# Iterating Fixed Point via Generalized Mann’s Iteration in Convexb-Metric Spaces with Application

Authors: A. Asif; M. Alansari; N. Hussain; M. Arshad; A. Ali
Journal: Complexity
 (2021)

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## Abstract

This manuscript investigates fixed point of single-valued Hardy-Roger’s typeF-contraction globally as well as locally in a convex b-metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom (F3) of the F-contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b-metric space is not necessarily complete, the convergence of the Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations.


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## Body

## 1. Introduction and Preliminaries

Various authors generalized metric space into many interesting generalizations (see [1–8]). Among them, Czerwik [9] introduced the idea of b-metric and proved fixed point theorems in it. It was further extended to partial b-metric and dislocated b-metric spaces in the past years [10, 11]. Chen et al. [12] introduced convex b-metric space and established various fixed point theorems.On the contrary, different authors extended Banach contraction to its generalizations (see [2–17]). Wardowski [18] introduced the idea of F-contraction which was later followed by many authors who delivered interesting results of F-contraction. One of them was presented by Cosentino et al. [19] who expanded F-contraction in F-contraction of Hardy-Roger’s type.In this article, we discussF-contraction in the frame of convex b-metric space using generalized Mann’s iteration algorithm. However, we have modified definition of F-contraction by eliminating two of its conditions and define F-Hardy-Roger’s contraction on closed ball in convex b-metric space. A detailed example explains the proved results.Some fundamental definitions related to our work are given below.Definition 1 (see [20]).
Letk≥1 be a real number. We denote by ℱ the family of all functions.
F:R+⟶R with the following properties:(F1)
F is strictly increasing.(F2)
for each sequencean⊂R+, limn⟶∞an=0 if and only if limn⟶∞Fan=−∞.(F3)
for each sequencean⊂R+, limn⟶∞an=0, there exists m∈0,1 such that limn⟶∞anmFan=0.(F4)
for each sequencean⊂R+ of positive numbers such that τ+Fkan≤Fkan−1 for all n∈N and some τ∈R+, then τ+Fknan≤Fkn−1an−1 for all n ∈ N.Definition 2 (see [20]).
LetE,d,k be a b-metric space where k≥1. A multivalued mapping T:E⟶CBE is called an F-contraction of Nadler type if there exist F∈ℱs and τ∈R+ such that τ+FkHTa,Tb≤Fda,bfor all a,b∈E with Ta=Tb.
Note that, in our theorems, we will considerℱb as the class of functions satisfying only (F1) and (F2) which modifies the definition of F-contraction.Definition 3 (see [9]).
Assume thatE=∅ with 1≤k∈R. If bk:E×E⟶0,+∞ satisfies the following axioms, for each a,b,c∈E.(1)
bka,b=0 if and only if a=b.(2)
bka,b=bkb,a.(3)
bka,b≤kbka,c+bkc,b.
Then the pairE,bk is known as b-metric space with k≥1.Definition 4 (see [9]).
Supposean is a sequence in E. Then(1)
an is convergent to a point a∈E if limn⟶∞dan,a=0.(2)
an is Cauchy if limn,m⟶∞dan,am=0.(3)
The spaceE,bk is complete if every Cauchy sequence an⊂E is convergent to a point a∈E.Definition 5 (see [12]).
Assume thatE=∅ and bk:E×E⟶0,+∞ is a mapping. Let I=0,1 with a continuous mapping η:E×E×I⟶E such that(1)bko,ηa,b;δ≤δbko,a+1−δbko,b,for each o∈E and a,b,δ∈E×E×I. η is said to be a convex structure on E.Definition 6 (see [12]).
Assume thatη:E×E×I⟶E is a convex structure on a b-metric space E,bk. Then E,bk,η is known as convex b-metric space.
Suppose thatE,bk,η is convex b-metric space with a self-mapping f. Then for an∈E and δn∈0,1, a generalize Mann’s iteration sequence anis defined as(2)an+1=ηan,fan;δn,n∈N.Definition 7 (see [12]).
LetE,d be a metric space. A self-mapping f on E is called an F-contraction of Hardy-Roger’s type if there exist F∈ℱ′ and τ∈R+ such that(3)τ+Fdfa,fb≤Fαda,by+βda,fa+γdb,fb+δda,fb+Ldb,Ta,for all a,b∈E with dfa,fb>0, where α+β+γ+2δ=1, γ≠1, and L≥0.
In the above definition,ℱ′ is the class of functions that satisfies (F1)-(F3).


