# Solution of Fractional Differential Equations Utilizing Symmetric Contraction

Authors: Aftab Hussain
Journal: Journal of Mathematics
 (2021)

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## Abstract

The aim of this paper is to present another family of fractional symmetricα-η-contractions and build up some new results for such contraction in the context of ℱ-metric space. The author derives some results for Suzuki-type contractions and orbitally T-complete and orbitally continuous mappings in ℱ-metric spaces. The inspiration of this paper is to observe the solution of fractional-order differential equation with one of the boundary conditions using fixed-point technique in ℱ-metric space.


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## Body

## 1. Preliminaries and Scope

Fixed-point theory has been promoted by a few particular works in the most recent decades [1–3]. One of the intriguing methodologies was presented in Karapinar et al.’s work [4] which starts a thought of interpolative kind of contractions and sest up shiny new fixed-point results in partial metric space. Recently, Jleli and Samet [5] introduced a new generalization of metric space and named it as ℱ-metric space.Definition 1.
(see [5]). Let ℱ be the set function f:0,+∞⟶−∞,+∞ that meets the following conditions:ℱ1f is nondecreasing; that is, for all 0<c<d, it implies fc≤fd.ℱ2 For each iteration dn⊂0,+∞, we have(1)limn⟶+∞dn=0,if and only iflimn⟶+∞fdn=−∞.
The generalized notion of metric space is as follows.Definition 2.
(see [5]). Let A≠∅ with D:A×A⟶0,+∞ be a given mapping. Suppose that there exists f,μ∈ℱ×0,+∞ such thatD1w,υ∈A×A, Dw,υ=0⇔w=υ.D2Dw,υ=Dυ,w for all w,υ∈A×A.D3 Each w,υ∈A×A, ∀N∈ℕ, N≥2, and for each uii=1N⊂A with u1,uN=w,υ, we have(2)Dw,υ>0impliesfDw,υ≤f∑i=1N−1dui,ui+1+μ.
Then, it is said thatD is an ℱ-metric on A.
Here, the pairA,D is called an ℱ-metric space and it is abbreviated as ℱ-MS. A sequence wn in A,D is ℱ-Cauchy, if limn,m⟶∞Dwn,wm=0. Furthermore, A,D is ℱ-complete, if every ℱ -Cauchy sequence is ℱ-convergent in A.
The following example is stated in [5].Example 1.
(see [5]). The set of natural numbers ℕ=A is an ℱ -MS if we define D by(3)Dw,υ=w−υ2,ifw,υ∈0,3×0,3,w−υ,ifw,υ∉0,3×0,3,for all w,υ∈A×A, ft=lnt, and μ=ln3. Moreover, D does not form a metric but it is an ℱ-MS.
Jleli and Samet proposed a simple Banach fixed-point theorem as follows.Theorem 1.
(see [5]). Let A,D be an ℱ-MS. Let g:A⟶A be a self-mapping. Suppose that the following conditions are met:(i)
A,D is ℱ-complete.(ii)
∃ a constant k∈0,1 such that(4)Dgw,gυ≤kDw,υ,w,υ∈A×A.
Then,g attains a unique fixed-point w∗∈A.
In 2012, Samet et al. introduced a class ofα-admissible mappings as follows.Definition 3.
(see [6]). Let T:A⟶A and α:A×A⟶0,+∞. T is said to be α-admissible if w,υ∈A, and αw,υ≥1 implies that αTw,Tυ≥1.
Next, Salimi et al. [7] modified the concept of α -admissible mapping as follows.Definition 4.
(see [7]). Let T:A⟶A and α,η:A×A⟶0,+∞ be two functions. T is called an α -admissible mapping with respect to η, if w,υ∈A, and αw,υ≥ηw,υ implies that αTw,Tυ≥ηTw,Tυ.
Ifηw,υ=1, then the above definition reduces to Definition 3. If αw,υ=1, then T is called an η-subadmissible mapping.Definition 5.
(see [8]). Consider a metric space A,d and assume that T:A⟶A and α,η:A×A⟶0,∞ are two functions. A mapping T is considered as α-η-continuous mapping in A,d whenever w∈A is given, and the sequence wn is as follows:(5)wn⟶wat∞,αwn,wn+1≥ηwn,wn+1,∀n∈ℕimpliesTwn⟶Tw.
For more details, see, for example, [9, 10].
A mappingT:A⟶A is called orbitally continuous in v∈A if limn⟶∞Tnw=v implies that limn⟶∞Tnw=Tv. T mapping is orbitally continuous on A if T is orbitally continuous ∀v∈A.


