℏ∗∈R such that Vℏ∗=ℏ∗. Furthermore, if R has the property (OH), then this fixed point is unique.


### 2.1. Periodic Point Result

LetR,⊥,τ be an OF-MS and V:R⟶R such that ℏ=Vℏ, then ℏ=Vnℏ, for every n∈ℕ. Although, the converse of this reality is not correct generally. The mapping V:R⟶R fulfilling
(53)FixV=FixVn,for all n∈ℕ is called to have property P.Definition 18.
LetR,⊥,τ be an OF-MS and V:R⟶R. The set
(54)Oℏ=ℏ,Vℏ,V2ℏ,⋯,Vnℏ,⋯,is said to be an orbit of R. A mapping V:R,⊥,τ⟶R,⊥,τ is said to be an orbitally O-continuous at ℏ if for each O-sequence Vnℏ in R,limn⟶∞Vnℏ=ς⟹limn⟶∞Vn+1ℏ=Vς. A mapping V:R⟶R is orbitally continuous on R if V is orbitally O-continuous at all ℏ∈R.Theorem 19.
LetR,⊥,τ be an O-complete OF-MS and V:R,⊥,τ⟶R,⊥,τ. Suppose that there exist I∈Ψ and ⅁>0 such that
(55)forallℏ∈RwithVℏ⊥V2ℏτVℏ,V2ℏ>0,⅁+IτVℏ,V2ℏ≤Iτℏ,Vℏ,and V is ⊥-preserving. Then, V has the property P supplied that V is orbitally continuous on R.Proof.
Letε>0 be fixed and there exists ξ:0,+∞⟶ℝ such that (D3) is satisfied for α≥0. Then, by (F2), there exists δ>0 such that
(56)0<ℏ<δimpliesξℏ<ξε−a.
Now, we manifest thatFixV≠∅. Define {ℏn} in R such that ℏn+1=Vnℏ. If there exists n0∈ℕ, such that ℏn0+1=ℏn0, then ℏn0 is a fixed point of V. Now, we assume that τℏn,ℏn+1>0, for all n∈ℕ. Using inequality (14), we get
(57)⅁+Iτℏn,ℏn+1=⅁+IτVℏn−1,V2ℏn−1≤Iτℏn−1,Vℏn−1.
This implies that(58)Iτℏn,ℏn+1≤Iτℏn−1,Vℏn−1−⅁≤Iτℏn−2,Vℏn−2−2⅁⋯≤Iτℏ0,ℏ1−n⅁.
Thus, we have(59)Iτℏn,ℏn+1≤Iτℏ0,ℏ1−n⅁,for all n∈ℕ. Now, by applying the limit as n⟶∞ and using (F2), we have
(60)limn⟶∞Iτℏn,ℏn+1=−∞⇔limn⟶∞τℏn,ℏn+1=0.
From the condition (F3), there exists 0<r<1 such that
(61)limn⟶∞τℏn,ℏn+1rIτℏn,ℏn+1=0.
From (59) and (61), we have
(62)τℏn,ℏn+1rIτℏn,ℏn+1−τℏn,ℏn+1rIτℏ0,ℏ1≤τℏn,ℏn+1rIτℏ0,ℏ1−nτ−τℏn,ℏn+1rIτℏ0,ℏ1≤−nτsndbℏn,ℏn+1r≤0.
Takingn⟶∞, we have
(63)limn⟶∞nτℏn,ℏn+1r=0.
Hence, there existsn1∈ℕ such that
(64)τℏn,ℏn+1≤1n1/r,for all n>n1. This yields
(65)∑i=nm−1τℏi,ℏi+1≤∑i=nm−11i1/r,for m>n. Since ∑i=n∞1/i1/r is convergent, so there exists N∈ℕ such that
(66)0<∑i=nm−11i1/r<∑i=n∞1i1/r<δ,for n>N. Hence, by (64) and (F1), we get
(67)ξ∑i=nm−1τℏi,ℏi+1≤ξ∑i=n∞1i1/r<ξε−a,
m>n≥N. Using (D3) and (67), we get
(68)τℏn,ℏm>0,m>n≥N⟹ξτℏn,ℏm≤ξ∑i=nm−1τℏi,ℏi+1+a<ξε,which, from (F1), gives that
(69)τℏn,ℏm<ε,m>n≥N.
Hence,ℏn is a Cauchy O-sequence in R,⊥,τ. As Vnℏ0:n∈ℕ⊆Oℏ0⊆R and R is O-complete, there exists ℏ∈R such that
(70)limn⟶∞Vnℏ0=ℏ.
Now, by the orbital⊥-continuity of V, we have
(71)ℏ=limn⟶∞Vn−1ℏ0=Vℏ.
Thus,V has a fixed point and FixVn=FixV is true for n=1. Suppose n>1. We assume on the contrary that ℏ∈FixVn but ℏ∈FixV, then τℏ,Vℏ=ω>0. Now,
(72)Iτℏ,Vℏ=IτVVn−1ℏ,V2Vn−1ℏ≤IτVn−1ℏ,Vnℏ−⅁≤IτVn−2ℏ,Vn−1ℏ−2⅁⋯≤Iτℏ,Vℏ−n⅁.
Thus, we have(73)Iτℏ,Vℏ≤Iτℏ,Vℏ−n⅁,for all n∈ℕ. Taking n⟶∞, we have
(74)limn⟶∞Iτℏ,Vℏ=−∞,which, from (F2) we get
(75)τℏ,Vℏ=0,which is contradiction. So, ℏ∈Vℏ.