## 2. Fixed Point Results ofF-Hardy-Roger’s Contractions on Closed Ball

This section examinesF-Hardy-Roger’s contraction for fixed point while executing the contraction only on closed ball rather than the whole convex b-metric. Also, an example is given to explain the proved theorem.Note that, in our theorems, we will considerℱb as the class of functions satisfying only (F1) and (F2) which modifies the definition of F-contraction and I=0,1.Definition 8.
LetE,bk,η is a convex b-metric space, a0 is some element in E, and ε>0; then the set Sεa0=a∈E:bka0,a≤ε is called a closed ball in E.
Not that,Sεa0 is a nonempty subset containing at least one element a0. Also, a subset of a complete b-metric space may not be complete. Hence a Cauchy sequence in the subset closed ball of a complete b-metric space may not converge in closed ball.Definition 9.
Assume thatF∈ℱb, E,bk,η be a convex b-metric space with k>1, δn−1∈0,1 for n∈N, and Sεa0 be a closed ball in E. Then f:E⟶E is known as F-Hardy-Roger’s contraction on closed ball if for α,β,γ∈0,∞ with kk2δn−1+αk+k2+1β+2γk<1, the following hold:(4)τ+Fkbkfa,fb≤Fαbka,b+βbka,fa+bkb,fb+γbka,fb+bkb,fa,for every a,b∈E.Theorem 1.
SupposeE,bk,η is a complete convex b-metric space with generalized Mann’s iteration an having δn−1∈0,1/4k2. Suppose f:E⟶E is an F-Hardy-Roger’s contraction on closed ball, and for all n∈N, λ=kδn−1+α+β+kγ+γδn−1/1−β−kγ and the following holds:(5)bka0,fa0≤λ1−kλε.
Thenf has a unique fixed point in E.Proof.
First we prove that the iterationan belongs to the closed ball. To do so, we will use mathematical induction. Since(6)bka0,a1=bka0,ηa0,fa0;δ0≤1−δ0bka0,fa0≤1−δ0λ1−kλε<ε,that is, bka0,a1<ε, therefore a1∈Sεa0. Suppose a2,a3,…,am∈Sεa0. Now for am+1, as(7)bkam,am+1≤1−δmbkam,fam,bkam,fam≤kbkam,fam−1+kbkfam−1,fam.
Then by (4),(8)τ+Fkbkfam−1,fam≤Fαbkam−1,am+βbkam−1,fam−1+bkam,fam+γbkam−1,fam+bkam,fam−1,and this implies that(9)Fbkam,fam=Fkbkam,fam−1+kbkfam−1,fam≤Fkbkam,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γbkam−1,fam+bkam,fam−1−τ≤Fkδm−1bkam−1,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γkbkam−1,am+kbkam,fam+δm−1bkam−1,fam−1−τ≤Fkδm−1bkam−1,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γk1−δm−1bkam−1,fam−1+kbkam,fam+δm−1bkam−1,fam−1−τ.
Using (F1), we write(10)1−β−kγbkam,fam≤kδm−1+α1−δm−1+β+k1−δm−1γ+γδm−1bkam−1,fam−1,and this implies that(11)bkam,fam<kδm−1+α+β+kγ+γδm−11−β−kγbkam−1,fam−1=λbkam−1,fam−1,that is, for all m∈N,(12)bkam,fam<λbkam−1,fam−1.