## 2. Fractional Symmetricα-η-Contraction of Type-I

In this segment, first we present a new fractional symmetricα-η-contraction of type-I.Definition 6.
LetT:A⟶A be an ℱ-metric space A,D and two functionsα,η:A×A⟶0,+∞. We consider that T is a fractional symmetric α-η-contraction of type-I along with constants λ∈0,1 and β,w^,γ∈0,1 such that, whenever αw,υ≥ηw,υ, we have(6)DTw,Tυp≤λSˇ1w,υ,where(7)Sˇ1w,υ=Dw,υp.Dw,Twp/β−w^β−γ,Dυ,Tυp/β−w^β−γ.Dw,Tw+Dυ,Tυp/w^−βw^−γ.Dw,Tυ+Dυ,Twp/γ−βγ−w^,where p∈1,∞, for all w,υ∈A\FixT.Example 2.
LetA=0,1,2,3 with grace of ℱ-metric D defined by(8)Dw,υ=w−υ2,ifw,υ∈A×A,w−υ,ifw,υ∉A×A,and consider ft=lnt and μ=ln3. Define T:A⟶A by(9)T0=0,T1=1,T2=T3=0,and α,η:A×A⟶0,+∞ by(10)αw,υ=1,ifw,υ∈A,0,otherwise,ηw,υ=12,ifw,υ∈A,0,otherwise.
Ifw,υ∈A, clearly αw,υ∈A≥ηw,υ such that(11)DT2,T3p=0≤λD2,3p⋅D2,T2p/β−w^β−γ⋅D3,T3p/β−w^β−γD2,T2+D3,T3p/w^−βw^−γD2,T3+D3,T2p/γ−βγ−w^=λ1⋅D2,0p/β−w^β−γD3,0p/β−w^β−γD2,0+D3,0p/w^−βw^−γ⋅D2,0+D3,0p/γ−βγ−w^=λ4p/β−w^β−γ⋅9p/β−w^β−γ⋅4+9p/w^−βw^−γ⋅4+9p/γ−βγ−w^≤λ4p/β−w^β−γ.9p/β−w^β−γ⋅4⋅9p/w^−βw^−γ⋅4⋅9p/γ−βγ−w^=λ4.9p/β−w^β−γ+p/w^−βw^−γ+p/γ−βγ−w^=λ.
By taking any value of constantsλ∈0,1 and β,w^,γ∈0,1, clearly, (6) holds for all p∈1,∞, w,υ∈A\FixT. Point out that T has two fixed points, which are 0 and 1.
Now, we initiate brand new fixed-point theorems for fractional symmetricα-η-contraction of type-I in the configuration of ℱ-complete ℱ-MS.Theorem 2.
LetA,D be a complete ℱ-metric space and T is a fractional symmetric α-η-contraction of type-I satisfying the following statements:(i)
T is an α-admissible mapping concerning η(ii)
There existsw0∈A to such an extent that αw0,Tw0≥ηw0,Tw0(iii)
T is α-η-continuous
At that point,T possesses a fixed point at A.Proof.
Considerw0 in A with the goal that αw0,Tw0≥ηw0,Tw0. For w0∈A, we build a chain wnn=1∞ in such a way that w1=Tw0 and w2=Tw1=T2w0. Proceeding with this exercise, wn+1=Twn=Tn+1w0, for every n∈ℕ. Presently, as long as mapping T is α-admissible with respect to η, at that time αw0,w1=αw0,Tw0≥ηw0,Tw0=ηw0,w1. Carrying on in this way, we get(12)αwn−1,wn≥ηwn−1,wn=ηwn−1,Twn−1,for alln∈ℕ.
Provided thatwn+1=wn for some n∈ℕ, then wn=w∗ is a fixed point of T. So, we assume that wn≠wn+1 accompanied by(13)DTwn−1,Twn=Dwn,Twn>0,for alln∈ℕ.
AsT is fractional symmetric α-η-contraction of type-I, a part of n∈ℕ, we have(14)Dwn,wn+1p=DTwn−1,Twnp≤λDwn−1,wnp⋅Dwn−1,Twn−1p/β−w^β−γ⋅Dwn,Twnp/β−w^β−γ⋅Dwn−1,Twn−1+Dwn,Twnp/w^−βw^−γ⋅Dwn−1,Twn+Dwn,Twn−1p/γ−βγ−w^=λDwn−1,wnp⋅Dwn−1,wnp/β−w^β−γ⋅Dwn,wn+1p/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1p/w^−βw^−γ⋅Dwn−1,wn+1+Dwn,wnp/γ−βγ−w^≤λDwn−1,wnp⋅Dwn−1,wnp/β−w^β−γ⋅Dwn,wn+1p/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1p/w^−βw^−γ⋅Dwn−1,wn+Dwn,wn+1p/γ−βγ−w^=λDwn−1,wnp⋅Dwn−1,wnp/β−w^β−γ⋅Dwn,wn+1p/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1p/w^−βw^−γ+p/γ−βγ−w^≤λDwn−1,wnp⋅Dwn−1,wnp/β−w^β−γ⋅Dwn,wn+1p/β−w^β−γ⋅Dwn−1,wn⋅Dwn,wn+1p/w^−βw^−γ+p/γ−βγ−w^=λDwn−1,wnp+p/β−w^β−γ+p/w^−βw^−γ+p/γ−βγ−w^⋅Dwn,wn+1p/β−w^β−γ+p/w^−βw^−γ+p/γ−βγ−w^=λDwn−1,wnp,which implies that(15)Dwn,wn+1p≤λDwn−1,wnp,and we deduce that(16)Dwn,wn+1≤λDwn−1,wn.
We conclude thatDwn−1,wn is a nonincreasing sequence with nonnegative terms. Thus, there is a nonnegative constant ϱ such that limn⟶∞Dwn−1,wn=ϱ. Note that ϱ≥0. From (16), we have(17)Dwn,wn+1≤λDwn−1,wn≤λnDw0,w1.
This provides(18)∑i=nm−1Dwi,wi+1≤λn1−λDw0,w1,m>n.
Considering for as much as(19)limn⟶+∞λn1−λDw0,w1=0,there subsist some N∈ℕ corresponding to(20)0<λn1−λDw0,w1<δ,n≥N.