## 2.1. Periodic Point Result

LetR,⊥,τ be an OF-MS and V:R⟶R such that ℏ=Vℏ, then ℏ=Vnℏ, for every n∈ℕ. Although, the converse of this reality is not correct generally. The mapping V:R⟶R fulfilling
(53)FixV=FixVn,for all n∈ℕ is called to have property P.Definition 18.
LetR,⊥,τ be an OF-MS and V:R⟶R. The set
(54)Oℏ=ℏ,Vℏ,V2ℏ,⋯,Vnℏ,⋯,is said to be an orbit of R. A mapping V:R,⊥,τ⟶R,⊥,τ is said to be an orbitally O-continuous at ℏ if for each O-sequence Vnℏ in R,limn⟶∞Vnℏ=ς⟹limn⟶∞Vn+1ℏ=Vς. A mapping V:R⟶R is orbitally continuous on R if V is orbitally O-continuous at all ℏ∈R.Theorem 19.
LetR,⊥,τ be an O-complete OF-MS and V:R,⊥,τ⟶R,⊥,τ. Suppose that there exist I∈Ψ and ⅁>0 such that
(55)forallℏ∈RwithVℏ⊥V2ℏτVℏ,V2ℏ>0,⅁+IτVℏ,V2ℏ≤Iτℏ,Vℏ,and V is ⊥-preserving. Then, V has the property P supplied that V is orbitally continuous on R.Proof.
Letε>0 be fixed and there exists ξ:0,+∞⟶ℝ such that (D3) is satisfied for α≥0. Then, by (F2), there exists δ>0 such that
(56)0<ℏ<δimpliesξℏ<ξε−a.
Now, we manifest thatFixV≠∅. Define {ℏn} in R such that ℏn+1=Vnℏ. If there exists n0∈ℕ, such that ℏn0+1=ℏn0, then ℏn0 is a fixed point of V. Now, we assume that τℏn,ℏn+1>0, for all n∈ℕ. Using inequality (14), we get
(57)⅁+Iτℏn,ℏn+1=⅁+IτVℏn−1,V2ℏn−1≤Iτℏn−1,Vℏn−1.
This implies that(58)Iτℏn,ℏn+1≤Iτℏn−1,Vℏn−1−⅁≤Iτℏn−2,Vℏn−2−2⅁⋯≤Iτℏ0,ℏ1−n⅁.
Thus, we have(59)Iτℏn,ℏn+1≤Iτℏ0,ℏ1−n⅁,for all n∈ℕ. Now, by applying the limit as n⟶∞ and using (F2), we have
(60)limn⟶∞Iτℏn,ℏn+1=−∞⇔limn⟶∞τℏn,ℏn+1=0.
From the condition (F3), there exists 0<r<1 such that
(61)limn⟶∞τℏn,ℏn+1rIτℏn,ℏn+1=0.
From (59) and (61), we have
(62)τℏn,ℏn+1rIτℏn,ℏn+1−τℏn,ℏn+1rIτℏ0,ℏ1≤τℏn,ℏn+1rIτℏ0,ℏ1−nτ−τℏn,ℏn+1rIτℏ0,ℏ1≤−nτsndbℏn,ℏn+1r≤0.
Takingn⟶∞, we have
(63)limn⟶∞nτℏn,ℏn+1r=0.
Hence, there existsn1∈ℕ such that
(64)τℏn,ℏn+1≤1n1/r,for all n>n1. This yields
(65)∑i=nm−1τℏi,ℏi+1≤∑i=nm−11i1/r,for m>n. Since ∑i=n∞1/i1/r is convergent, so there exists N∈ℕ such that
(66)0<∑i=nm−11i1/r<∑i=n∞1i1/r<δ,for n>N. Hence, by (64) and (F1), we get
(67)ξ∑i=nm−1τℏi,ℏi+1≤ξ∑i=n∞1i1/r<ξε−a,
m>n≥N. Using (D3) and (67), we get
(68)τℏn,ℏm>0,m>n≥N⟹ξτℏn,ℏm≤ξ∑i=nm−1τℏi,ℏi+1+a<ξε,which, from (F1), gives that
(69)τℏn,ℏm<ε,m>n≥N.
Hence,ℏn is a Cauchy O-sequence in R,⊥,τ. As Vnℏ0:n∈ℕ⊆Oℏ0⊆R and R is O-complete, there exists ℏ∈R such that
(70)limn⟶∞Vnℏ0=ℏ.
Now, by the orbital⊥-continuity of V, we have
(71)ℏ=limn⟶∞Vn−1ℏ0=Vℏ.
Thus,V has a fixed point and FixVn=FixV is true for n=1. Suppose n>1. We assume on the contrary that ℏ∈FixVn but ℏ∈FixV, then τℏ,Vℏ=ω>0. Now,
(72)Iτℏ,Vℏ=IτVVn−1ℏ,V2Vn−1ℏ≤IτVn−1ℏ,Vnℏ−⅁≤IτVn−2ℏ,Vn−1ℏ−2⅁⋯≤Iτℏ,Vℏ−n⅁.
Thus, we have(73)Iτℏ,Vℏ≤Iτℏ,Vℏ−n⅁,for all n∈ℕ. Taking n⟶∞, we have
(74)limn⟶∞Iτℏ,Vℏ=−∞,which, from (F2) we get
(75)τℏ,Vℏ=0,which is contradiction. So, ℏ∈Vℏ.