Note thatλ<1. Consequently,(13)bkam,fam<λbkam−1,fam−1<λbkam−2,fam−2<⋯<λbka0,fa0,and hence bkam,fam<λmbka0,fa0 for all m∈N. Thus by (7), bkam,am+1<λmbka0,fa0 for all m∈N. By triangular inequality(14)bka0,am+1≤kbka0,a1+k2bka1,a2+⋯+km+1bkam,am+1<kbka0,a1+kλbka0,a1+⋯+kλmbka0,a1=k1−kλm1−kλbka0,a1=k1−kλm1−kλλ1−kλε<ε.
Henceam+1∈Sεa0. Therefore, we conclude that an∈Sεa0 for all n∈N. Now from (4), (7), and (12), we write(15)Fbkan,an+1≤Fbkan−1,fan−1−τ≤Fbkan−2,fan−2−2τ≤⋯≤Fbka0,fa0−nτ.
Exerting limitn⟶∞, we obtain(16)limn⟶∞Fbkan,fan=−∞.
Consequently by (F2), we acquire(17)limn⟶∞bkan,fan=0.
Hence,limn⟶∞bkan,an+1=limn⟶∞bkan,fan=0. It remains to prove that the sequence an is a Cauchy sequence. Suppose on the contrary that an is not Cauchy. Then we can find a positive real number ε0 and two subsequences aμu and aωu of an while μu>ωu>u with μu as the smallest natural index such that(18)bkaμu,aωu≥σ0,bkaμu−1,aωu<σ0.
Using the above inequalities, we can write(19)σ0≤bkaμu,aωu≤kbkaμu,aωu+1+bkaωu+1,aωuσ0k≤limu⟶∞supbkaμu,aωu+1.
Now,(20)bkaμu,aωu+1≤bkηaμu−1,faμu−1,δμu−1,aωu+1=δμu−1bkaμu−1,aωu+1+1−δμu−1bkfaμu−1,aωu+1≤δμu−1bkaμu−1,aωu+1+1−δμu−1kbkfaμu−1,faωu+1+bkfaωu+1,aωu+1.
Since, by (4),(21)Fkbkfaμu−1,faωu+1≤Fαbkaμu−1,aωu+1+βbkaμu−1,faμu−1+bkfaωu+1,aωu+1+γbkaμu−1,faωu+1+bkaωu+1,faμu−1−τ,and therefore,(22)Fδμu−1bkaμu−1,aωu+1+1−δμu−1kbkfaμu−1,faωu+1+kbkfaωu+1,aωu+1≤Fδμu−1bkaμu−1,aωu+1+1−δμu−1αbkaμu−1,aωu+1+βbkaμu−1,faμu−1+bkfaωu+1,aωu+1+γbkaμu−1,faωu+1+bkaωu+1,faμu−1+kbkfaωu+1,aωu+1−τ,using (F1) and (20), we write(23)bkaμu,aωu+1≤γμu−1bkaμu−1,aωu+1+1−δμu−1αbkaμu−1,aωu+1+βbkaμu−1,faμu−1+bkfaωu+1,aωu+1+γbkaμu−1,faωu+1+bkaωu+1,faμu−1+kbkfaωu+1,aωu+1≤δμu−1kbkaμu−1,aωu+kbkaωu,aωu+1+1−δμu−1αkbkaμu−1,aωu+kbkaωu,aωu+1+βbkaμu−1,faμu−1+bkfaωu+1,aμu+1+γkbkaμu−1,aωu+k2bkaωu,aωu+1+k2bkaωu+1,faωu+1kbkaωu,aωu+1+k2bkaμu−1,aωu+k2bkaμu−1,faμu−1+kbkfaωu+1,aωu+1.
Exerting limitu⟶∞, we obtain(24)limu⟶∞supbkaμu,aωu+1≤limu⟶∞supδμu−1kbkaμu−1,aωu+1−δμu−1αkbkaμu−1,aωu+γkbkaμu−1,aωu+k2bkaμu−1,aωu≤limu⟶∞supδμu−1k+1−δμu−1α+2k2γbkaμu−1,aωu<δμu−1k+1−δμu−1α+2k2γσ0≤δμu−1k+α+2k2γσ0,and hence, by hypothesis, we get(25)limu⟶∞supbkaμu,aωu+1<σ0k.
This shows that(26)σ0k≤limu⟶∞supbkaμu,aωu+1<σ0k,which is a contradiction. Hence, an is a Cauchy sequence. The completeness of E assures the existence of an element a∗ in E such that(27)limn⟶∞bkan,a∗=0.