Letε>0 be fixed and let f,μ∈ℱ×0,∞ be cognate and D3 is satisfied. By ℱ2, there exists δ>0 which connotes that(21)0<t<δimpliesft<fε−μ.
Hence, by (21) and ℱ1, we get(22)f∑i=nm−1Dwi,wi+1≤fλn1−λDw0,w1<fε−μ,where m,n∈ℕ with the goal that m>n≥N together with Dwn,wm>0. Therefore, by using D3 and (22), we have(23)fDwm,wn≤f∑i=nm−1Dwi,wi+1+μ<fε,which implies that by ℱ1, we have(24)Dwm,wn<ε,form>n≥N.
Consequently,wn is an ℱ-Cauchy sequence. Meanwhile A,D is an ℱ-complete metric space and there exists w∗∈A such that wn is ℱ-convergent to w∗; that is,(25)limn⟶∞Dwn,w∗=0,and T is α-η-continuous as well as αwn−1,wn≥ηwn−1,wn, each one of n∈ℕ at that point wn+1=Twn⟶Tw∗ as n⟶∞; in other words, w∗=Tw∗. Now we are going to prove that w∗ is a fixed point of T. We argue by contradiction by supposing that DTw∗,w∗>0. By D3, we have(26)fDTw∗,w∗p≤fDTw∗,Twnp+DTwn,w∗p+μ,n∈ℕ.
By using (ℱ1) and the contractive condition, we have(27)fDTw∗,w∗≤fλDw∗,wnp⋅DTw∗,w∗p/β−w^β−γ⋅Dwn,Twnp/β−w^β−γ⋅DTw∗,w∗+Dwn,Twnp/w^−βw^−γ⋅DTwn,w∗+Dwn,Tw∗p/γ−βγ−w^+Dwn+1,w∗p+μ,for all n∈ℕ. In other words, by using (ℱ2) and (25), we get(28)limn⟶∞fλDw∗,wnp+Dwn+1,w∗p+μ=−∞,which gives a contradiction. Therefore DTw∗,w∗=0; hence w∗ possesses a fixed point of T.Theorem 3.
LetA,D be an ℱ-complete ℱ-metric space and let T be a fractional symmetric α-η-contraction of type-I fulfilling the accompanying affirmations:(i)
T is an α-admissible mapping concerning η(ii)
There exists aw0∈A to such extent that αw0,Tw0≥ηw0,Tw0(iii)
An iterationwn in A is such that αwn,wn+1≥ηwn,wn+1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥ηwn,w∗ holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.Proof.
On closing lines of the proof of Theorem2, we acquire αwn,w∗≥ηwn,w∗ for each n∈ℕ. Using (D3), we have(29)fDTw∗,w∗p≤fDTw∗,Twnp+Dwn,w∗p+μ.
From (6) connecting ℱ1, we have(30)fDTw∗,w∗p≤fDTw∗,Twnp+DTwn,w∗p+μ≤fλDw∗,wnp⋅DTw∗,w∗p/β−w^β−γ⋅Dwn,Twnp/β−w^β−γ⋅DTw∗,w∗+Dwn,Twnp/w^−βw^−γ⋅DTwn,w∗+Dwn,Tw∗p/γ−βγ−w^+Dwn+1,w∗p+μ.
Employ (25) with certitude that(31)limn⟶∞Dwn,w∗=0,together withlimn⟶∞Dwn+1,w∗=0,and we obtain(32)fDw∗,Tw∗p≤fDw∗,Tw∗p+μ,and making use of ℱ2, we have(33)limn⟶∞fDw∗,Tw∗p+μ=−∞,which is a contradiction. Therefore, Dw∗,Tw∗=0; in other words, w∗ possesses a fixed point of T.Example 3.
LetA=0,1⊂ℝ with an ℱ-metric D:A×A⟶0,∞ by(34)Dw,υ=w−υ2,ifw,υ∈A×A,w−υ,ifw,υ∉A×A,accompanied by ft=lnt together with μ=ln1,p=1. Define T:A⟶A along with(35)Tw=w2e6,ifw∈A,0,ifw∉A,and α,η:A×A⟶0,+∞ by(36)αw,υ=2,ifw,υ∈0,∞,0,otherwise,ηw,υ=1,ifw,υ∈0,∞,0,otherwise.(i)
Case I. Ifw=υ, clearly Dw,υ=0. Hence, every condition of Theorem 2 is satisfied.(ii)
Case II. Ifw,υ∈A, clearly T is an α-admissible mapping with respect to η, whenever αw,υ≥ηw,υ, such that(37)DTw,Tυ=1e6w2−υ2≤λυ−w2⋅w−we1/β−w^β−γ⋅υ−υe1/β−w^β−γ⋅w−we+υ−υe1/w^−βw^−γ⋅w−υe+υ−we1/γ−βγ−w^,(iii)
By taking constantλ∈0,1, and β,w^,γ∈0,1, for all w,υ∈A\FixT.(iv)
Case III. If anyw,υ∉A, then we have(38)DTw,Tυ=0≤λυ−w⋅w1/β−w^β−γ⋅υ1/β−w^β−γ⋅w+υ1/w^−βw^−γ⋅w+υ1/γ−βγ−w^.
Therefore, whole constraints of Theorem2 are satisfied. Hence, T is fractional symmetric α-η-contraction of type-I.Definition 7.
Consider anℱ-metric space A,D and two functionsα,η:A×A⟶0,+∞. Then an ℱ-metric space on A is said to be α-η-complete if and only if every ℱ-Cauchy sequence wn, along with(39)αwn,wn+1≥ηwn,wn+1each one of then∈ℕ.ℱ-converges in A.Remark 1.
Theorems2 and 3 also hold for α-η-complete ℱ -metric space instead of ℱ-complete ℱ-metric space (for details, see [10]).