## 3. Applications

Consider a nonlinear differential equation of fractional order(76)CDηℏt=ft,ℏt.0<t<1,1<η≤2 via the integral boundary conditions
(77)ℏ0=0,ℏ1=∫0πℏsds,0<π<1,where CDη represents the Caputo fractional derivative of order η given by
(78)CDηft=1Γj−η∫0tt−sj−η−1fjsds,j−1<η<j,j=η+1, where η represents the integer part of the real number η and f:0,1×ℝ⟶ℝ is continuous. We take
(79)R≪ℏ:ℏ∈C0,1,ℝ,with supremum norm ℏ∞=supt∈0,1ℏt. Thus, R,ℏ∞ is a Banach space. Recall, the Riemann–Liouville fractional integral of order η is given by
(80)Iηft=1Γη∫0tt−sη−1fsds,withη>0.Lemma 20 (see [15]).
The Banach spaceR,·∞ equipped with the F-metric d given by
(81)dℏ,ς=ℏ−ς∞=supt∈0,1ℏt−ςt,and orthogonal relation ℏ⊥ς⇔ℏς≥0, where ℏ,ς∈R, is an orthogonal F-metric space.Theorem 21.
Suppose that(i)
there exists⅁∈1,∞ such that(82)ft,ℏ−ft,ς≤Γη+15e−⅁ℏ−ς,for all t∈0,1(ii)
there existsL:R,⊥,d⟶R,⊥,d is defined by(83)Vℏt=1Γη∫0tt−sη−1fs,ℏsds−2t2−λ2Γη∫011−sη−1fs,ℏsds+2t2−λ2Γη∫0λ∫0ss−mη−1fm,ℏmdmds,for all ℏ,ς∈R such that ℏtςt≥0,where (0<λ<1). Then, (76) has at least one solutionProof.
It is well known thatℏ∈R is a solution of (76) iff ℏ∈R is a solution of the integral equation
(84)ℏt=1Γη∫0tt−sη−1fs,ℏsds−2t2−λ2Γη∫011−sη−1fs,ℏsds+2t2−λ2Γη∫0λ∫0ss−mη−1fm,ℏmdmds.
Then, problem (76) is equivalent to find ℏ∈R which is a fixed point of V . Suppose that a relation on R, by
(85)ℏ⊥ς⇔ℏtςt≥0,for all t∈0,1. With this relation, R is orthogonal because for all ℏ∈R, there exists ςt=0, for all t∈0,1 such that ℏtςt=0. We examine
(86)dℏ,ς=ℏ−ς∞=supt∈0,1ℏt−ςt,for all ℏ,ς∈R. So, the triplet R,⊥,d is a complete OF-MS. It is clear from the definition that V is ⊥-continuous. We first prove that V is ⊥-preserving. Let ℏt⊥ςt, for all t∈0,1. Now, we get
(87)Vℏt=1Γη∫0tt−sη−1fs,ℏsds−2t2−λ2Γη∫011−sη−1fs,ℏsds+2t2−λ2Γη∫0λ∫0ss−mη−1fm,ℏmdmds>0,which yields that Vℏt⊥Vςt, i.e., V is ⊥-preserving. Next, for all t∈0,1,ℏt⊥ςt, we have