However, this is not clear whether the convergent pointa∗ lies inside the closed ball Sεa0 or not. Therefore, before proceeding to the proof of fixed point, we first prove that a∗∈Sεa0. By triangular inequality, we write(28)bka0,a∗≤kbka0,an+kbkan,a∗≤kbkan,a∗+k2bka0,a1+k3bka1,a2+⋯+kn+1bkan−1,an<kbkan,a∗+k2bka0,a1+kλbkbka0,a1+⋯+kλn−1bka0,a1=kbkan,a∗+k21−kλn−11−kλbka0,a1≤kbkan,a∗+k21−kλn−11−kλλ1−kλε,exerting limit n⟶∞, we get bka0,a∗<ε. That is, a∗∈Sεa0. Next, we prove that a∗ is the fixed point of f:(29)bkan,fa∗≤kbkan,fan+bkfan,fa∗.
As(30)Fkbkfan,fa∗≤Fαbka∗,an+βbkan,fan+bka∗,fa∗+γbkan,fa∗+bka∗,fan−τ,therefore, using above equation and (29),(31)Fbkan,fa∗≤Fkbkan,fan+αbka∗,an+βbkan,fan+bka∗,fa∗+γbkan,fa∗+bka∗,fan−τ≤Fkbkan,fan+αbka∗,an+βbkan,fan+bka∗,fa∗+γkbkan,a∗+kbka∗,fa∗+kbka∗,an+kbkan,fan−τ,and utilizing (F1), we obtain(32)Fbkan,fa∗−β+kγbka∗,fa∗≤Fα+2γkbka∗,an+k+β+kγbkan,fan−τ≤Fα+2γkbka∗,an+k+β+kγλnbka0,fa0−τ.
Clearly,(33)limn⟶∞α+2γkbka∗,an+k+β+kγλnbka0,fa0=0,and therefore, by (F2), we get(34)limn⟶∞Fbkan,fa∗−β+kγbka∗,fa∗=−∞,and consequently, by (F2) again, we obtain(35)limn⟶∞bkan,fa∗−β+kγbka∗,fa∗=0,that is, 1−β−kγbka∗,fa∗=0, Hence a∗ is the fixed point of f. It remains to prove that a∗ is the only fixed point f. Suppose on the contrary that a∗∗ be another fixed point of f. Then(36)Fkbka∗,a∗∗=Fkbkfa∗,fa∗∗≤Fαbka∗,a∗∗+βbka∗,fa∗+bka∗∗,fa∗∗+γbka∗,fa∗∗+bka∗∗,fa∗−τ=Fα+2γbka∗,a∗∗−τ,which is a contradiction because k≥1, F is strictly increasing, and by hypothesis, α+2γ<1. Hence, bka∗,a∗∗=0. That is, a∗=a∗∗.Example 1.
SupposeE=0,+∞ and ε=a0=1/2, then Sε1/2=I. Let f:E⟶E be defined by(37)fa=e−3/4,ifa=12,a6,otherwise,and choose a mapping bk:E×E⟶0,+∞ defined as(38)bka,b=x−y2,if bothx,y∈0,1,x−y,otherwise.
Demonstrateη:E×E×I⟶E as ηa,b;δ=δa+1−δb for all a,b∈E. Choose an=ηan−1,fan−1;δn−1, k=2, and fix δn−1=1/16=1/4k2. Observe that δa+1−δb∈0,1 whenever a,b∈0,1. Now, consider o,a,b∈E, then we have two cases:(i)
Wheno∉0,1, we get(39)bko,ηa,b;δ=δo−a+1−δo−b≤δo−a+1−δo−b=δbko,a+1−δbko,b.(ii)
Wheno∈0,1. We have the following subcases:(a)
If botha,b∈0,1, then obviously ηa,b;δ=δa+1−δb∈0,1, and hence(40)bko,ηa,b,γ=γo−a+1−γo−b2≤γo−a+1−γo−b2=γo−a2+1−γo−b2+2γ1−γo−ao−b≤γo−a2+1−γo−b2+γ1−γo−a2+o−b2=γo−a2+1−γo−b2=γbko,a+1−γbko,b.(b)
If only one ofa and b is in 0,1, say a is in 0,1, then obviously ηa,b;δ=δa+1−δb∉0,1, and hence,(41)bko,ηa,b;δ=δo−a+1−δo−b≤δo−a+1−δo−b≤δo−a2+1−δo−b=δbko,a+1−δbko,b.The same can be done forb in 0,1 and a not in 0,1.(c)