## 3. Fractional Symmetricα-η-Contraction of Type-II

In this section, a fractional symmetricα-η-contraction of type-II is introduced and in the structure of ℱ-complete ℱ-metric space. Using this notion, we shall provide a fixed-point theorem.Definition 8.
Consider a self-mapT:A⟶A on an ℱ-metric space A,D and two functions α,η:A×A⟶0,+∞. We presume that T is a fractional symmetric α-η-contraction of type-II provided that there are constants λ∈0,1 and β,w^,γ∈0,1 such that, whenever αw,υ≥ηw,υ, we own(40)DTw,Tυp≤λSˇ2w,υ,where(41)Sˇ2w,υ=Dw,υ⋅Dw,Twpβ/β−w^β−γ⋅Dυ,Tυpβ/β−w^β−γ⋅Dw,Tw+Dυ,Tυpw^/w^−βw^−γ⋅Dw,Tυ+Dυ,Twpγ/γ−βγ−w^,where p∈1,∞, for all w,υ∈A\FixT.
Now we show and demonstrate our next theorem.Theorem 4.
LetA,D be an ℱ-complete ℱ-metric space and let T be a fractional symmetric α-η-contraction of type-II fulfilling the accompanying affirmations:(i)
T is an α-admissible mapping concerning η(ii)
There existsw0∈A to such an extent that αw0,Tw0≥ηw0,Tw0(iii)
T is α-η-continuous
After that,T possesses a fixed point in A.Proof.
Considerw0 in A correspondent to αw0,Tw0≥ηw0,Tw0. For w0∈A, we build an iteration wnn=1∞in such a way that w1=Tw0 and w2=Tw1=T2w0. Proceeding with this exercise, wn+1=Twn=Tn+1w0, for all n∈ℕ. Now, as long as mapping T is α-admissible with respect to η, at that time αw0,w1=αw0,Tw0≥ηw0,Tw0=ηw0,w1. Carrying on in this way, we own(42)αwn−1,wn≥ηwn−1,wn=ηwn−1,Twn−1,for alln∈ℕ.
Provided thatwn+1=wn for some n∈ℕ, then wn=w∗ is a fixed point of T. So, we assume that wn≠wn+1 accompanied by(43)DTwn−1,Twn=Dwn,Twn>0,everyn∈ℕ.
AsT is fractional symmetric α-η-contraction of type-II, a part of n∈ℕ, we own(44)Dwn,wn+1p=DTwn−1,Twnp≤λDwn−1,wnp⋅Dwn−1,Twn−1pβ/β−w^β−γ⋅Dwn,Twnpβ/β−w^β−γ⋅Dwn−1,Twn−1+Dwn,Twnpw^/w^−βw^−γ⋅Dwn−1,Twn+Dwn,Twn−1pγ/γ−βγ−w^=λDwn−1,wnp⋅Dwn−1,wnpβ/β−w^β−γ⋅Dwn,wn+1pβ/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^/w^−βw^−γ⋅Dwn−1,wn+1+Dwn,wnpγ/γ−βγ−w^≤λDwn−1,wnp⋅Dwn−1,wnpβ/β−w^β−γ⋅Dwn,wn+1pβ/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^/w^−βw^−γ⋅Dwn−1,wn+Dwn,wn+1pγ/γ−βγ−w^=λDwn−1,wnp⋅Dwn−1,wnpβ/β−w^β−γ⋅Dwn,wn+1pβ/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^/w^−βw^−γ+pγ/γ−βγ−w^≤λDwn−1,wnp⋅Dwn−1,wnpβ/β−w^β−γ⋅Dwn,wn+1pβ/β−w^β−γ⋅Dwn−1,wn⋅Dwn,wn+1pw^/w^−βw^−γ+pγ/γ−βγ−w^=λDwn−1,wnp+pβ/β−w^β−γ+pw^/w^−βw^−γ+pγ/γ−βγ−w^⋅Dwn,wn+1pβ/β−w^β−γ+pw^/w^−βw^−γ+pγ/γ−βγ−w^=λDwn−1,wnp,which implies that(45)Dwn,wn+1p≤λDwn−1,wnp,and we deduce that(46)Dwn,wn+1≤λDwn−1,wn.
We conclude thatDwn−1,wn is a nonincreasing sequence with nonnegative terms. As a result, there is a nonnegative constant ρ such that limn⟶∞Dwn−1,wn=ρ. We shall indicate that ρ>0. Indeed, from (46), we derive that(47)Dwn,wn+1≤λDwn−1,w≤λnDw0,w1.
The rest of the test follows the same lines of Theorem2.Theorem 5.
Consider anℱ-complete ℱ-metric space A,D and let T be a fractional symmetric α-η-contraction of type-II meeting the following assertions:(i)
T is an α-admissible mapping with respect to η(ii)
There existsw0∈A such that αw0,Tw0≥ηw0,Tw0(iii)
An iterationwn in A is such that αwn,wn+1≥ηwn,wn+1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥ηwn,w∗ holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.Proof.
Similar to the lines of Theorem3, since, by (iii), αwn,w∗≥ηwn,w∗ holds for every n∈ℕ. Using (D3), we meet(48)fDTw∗,w∗≤fDTw∗,Twn+Dwn,w∗+μ.
From (40) and ℱ1, we have(49)fDTw∗,w∗p≤fDTw∗,Twnp+DTwn,w∗p+μ≤fλDw∗,wnp⋅DTw∗,w∗pβ/β−w^β−γ⋅Dwn,Twnpβ/β−w^β−γ⋅DTw∗,w∗+Dwn,Twnpw^/w^−βw^−γ⋅DTwn,w∗+Dwn,Tw∗pγ/γ−βγ−w^+Dwn+1,w∗p+μ.
Making use of (25), we get(50)limn⟶∞Dwn,w∗=0togetherlimn⟶∞Dwn+1,w∗=0,and we procure(51)fDw∗,Tw∗≤fDw∗,Tw∗+μ.
Usingℱ2, we have(52)limn⟶∞fDw∗,Tw∗p+μ=−∞,which is a logical inconsistency. Along these lines Dw∗,Tw∗=0; that is, w∗ possesses a fixed point of T.