(88)Vℏt−Vςt=1Γη∫0tt−sη−1fs,ℏsds−2t2−λ2Γη∫011−sη−1fs,ℏsds+2t2−λ2Γη∫0λ∫0ss−mη−1fm,ℏmdmds−1Γη∫0tt−sη−1fs,ςsds+2t2−λ2Γη∫011−sη−1fs,ςsds−2t2−λ2Γη∫0λ∫0ss−mη−1fm,ςmdmds≤1Γη∫0tt−sη−1fs,ℏs−fs,ςsds+2t2−λ2Γη∫011−sη−1fs,ℏs−fs,κsds+2t2−λ2Γη∫0λ∫0ss−mη−1fm,ςm−fm,ℏmdmds,which implies that
(89)Jℏt−Jςt≤1Γη∫0tt−sη−1Γη+15e−πℏs−ςsds+2t2−λ2Γη∫011−sη−1Γη+15e−πℏs−ςsds+2t2−λ2Γη∫0λ0ss−mη−1Γη+15e−πςm−ℏmdmds≤Γη+15e−⅁ℏs−ςs∞supt∈0,11Γη∫0tt−sη−1ds+2t2−λ2Γη∫011−sη−1ds+2t2−λ2Γη∫0λ∫0ss−mη−1dmds≤e−⅁ℏ−ς∞.
From the above inequality, we obtain that(90)dVℏ,Vς≤e−⅁dℏ,ς.
Taking natural log function on both sides, we have(91)lndVℏ,Vς≤lne−⅁dℏ,ς,that is
(92)lndVℏ,Vς≤lne−⅁+lndℏ,ς,that is
(93)lndVℏ,Vς≤−⅁+lndℏ,ς.
Thus,(94)⅁+lndVℏ,Vς≤lndℏ,ς,where ⅁>1. Now, consider I:ℝ+⟶ℝ given by It=lnt for each t>0, then I∈Ψ. Thus,
(95)⅁+IdVℏ,Vς≤Idℏ,ς,for all ℏ,ς∈R and dVℏ,Vς>0.
Now, let (ℏn) be a Cauchy O-sequence converging in R. Thus, we get ℏntℏn+1t≥0, for all t∈0,1 and for n∈ℕ.We will have two possible cases to discuss:Case 1.
Ifℏnt≥0, for all n∈ℕ and t∈0,1. Then, for all t∈0,1, there exists a sequence of positives that converges to ℏt. Thus, we obtain ℏt≥0, for all t∈0,1, i.e., ℏnt⊥ℏt, for all n∈ℕ and t∈0,1.Case 2.
Ifℏnt≤0, for all n∈ℕ, has to be thrown out. Hence, by Corollary 13, Equation (76) has a unique solution.


## 4. Conclusion

In this article, we introduced the notion ofα,⊥F-contraction in the context of orthogonal F-metric space and established some new fixed point theorems in this newly introduced space. We have given soeme examples to manifest the authenticity of the obtained results. As application of our foremost result, we looked into the solution of a nonlinear fractional differential equation.In this direction, our future work will focus on studying the fixed points forα,⊥F-contraction in the context of orthogonal F-metric space endowed with a graph.



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