If botha,b∉0,1, then obviously ηa,b;δ=δa+1−δb∉0,1, and hence(42)bko,a,b;δ=δo−a+1−δo−b≤δo−a+1−δo−b=δo−a+1−δo−b=δbko,a+1−δbko,b.
From all the possible cases, it is clear thatE,bk,η is a convex b-metric space with k=2. Next, fix β=γ=0 and α=1/16. Observe that(43)kk2δn−1+αk+k2+1β+2γk=kk214k2+αk<1.
Also(44)λ1−kλε=k2δm−1+α+β+kγ+γδm−11−β−kγ1−kk2δm−1+α+β+kγ+γδm−11−β−kγ12=k2δm−1+α1−kk2δm−1+α12=5161−5812>0.00076≅bk12,e−3/4=bka0,fa0,that is, bka0,fa0<λ1−kλε. Now checking for the contractive condition, we again have two cases:Case 1: if botha,b∈0,1, then(45)lnkbkfa,fb=lnka6−b62=lnk136a−b2≤lnαa−b2=lnαbka,b+βbka,fa+bkb,fb+γbka,fb+bkb,fa,that is,(46)Fkbkfa,fb≤Fαbka,b+βbka,fa+bkb,fb+γbka,fb+bkb,fa.Thus, the inequality (4) is satisfied for τ∈0,ln4k2+2/4k2+1 and for all a,b∈0,1.Case 2: if botha,b∉0,1, i.e., say a=12 and b=6, then(47)lnkbkfa,fb=lnk2−1>ln11612−6=lnαbka,b=lnαbka,b+βbka,fa+bkb,fb+γbka,fb+bkb,fa.
Hence the inequality (4) does not hold for all the elements of E. Hence the contractive condition (4) holds true only for a,b∈0,1. Observe that Fx=lnx satisfies (F1) and (F2). Moreover(48)an=δn−1an−1+1−δn−1fan−1=δn−1an−1+1−δn−1an−16=56δn−1+16an−1=1796an−1.
Similarly(49)an−1=1796an−2,an−2=1796an−3,…,a1=1796a0.
Therefore,an=17/96na0 and fan=1/617/96na0. Letting n⟶∞, we get an⟶0 and fan⟶0. That is, 0 is the fixed point of f. For uniqueness, suppose on contrary that r is another fixed point of f then bk0,r>0, say bk0,r=δ. Hence(50)δ=bk0,r=bkf0,fr=0−r52=r225=125bk0,r=125δ,which is a contradiction. Therefore, 0 is the only fixed point of f.
Takingβ=0 in Theorem 1, we obtain the following results.Corollary 1.
Assume thatF∈ℱb, E,bk,η is a complete convex b-metric space with k>1 and an is a generalized Mann’s iteration having δn−1∈0,1/4k2 for all n∈N. Suppose Sεa0 be a closed ball in E and for α,γ∈0,∞ with kk2δn−1+αk+k2+12γk<1, λ=kδn−1+α+kγ+γδn−1/1−kγ. If f:E⟶E is a mapping such that for a,b∈0,1, the following holds:(51)τ+Fkbkfa,fb≤Fαbka,b+γbka,fb+bkb,fa;bka0,fa0≤λ1−kλε.Thenf has a unique fixed point in E.Takingγ=0 in Theorem 1, we obtain the following results.Corollary 2.
Assume thatF∈ℱb, E,bk,η is a complete convex b-metric space with k>1 and an is a generalized Mann’s iteration having δn−1∈0,1/4k2 for all n∈N. Suppose Sεa0 be a closed ball in E and for α,β∈0,∞ with kk2δn−1+αk+k2+1β<1, λ=kδn−1+α+β/1−β. If f:E⟶E is a mapping such that for a,b∈0,1, the following holds:(52)τ+Fkbkfa,fb≤Fαbka,b+βbka,fa+bkb,fb;bka0,fa0≤λ1−kλε.
Thenf has a unique fixed point in E.