## 4. Fractional Symmetricα-η-Contraction of Type-III

In this section, fractional symmetricα-η-contraction of type-III is considered in the environment of ℱ-complete ℱ-metric space. After stating a fixed-point theorem for such maps, we set up fractional symmetric α-η-contraction of type-III as follows.Definition 9.
Consider anℱ-metric space A,D with a self-map T:A⟶A and two functions α,η:A×A⟶0,+∞. We say that T is fractional symmetric α-η-contraction of type-III along with constants λ∈0,1 and β,w^,γ∈0,1 such that, whenever αw,υ≥ηw,υ, we have(53)DTw,Tυp≤λSˇ3w,υ,where(54)Sˇ3w,υ=λmaxDw,υ,Dw,Twpβ2/β−w^β−γ⋅Dυ,Tυpβ2/β−w^β−γ⋅Dw,Tw+Dυ,Tυpw^2/w^−βw^−γ⋅Dw,Tυ+Dυ,Twpγ2/γ−βγ−w^,where p∈1,∞, for all w,υ∈A\FixT.
Now we declare and demonstrate our next theorem.Theorem 6.
Let anℱ-complete A,D be an ℱ-metric space along with T being a fractional symmetric α-η-contraction of type-III which meets the following assertions:(i)
T is an α-admissible mapping concerning η(ii)
There existsw0∈A such that αw0,Tw0≥ηw0,Tw0(iii)
T is α-η-continuous
After that,T possesses a fixed point in A.Proof.
Considerw0 in A with the aim that αw0,Tw0≥ηw0,Tw0. Take any w0∈A; we erect an recapitulate wnn=1∞ in such a way that w1=Tw0 and w2=Tw1=T2w0. Continuing with this practice, wn+1=Twn=Tn+1w0, every n∈ℕ. As long as mapping T is α-admissible with respect to η, at that time αw0,w1=αw0,Tw0≥ηw0,Tw0=ηw0,w1. Carrying on in this way, we find(55)αwn−1,wn≥ηwn−1,wn=ηwn−1,Twn−1,for alln∈ℕ.
Provided thatwn+1=wn for some n∈ℕ, then wn=w∗ is a fixed point of T. So, we assume that wn≠wn+1 accompanied by(56)DTwn−1,Twn=Dwn,Twn>0,eachn∈ℕ.
AsT is fractional symmetric α-η-contraction of type-III, a part of n∈ℕ, we own(57)Dwn,wn+1p=DTwn−1,Twnp≤λmaxDwn−1,wnp⋅Dwn−1,Twn−1pβ2/β−w^β−γ⋅Dwn,Twnpβ2/β−w^β−γ⋅Dwn−1,Twn−1+Dwn,Twnpw^2/w^−βw^−γ⋅Dwn−1,Twn+Dwn,Twn−1pγ2/γ−βγ−w^=λmaxDwn−1,wnp⋅Dwn−1,wnpβ2/β−w^β−γ⋅Dwn,wn+1pβ2/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^2/w^−βw^−γ⋅Dwn−1,wn+1+Dwn,wnpγ2/γ−βγ−w^≤λmaxDwn−1,wnp⋅Dwn−1,wnpβ2/β−w^β−γ⋅Dwn,wn+1pβ2/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^2/w^−βw^−γ⋅Dwn−1,wn+Dwn,wn+1pγ2/γ−βγ−w^=λmaxDwn−1,wnp⋅Dwn−1,wnpβ2/β−w^β−γ⋅Dwn,wn+1pβ2/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^2/w^−βw^−γ+pγ2/γ−βγ−w^≤λmaxDwn−1,wnp⋅Dwn−1,wnpβ2/β−w^β−γ⋅Dwn,wn+1pβ2/β−w^β−γ⋅Dwn−1,wn⋅Dwn,wn+1pw^2/w^−βw^−γ+pγ2/γ−βγ−w^=λmaxDwn−1,wnp,Dwn−1,wnp+pβ2/β−w^β−γ+pw^2/w^−βw^−γ+pγ2/γ−βγ−w^⋅Dwn,wn+1pβ2/β−w^β−γ+pw^2/w^−βw^−γ+pγ2/γ−βγ−w^=λmaxDwn−1,wnp,Dwn,wn+1p,.
Provided thatmaxDwn,wn+1,Dwn−1,wn=Dwn,wn+1, at that time,(58)Dwn,wn+1≤λDwn,wn+1,which is a contradiction. We deduce that(59)Dwn,wn+1≤λDwn−1,wn.
We conclude thatDwn−1,wn is a nonincreasing sequence with nonnegative terms. As a result, there is a nonnegative constant ρ such that limn⟶∞Dwn−1,wn=ρ. We shall indicate that ρ>0. Indeed, from (59), we derive that(60)Dwn,wn+1≤λDwn−1,wn≤λnDw0,w1.
Pause the proof and go behind the closing lines of Theorem2.Theorem 7.
Consider anℱ-complete ℱ-metric space A,D and let T be a fractional symmetric α-η-contraction of type-III meeting the following assertions:(i)
T is an α-admissible mapping with respect to η(ii)
There existsw0∈A such that αw0,Tw0≥ηw0,Tw0(iii)
An iterationwn in A is such that αwn,wn+1≥ηwn,wn+1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥ηwn,w∗ holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.Proof.
Similar to the same lines of Theorem3, considering (iii), αwn,w∗≥ηwn,w∗ for all n∈ℕ. By (D3), we have(61)fDTw∗,w∗≤fDTw∗,Twn+Dwn,w∗+μ.
Using (40) along with ℱ1, we have(62)fDTw∗,w∗p≤fDTw∗,Twnp+DTwn,w∗p+μ≤fλDw∗,wnp,DTw∗,w∗pβ2/β−w^β−γ⋅Dwn,Twnpβ2/β−w^β−γ⋅DTw∗,w∗+Dwn,Twnpw^2/w^−βw^−γ⋅DTwn,w∗+Dwn,Tw∗pγ2/γ−βγ−w^+Dwn+1,w∗p+μ.
Using (25) the factuality is(63)limn⟶∞Dwn,w∗=0as long aslimn⟶∞Dwn+1,w∗=0,and we obtain(64)fDw∗,Tw∗p≤fDw∗,Tw∗p+μ.
Utilizingℱ2, we have(65)limn⟶∞fDw∗,Tw∗+μ=−∞,which is a logical inconsistency. Along these lines, Dw∗,Tw∗=0; that is, w∗ possesses a fixed point of T.