## 3. Fixed Point Results of Globally Contractive Mappingsf on Convex b-Metric Space

In this section, fixed point is instigated forF-contraction of Hardy-Roger’s type while the contractive condition holds true on the whole b-metric space rather than locally only on closed ball.Theorem 2.
SupposeF∈ℱb, E,bk,η is a complete convex b-metric space with k>1 and an is a generalized Mann’s iteration having δn−1∈0,1/4k2 for all n∈N. If for α,β,γ∈0,∞ with kkδn−1+α+k+δn−1+1β+γk<1 and f:E⟶E such that the following hold:(53)τ+Fkbkfa,fb≤Fαbka,b+βbka,fa+bkb,fb+γbka,fb+bkb,fa,for every a,b∈E. Then f has a unique fixed point in E.Proof.
We know by hypothesis(54)bkam,am+1≤1−δmbkam,fam,bkam,fam≤kbkam,fam−1+kbkfam−1,fam.
By (53),(55)τ+Fkbkfam−1,fam≤Fαbkam−1,am+βbkam−1,fam−1+bkam,fam+γbkam−1,fam+bkam,fam−1,and this implies that(56)Fbkam,fam=Fkbkam,fam−1+kbkfam−1,fam≤Fkbkam,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γbkam−1,fam+bkam,fam−1−τ≤Fkδm−1bkam−1,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γkbkam−1,am+kbkam,fam+δm−1bkam−1,fam−1−τ≤≤Fkδm−1bkam−1,fam−1+α1−δm−1bkam−1,fam−1+βbkam−1,fam−1+bkam,fam+γk1−δm−1bkam−1,fam−1+kbkam,fam+δm−1bkam−1,fam−1−τ,using (F1), we write(57)1−β−kγbkam,fam≤kδm−1+α1−δm−1+β+k1−δm−1γ+γδm−1bkam−1,fam−1,and this implies that(58)bkam,fam<kδm−1+α+β+kγ+γδm−11−β−kγbkam−1,fam−1=λbkam−1,fam−1,where λ=kδm−1+α+β+kγ+γδm−1/1−β−kγ. Or we can say that for all m∈N,(59)bkam,fam<λbkam−1,fam−1.
Note thatλ<1. Now from (53), (54), and (59), we write(60)Fbkan,an+1≤Fbkan−1,fan−1−τ≤Fbkan−2,fan−2−2τ≤⋯≤Fbka0,fa0−nτ,exerting limit n⟶∞, we obtain(61)limn⟶∞Fbkan,fan=−∞,consequently by (F2), we acquire(62)limn⟶∞bkan,fan=0.
Hence,(63)limn⟶∞bkan,an+1=limn⟶∞bkan,fan=0.
The Cauchyness of the sequencean can be proved by following the same steps as in equation (59). Afterwards, on contrary to the previous of the theorem, we do not need to prove convergence as every Cauchy sequence in complete b-metric space is Convergent. Hence there exist an element a∗ in E such that limn⟶∞an⟶a∗. Furthermore, a∗ can be proved as the unique fixed point of f in the similar way as done in Theorem 1.Corollary 3.
SupposeF∈ℱb, E,bk,η is a complete convex b-metric space with k>1 and an is a generalized Mann’s iteration having δn−1∈0,1/4k2 for all n∈N. If for α,β∈0,∞ with kkδn−1+αk+k2+1β<1 and f:E⟶E, such that the following hold:(64)τ+Fkbkfa,fb≤Fαbka,b+βbka,fa+bkb,fb,or every a,b∈E. Then f has a unique fixed point in E.