## 5. Fractional Symmetricα-η-Contraction of Type-IV

In this part, we propose a new notion, fractional symmetricα-η-contraction of type-IV, in the framework of ℱ-complete ℱ-metric space.(i)
T is an α-admissible mapping concerning η(ii)
There existsw0∈A which connotes that αw0,Tw0≥ηw0,Tw0(iii)
T is α-η-continuousDefinition 10.
Consider anℱ-metric space A,D with a self-map T:A⟶A and two functions α,η:A×A⟶0,+∞. We say that T is a fractional symmetric α-η-contraction type-IV along with constants λ∈0,1 and β,w^,γ∈0,1 with β+w^+γ<1 such that, whenever αw,υ≥ηw,υ, we have(66)DTw,Tυp≤λSˇ4w,υ,where(67)Sˇ4w,υ=λDw,υpβ3/β−w^β−γ⋅Dw,Twpβ3/β−w^β−γ⋅Dw,Tw+Dυ,Tυpw^3/w^−βw^−γ⋅Dw,Tυ+Dυ,Twpγ3/γ−βγ−w^,where p∈1,∞, for all w,υ∈A\FixT.
Now we declare and demonstrate our next theorem.Theorem 8.
Let anℱ-complete A,D be an ℱ-metric space along with T being a fractional symmetric α-η-contraction of type-IV that meets the following assertions:After that,T possesses a fixed point in A.Proof.
Considerw0 in A with the aim that αw0,Tw0≥ηw0,Tw0. Take any w0∈A; we build a chain wnn=1∞ in such a way that w1=Tw0 and w2=Tw1=T2w0. Proceeding with this exercise, wn+1=Twn=Tn+1w0, for every n∈ℕ. As long as mapping T is α-admissible with respect to η, at that time αw0,w1=αw0,Tw0≥ηw0,Tw0=ηw0,w1. Carrying on in this way, we get(68)αwn−1,wn≥ηwn−1,wn=ηwn−1,Twn−1,eachn∈ℕ.
Provided thatwn+1=wn for some n∈ℕ, then wn=w∗ is a fixed point of T. So, we assume that wn≠wn+1 accompanied by(69)DTwn−1,Twn=Dwn,Twn>0,for alln∈ℕ.
AsT is fractional symmetric α-η-contraction of type-IV, a part of n∈ℕ, we have(70)Dwn,wn+1p=DTwn−1,Twnp≤λDwn−1,wnpβ3/β−w^β−γ⋅Dwn−1,Twn−1pβ3/β−w^β−γ⋅Dwn−1,Twn−1+Dwn,Twnpw^3/w^−βw^−γ⋅Dwn−1,Twn+Dwn,Twn−1pγ3/γ−βγ−w^=λDwn−1,wnpβ3/β−w^β−γ⋅Dwn,wn+1pβ3/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^3/w^−βw^−γ⋅Dwn−1,wn+1+Dwn,wnpγ3/γ−βγ−w^≤λDwn−1,wnpβ3/β−w^β−γ⋅Dwn,wn+1pβ3/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pw^3/w^−βw^−γ⋅Dwn−1,wn+Dwn,wn+1pγ3/γ−βγ−w^=λDwn−1,wnpβ3/β−w^β−γ⋅Dwn,wn+1pβ3/β−w^β−γ⋅Dwn−1,wn+Dwn,wn+1pγ3/γ−βγ−w^≤λDwn−1,wnpβ3/β−w^β−γ⋅Dwn,wn+1pβ3/β−w^β−γ⋅Dwn−1,wn⋅Dwn,wn+1pw^3/w^−βw^−γ+pγ3/γ−βγ−w^=λDwn−1,wnpβ3/β−w^β−γ+pw^3/w^−βw^−γ+pγ3/γ−βγ−w^⋅Dwn,wn+1pβ3/β−w^β−γ+pw^3/w^−βw^−γ+pγ3/γ−βγ−w^=λDwn−1,wn⋅Dwn,wn+1pβ+w^+γ≤λmaxDwn−1,wnp,Dwn,wn+1p.
On condition thatmaxDwn,wn+1,Dwn−1,wn=Dwn,wn+1, at that time,(71)Dwn,wn+1≤λDwn,wn+1,which is a contradiction. We deduce that(72)Dwn,wn+1≤λDwn−1,wn.
Let up the closing lines of Theorem2.Theorem 9.
Consider anℱ-complete ℱ-metric space A,D and suppose that T is a fractional symmetric α-η-contraction of type-IV fulfilling the accompanying affirmations:(i)
T is an α-admissible mapping concerning η(ii)
There existsw0∈A to such an extent that αw0,Tw0≥ηw0,Tw0(iii)
An iterationwn in A is analogous to αwn,wn+1≥ηwn,wn+1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥ηwn,w∗ holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.
Whetherηw,υ=1, in Theorems 2, 3, 4, and 5, we introduce the following corollaries.Corollary 1.
Consider anℱ-complete ℱ-metric space A,D and suppose that T is a fractional symmetric α-η-contraction of type-I fulfilling the accompanying affirmations:(i)
T is an α-admissible mapping(ii)
There subsistw0∈A parallel to αw0,Tw0≥1(iii)
T is α-η-continuous
Afterwards,T possesses a fixed point in A.Corollary 2.
Consider anℱ-complete ℱ-metric space A,D and let T be fractional symmetric α-η-contraction of type-I fulfilling the accompanying affirmations:(i)
T is an α-admissible mapping(ii)
There subsistw0∈A parallel to αw0,Tw0≥1(iii)
An iterationwn in A is analogous to αwn,wn+1≥1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥1 holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.Corollary 3.
Let anℱ-complete A,D be ℱ-metric space and let T be a fractional symmetric α-η-contraction of type-II meeting the accompanying affirmations:(i)
T is an α-admissible mapping(ii)
There subsistw0∈A parallel to αw0,Tw0≥1(iii)
T is α-η-continuous
ThenT gets a fixed point in A.Corollary 4.
Let anℱ-complete A,D be ℱ-metric space and let T be a fractional symmetric α-η-contraction of type-II meeting the accompanying affirmations:(i)
T is an α-admissible mapping(ii)
There subsistw0∈A parallel to αw0,Tw0≥1(iii)
An iterationwn in A is analogous to αwn,wn+1≥1 escorted by wn⟶w∗ at the same time n⟶∞; after that αwn,w∗≥1 holds for each n∈ℕ
Afterwards,T possesses a fixed point in A.
In similar fashion, we can deduceCorollaries 1, 2, 3, and 4 for fractional symmetric α-η-contraction of type-III and that of type-IV, respectively.