## 4. Application

In Physics and Engineering, many problems are modelled in to second-order linear differential equations, i.e., system of an object with mass attached to a vertical spring, an inductor, a capacitor connected in series, and an electric circuit with a resistor. The differential equation discussed in this section shows the engineering problem of activation of spring affected by external forces. The differential equation will be reformulated to fixed point problems and the existence of the fixed point will assure the existence of the solution of differential equation (see [1, 17]).Suppose the boundary value problem for second-order differential equation be(65)d2ads2+cndads=As,as,a0=0,a′0=l,where A:I×R+⟶R is a continuous function. Equation (65) can be equivalently written as(66)as=∫0sBs,rAr,ardr,s∈I,where Bs,r is the green function stated as(67)Bs,r=s−reτs−r,if0≤r≤S≤1,0,if0≤s≤r≤1.Withτ>0 formulated in terms of m and c in (66). Suppose E:=C0,1,R+ is the collection of all continuous functions from 0,1 into R+. Define(68)aτ=Sups∈Iase−2τs,for a∈E, and bk:E×E⟶R+ as(69)bkf,g=fτ−gτ2.ConsiderT:E⟶E is defined by(70)Tas=∫0sBs,rAr,ardr,for all a∈E and s∈I. Note that the existence of fixed point of T guarantees the existence of solution of equation (66).Before starting our main theorem, we first prove the following important lemma.Lemma 1.
SupposeE:=C0,1,R+, bk is defined by (69) and η:E×E×0,1/4k2 is stated as ηf,g,δp=δfp+1−δgp for all p∈I, then E,bk,η is a convex b-metric space.Proof.
Forf,g,h∈E, τ>0, and δ∈0,1,(71)bkf,ηg,h,δ=bkfτ−δg+1−δhτ2≤bkfτ−δgτ−1−δhτ2=δfτ−gτ+1−δfτ−hτ2≤δfτ−gτ+1−δfτ−hτ2=δfτ−gτ2+1−δfτ−hτ2+2δ1−δfτ−gτfτ−hτ≤δfτ−gτ2+1−δfτ−hτ2+δ1−δfτ−gτ2+fτ−hτ2=δfτ−gτ2+1−δfτ−hτ2=δbkf,g+1−δbkf,h.
Hencebk is convex b-metric space with k=2.Theorem 3.
Suppose the integral equation (66) and let the following condition holds:(1)
A is an increasing function.(2)
There existτ>0 such as(72)Ar,a≤λτ2e−τka,whereλ∈0,1, r∈I, and ar∈R+.(3)
Forα,β,γ∈0,∞ with kkδn−1+α+k+δn−1+1β+γk<1 and the norm ⋅τ defined above, the following hold:(73)λe−τ2aτ−bτ2≤αaτ−bτ2+β1−λe−τk2+γ1+λe−τ2k⋅aτ2+bτ2−4γλe−τkaτbτ,for δn−1∈0,1/4k2. Then the equation (69) has a solution.Proof.
By (70),(74)Tar=∫0sBs,rAr,ardr≤∫0sBs,rAr,ardr≤∫0sBs,rλτ2e−τkardr=∫0sBs,rλτ2e−τke2τre−2τrardr≤∫0sλτ2e−τks−reτs−re2τraτdr=λτ2ke−τ+τsaτ∫0ss−reτrdr=λτ2ke−τ+τsaτ−sτ−1τ2+eτsτ2=λe−τkaτe2τs1−τse−τs−e−τs.
As1−τse−τs−e−τs<1, then(75)Tarτ≤λe−τkaτ.
Similarly, we obtain(76)Tbrτ≤λe−τkbτ.
Now(77)bkTa,Tb=Taτ−Tbτ2≤λe−τ2kaτ−bτ2,or(78)kbkTa,Tb≤λe−τ2aτ−bτ2.
By hypothesis, we write(79)kbkTa,Tb≤αaτ−bτ2+β1−λe−τk2+γ1+λe−τ2k⋅aτ2−bτ2−4γλe−τkaτbτ=αaτ−bτ2+β1−λe−τk2aτ2−bτ2+γaτ2−bτ21+λe−τ2k−4λe−τkaτbτ=αaτ−bτ2+βaτ21−λe−τk2+bτ21−λe−τk2+γaτ2+λe−τk2bτ2−2λe−τkaτbτ+bτ2+λe−τk2aτ2−2λe−τkbτaτ=αaτ−bτ2+βaτ−λe−τkaτ2+bτ−λe−τkbτ2+γaτ−λe−τkbτ2+bτ−λe−τkaτ2=αbka,b+βbka,Ta+bkb,Tb+γbka,Ta+bkb,Tb⋅
Therefore,(80)lnkbkTa,Tb≤lnαbka,b+βbka,Ta+bkb,Tb+γbka,Tb+bkb,Ta,and hence,(81)τ+FkbkTa,Tb≤Fαbka,b+βbka,Ta+bkb,Tb+γbka,Tb+bkb,Ta,and for(82)τ=0,ρ,whereρ=lnαbka,b+βbka,Ta+bkb,Tb+γbka,Tb+bkb,TakbkTa,Tb.
Therefore, all the conditions of the Theorem2 are satisfied, so the integral equation (66) has a unique solution. Consequently, the differential equation (65) has a unique solution.


## 5. Conclusion

This paper has introducedF-Hardy-Roger’s contraction on closed ball in convex b-metric space, and generalized Mann’s iteration theorem algorithm is used to find the fixed point. The existence of the limit point inside the closed ball is established despite of the fact that the subset closed ball in a complete b-metric space is not complete. The convergence of the chosen sequence is insured inside the closed ball without completeness of the ball. Hence, a new approach is used to investigate fixed point for F-Hardy-Roger’s contractions (and can extend similarly to others contractions). The results are followed by an interesting example in R2. Furthermore, an application of our results in finding a unique solution to second-order differential equation with boundary valued is described along with an important lemma. The paper furthered the research already done on the topic of F-contractions and fixed point theory. In future, these results will be formulated in the structure of Hilbert spaces and orthogonal partial b-metric spaces, and its application in convex minimization and fractional differential equations will be investigated in continuation to the work already done in [21–25].



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