## 6. Consequences

As a consequence of our results, we derive some effect for Suzuki-type fractional symmetric contractions and orbitallyT-complete and orbitally continuous mappings in ℱ-metric spaces.Theorem 10.
Consider anℱ-metric space A,D and let T be a continuous self-mapping on A. Assume that there exists r∈0,1 in addition to β,w^,γ∈0,1 such that(73)Dw,Tw≤Dw,υimpliesDTw,Tυp≤rSˇ1w,υ,where(74)Sˇ1w,υ=Dw,υp⋅Dw,Twp/β−w^β−γ⋅Dυ,Tυp/β−w^β−γ⋅Dw,Tw+Dυ,Tυp/w^−βw^−γ⋅Dw,Tυ+Dυ,Twp/γ−βγ−w^,where p∈1,∞, for all w,υ∈A\FixT.
At that time,T possesses a fixed point in A.Proof.
Describeα,η:A×A⟶0,+∞ by(75)αw,υ=Dw,υ,ηw,υ=Dw,Tw,for allw,υ∈A,and β,w^,γ∈0,1, and r∈0,1. It is clear that(76)ηw,υ≤αw,υ,for allw,υ∈A,which means that conditions (i)-(iii) of our Theorem 2 hold true. Let(77)ηw,Tw≤αw,υthenDw,Tw≤Dw,υ,which implies contractive condition:(78)DTw,Tυp≤rSˇ1w,υ.
Finally, every constraint of Theorem2 holds true. Hence, T possesses a fixed point in A.Theorem 11.
Consider anℱ-metric space A,D and let T be a self-mapping of A. Suppose that the following assertions hold:(i)
A,D is an orbitally T-complete ℱ-metric space.(ii)
There existsr∈0,1 in addition to β,w^,γ∈0,1 such that(79)DTw,Tυp≤rSˇ1w,υ,(i)
where(80)Sˇ1w,υ=Dw,υp⋅Dw,Twp/β−w^β−γ⋅Dυ,Tυp/β−w^β−γ⋅Dw,Tw+Dυ,Tυp/w^−βw^−γ⋅Dw,Tυ+Dυ,Twp/γ−βγ−w^,(ii)
p∈1,∞, for all w,υ∈Oω for some ω∈A, where Oω is an orbit of ω,(iii)
ifυn is a sequence such that υn⊆Oω with υn⟶υ∗ as n⟶∞, then υ∗∈Oω.
Then,T possesses a fixed point.Proof.
Describeα,η:A×A⟶0,+∞, by αw,υ=3 on Oω×Oω and αw,υ=0; otherwise, ηw,υ=1 for all w,υ∈A (see Remark 6 [11]). Then A,D is an α-η-complete ℱ-metric and T is an α-admissible mapping with respect to η. Let αw,υ≥ηw,υ; then w,υ∈Oω, and then, from (ii), we have(81)DTw,Tυp≤rSˇ1w,υ,where(82)Sˇ1w,υ=Dw,υp⋅Dw,Twp/β−w^β−γ⋅Dυ,Tυp/β−w^β−γ⋅Dw,Tw+Dυ,Tυp/w^−βw^−γ⋅Dw,Tυ+Dυ,Twp/γ−βγ−w^.
That is,T is a fractional symmetric α-η-contraction of type-I. Let υn be a sequence commensurate with αυn,υn+1≥ηυn,υn+1 together with υn⟶υ∗ for n⟶∞. So, υn⊆Oω. From (iii), υ∗∈Oω; that is, αυn,υ∗≥ηυn,υ∗. Hence, every norm of Theorem 3 holds true. Thus, T possesses a fixed point.Theorem 12.
Consider anℱ-metric space A,D and let T be a self-mapping of A. Suppose that the following assertions hold:(i)
For allw,υ∈Oω, there exists r∈0,1 along with β,w^,γ∈0,1, such that(83)DTw,Tυp≤rSˇ1w,υ,(ii)
where(84)Sˇ1w,υ=Dw,υp⋅Dw,Twp/β−w^β−γ⋅Dυ,Tυp/β−w^β−γ⋅Dw,Tw+Dυ,Tυp/w^−βw^−γ⋅Dw,Tυ+Dυ,Twp/γ−βγ−w^,(iii)
p∈1,∞, for some ω∈A,(iv)
the operatorT is orbitally continuous.
Afterwards,T possesses a fixed point.Proof.
Describeα,η:A×A⟶0,+∞, by αw,υ=3 on Oω×Oω and αw,υ=0; otherwise, ηw,υ=1 (see Remark 1.1 [12]), and we know that T is an α-η-continuous mapping. Let αw,υ≥ηw,υ; then w,υ∈Oω. So Tw,Tυ∈Oω; that is, αTw,Tυ≥ηTw,Tυ. Therefore, T is an α-admissible mapping with respect to η. From (i), we have(85)DTw,Tυp≤rSˇ1w,υ,where(86)Sˇ1w,υ=Dw,υp⋅Dw,Twp/β−w^β−γ⋅Dυ,Tυp/β−w^β−γ⋅Dw,Tw+Dυ,Tυp/w^−βw^−γ⋅Dw,Tυ+Dυ,Twp/γ−βγ−w^,and aforesaid T is a fractional symmetric α-η-contraction of type-I. Hence, each constraint of Theorem 2 holds true. Thus, T gets a fixed point.Theorem 13.
Consider anℱ-metric space A,D and let T be a self-mapping of A. Suppose that the following assertions hold:(i)
A,D is an orbitally T-complete ℱ-metric space;(ii)
there subsistr∈0,1 parallel to β,w^,γ∈0,1 such that(87)DTw,Tυp≤rSˇ2w,υ,(iii)
where(88)Sˇ2w,υ=Dw,υp⋅Dw,Twpβ/β−w^β−γ⋅Dυ,Tυpβ/β−w^β−γ⋅Dw,Tw+Dυ,Tυpw^/w^−βw^−γ⋅Dw,Tυ+Dυ,Twpγ/γ−βγ−w^,(iv)
p∈1,∞, for all w,υ∈Oω for some ω∈A, where Oω is an orbit of ω;(v)
ifυn is a sequence such that υn⊆Oω with υn⟶υ∗ as n⟶∞, then υ∗∈Oω.
Then,T possesses a fixed point.Theorem 14.
Consider anℱ-metric space A,D and let T be a self-mapping of A. Suppose that the following assertions hold:(i)
for allw,υ∈Oω, there subsist r∈0,1 along with β,w^,γ∈0,1, such that(89)DTw,Tυp≤rSˇ2w,υ,(ii)
where(90)Sˇ2w,υ=Dw,υp⋅Dw,Twpβ/β−w^β−γ⋅Dυ,Tυpβ/β−w^β−γ⋅Dw,Tw+Dυ,Tυpw^/w^−βw^−γ⋅Dw,Tυ+Dυ,Twpγ/γ−βγ−w^,(iii)
p∈1,∞, for some ω∈A;(iv)
the operatorT is orbitally continuous.
Afterwards,T possesses a fixed point.
Theorems10, 11, and 12 can be derived easily for fractional symmetric contraction of type-III and that of type-IV, respectively.


## 7. Application to Fractional-Order Differential Equations

The local and nonlocal fractional differential equations have been recently proved to be significant tools in the modeling of many phenomena in numerous fields of science and building. The fractional-order differential equations have numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth. For more details, see [2, 13–19]. Our aim is to give the existence and uniqueness of bounded solution of local fractional-order differential equation given in (93). Consider a function f:0,∞⟶ℝ. The conformable derivative of order α of f at t>0 is defined by [20](91)Dαft=limε⟶0ft+εt1−α−ftε.The conformable fractional integral associated with (91) is defined by [20, 21](92)I0αft=∫0tsα−1fsds.We consider the following boundary value problem of a conformable fractional-order differential equation:(93)Dαwt=λft,wt,t∈0,1,1<α<2,withw0=0,w1=∫01wsds.The integral representation of the solution to the boundary value problem (93) is(94)wt=λ∫01Gt,sfs,wsds,where Gt,s is a Green’s function defined by(95)Gt,s=−2tsα+sα−1,0≤s≤t≤1,−2tsα,0≤t≤s≤1,and ∫01wsds denotes the Riemann integrable of w with respect to s and f:0,1×ℝ⟶ℝ is a continuous function.(96)wt=c0+c1t+λ∫0tsα−1fs,wsds,w0=0⇒c0=0,w1=c1+λ∫01sα−1fs,wsds,∫01wsds=∫01c1sds+λ∫01∫0szα−1fz,wzdzds=12c1+λ∫01∫z1zα−1fz,wzdsdz=12c1+λ∫011−zzα−1fz,wzdsdz=12c1+λ∫01sα−1−sαfs,wsds,12c1=−λ∫01sα−1fs,wsds+λ∫01sα−1−sαfs,wsds=−λ∫01sαfs,wsdsc1=−2λ∫01sαfs,wsds.So,(97)wt=−2tλ∫01sαfs,wsds++λ∫0tsα−1fs,wsds,=−2tλ∫0tsαfs,wsds−2tλ∫t1sαfs,wsds+λ∫0tsα−1fs,wsds=λ∫0t−2tsα+sα−1fs,wsds+λ∫t1−2tsαfs,wsds=λ∫01Gt,sfs,wsds.LetCI be the linear space of all continuous functions defined on I=0,1, and let Dw,υ=w−υ∞=maxt∈Iwt−υt for all w,υ∈CI. Then, CI,D is an ℱ-complete metric space.We consider the following conditions:(a)
There existsr∈0,1, and ζ: ℝ2⟶ℝ is a function for each a,b∈ℝ with ζa,b≥ξa,b, such that(98)fs,wsds−fs,υsds≤rws−υs⋅ws−Twsβ/β−w^β−γ⋅υs−Tυsβ/β−w^β−γ⋅ws−Tws+υs−Tυsw^/w^−βw^−γ⋅ws−Tυs+υs−Twsγ/γ−βγ−w^,whereβ,w^,γ∈0,1.(b)
There existsw1∈CI such that(99)ζw1t,∫01Gt,sfs,w1sds≥ξw1t,∫01Gt,sfs,w1sds,for allt∈I;(c)
For eachw,υ∈CI, there exists w1,v1∈CI such that(100)ζwt,υt≥ξwt,υtimpliesζ∫01Gt,sfs,w1sds,∫01Gt,sfs,υ1sds≥ξ∫01Gt,sfs,w1sds,∫01Gt,sfs,υ1sds,for allt∈I;(d)
For any cluster pointw of a sequence wn of points in CI with(101)ζwn,wn+1≥ξwn,wn+1,limn⟶∞infζwn,w≥limn⟶∞infξwn,w.Theorem 15.
Suppose that conditions (a)-(d) are satisfied. Then, (93) has at least one solution w∗∈CI.Proof.
We know thatw∈CI is a solution of (93) if and only if w∈CI is a solution of the fractional-order integral equation(102)wt=λ∫01Gt,sfs,wsds,for allλ,t∈I.
We define a mapT:CI⟶CI by(103)Twt=λ∫01Gt,sfs,wsds,for allt∈I.
Then, problem (93) is equivalent to finding w∗∈CI, that is, a fixed point of T. Let w,υ∈CI, such that ζwt,υt≥0, for all t∈I. For using (a), we get(104)Twt−Tυt=λ∫01Gt,sfs,wsds−fs,υsds≤λ∫01Gt,sfs,wsds−fs,υsds≤λ∫01Gt,srdsws−υs⋅ws−Twsβ/β−w^β−γ⋅υs−Tυsβ/β−w^β−γ⋅ws−Tws+υs−Tυsw^/w^−βw^−γ⋅ws−Tυs+υs−Twsγ/γ−βγ−w^≤maxt∈I∫01Gt,sdsrws−υs∞⋅ws−Tws∞β/β−w^β−γ⋅υs−Tυs∞β/β−w^β−γ⋅ws−Tws∞+υs−Tυs∞w^/w^−βw^−γ⋅ws−Tυs∞+υs−Tws∞γ/γ−βγ−w^≤rws−υs∞⋅ws−Tws∞β/β−w^β−γ⋅υs−Tυs∞β/β−w^β−γ⋅ws−Tυs∞pγ−qγ⋅ws−Tws∞+υs−Tυs∞w^/w^−βw^−γ⋅ws−Tυs∞+υs−Tws∞γ/γ−βγ−w^.
Thus,(105)DTw,Tυ<ws−υs⋅ws−Twsβ/β−w^β−γ⋅υs−Tυsβ/β−w^β−γ⋅ws−Tws+υs−Tυsw^/w^−βw^−γ⋅ws−Tυs+υs−Twsγ/γ−βγ−w^,for all w,υ∈CI such that ζwt,υt≥ξwt,υt for all t∈I. We define α:CI×CI⟶0,∞ by(106)αw,υ=1,ifζwt,υt≥0,t∈I,0,otherwise,ηw,υ=12,ifξwt,υt≥0,t∈I,0,otherwise.
Then, for allw,υ∈CI, αw,υ≥ηw,υ, we have(107)DTw,Tυ≤rws−υs⋅ws−Twsβ/β−w^β−γ⋅υs−Tυsβ/β−w^β−γ⋅ws−Tws+υs−Tυsw^/w^−βw^−γ⋅ws−Tυs+υs−Twsγ/γ−βγ−w^.
Obviously,αw,υ≥ηw,υ for all w,υ∈CI. If αw,υ≥ηw,υ for each w,υ∈CI, then ζwt,υt≥ξwt,υt. From (c), we have ζTwt,Tυt≥ξTwt,Tυt and so αTw,Tυ≥ηTw,Tυ. Thus, T is an α-admissible map concerning η. From (b), there subsist w1∈CI parallel to αw1,Tw1=ηw1,Tw1. By (d), we have that, for any cluster point w of a sequence wn of points in CI with αwn,wn+1=ηwn,wn+1, limn⟶∞infαwn,w=limn⟶∞infηwn,w . By applying Theorem 2, if T has a fixed point in CI, there exists w∗∈CI such that Tw∗=w∗, and w∗ is a solution of (93).


### 7.1. Applications

The fractional-order differential equations emerge in various areas of engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, and fitting of experimental data.


### 7.2. Open Problem

What are the conditions for making a power of the contraction a nonnegative real number for fixed point and coincidence fixed point for two or more maps in various spaces?


## 7.1. Applications

The fractional-order differential equations emerge in various areas of engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, and fitting of experimental data.


## 7.2. Open Problem

What are the conditions for making a power of the contraction a nonnegative real number for fixed point and coincidence fixed point for two or more maps in various spaces?


## 8. Conclusion

The aim of this paper is to produce four new classes of type contractions. This research focuses on new idea of fractional symmetricα-η-contractions of type-I, type-II, type-III, and type-IV in the structure of ℱ-metric space, which is different from and more general than ordinary metric. This paper will open a new conspiracy of fractional fixed-point theory. We develop here Suzuki-type fixed point results in orbitally complete ℱ-metric space. These new investigations and applications would enhance the impact of new setup.



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*Source: 5510971-2021-05-22.xml*