# Higher Order of Convergence with Multivalued Contraction Mappings

Authors: Jia-Bao Liu; Asma Rashid Butt; Shahzad Nadeem; Shahbaz Ali; Muhammad Shoaib
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we establish some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Furthermore, we useQ- and R-order of convergence. Our main results extend many previous existing results in the literature. Consequently, to substantiate the validity of proposed method, we give its application in integral inclusion.


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## Body

## 1. Introduction

A well-known mathematician, Banach, gave a main source of the existence of fixed points. An iterative method was used in the Banach contraction principle which converges to the fixed point linearly. In order to obtain higher order of convergence, Proinov [1] generalized Banach contraction theorem by generalizing the contractive condition which involves a gauge function of order r≥1. Later, Kiran and Kamran [2] extended his work and generalized using multivalued maps from a complete metric space into the space of all nonempty proximinal closed spaces. In this context, we study the multivalued contraction mapping involved in metric space from [3–7]. And some details for multivalued mapping and their fixed points are included in [1, 2, 8–15]. Recently, Petrusel investigated the local fixed point results for graphic contractions and multivalued locally contractive operators and proposed their application in optimization theory in [16, 17]. Sow proposed the strong convergence of a modified Mann algorithm for multivalued quasi-nonexpansive mappings and monotone mappings with an application in [18, 19]. (Proinov, General local convergence theory for a class of iterative processes and its applications to Newton’s method. Journal of Complexity, 25, 1, 38–62). We extend some results of Proinov to the case of multivalued maps from a metric space V into the space of all bounded closed subspace of V. We recall some notations which we used in this paper. Let V,d be a metric space and A be a subset of V; distance from every v∈V to subset A is defined as there is a element a∈A such that dv,a=dv,A. Now, distance from the point to set is defined as v∈V and A⊂V, where dv,A=infdv,w:w∈A. We denote CBV as the class of all nonempty closed and bounded subsets of V. Let H be the generalized Hausdorff metric on CBV generated by metric d as(1)HA,B=maxsupv∈Adv,B,supw∈Bdw,A,for each A,B∈CBV. The fixed point of T:V⟶CBV is a point v∈V if v∈Tv. Throughout this paper, L denotes an interval on ℝ+ containing 0, this interval is represented as 0,A,0,a or 0,∞, and for polynomial, we used Sbe=1+e+⋯+eb−1. We use the order of convergence as Q-order and R-order with r≥1 for iterative process:(2)vb+1∈Tvb,b=0,1,2,…,and use gauge function ϱ on L, that is, ϱ:L⟶L. In this paper, we present some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Also, we use Q- and R-order of convergence. In the last section, we applied our proposed method in integral theory.


## 2. Preliminaries

In this section, we take some results and definitions from [1, 19].Definition 1.
A gauge mapping with orderr which is greater than 1 defines with ϱ:L⟶L on L if it fulfills the following conditions:(1)
ϱγe≤γrϱe for each γ∈0,1ande∈L(2)
ϱe≤e for each e∈L
The first condition of the above definition can easily show that it is equivalent to the conditionϱ0=0 and ϱe/er is nondecreasing on L−0. Fixed point theory in metric space is full of fixed point theorems in different classes of contraction, which can be obtained by different properties of gauge function ϱ.Example 1.
ϱ:L⟶L where L=0,∞ and ϱe=γe where γ∈0,1. Then,(1)
ϱγe=γ2e=γ.γe=γγe=γϱe, where e∈L⇒ϱγe=γϱe(2)
ϱe=γe<e because ϱ∈0,1Lemma 1.
A mappingϱ of order r on L fulfills the gauge condition and there is another mapping π on L which is nondecreasing and nonnegative such that(3)ϱe=eπe,for eache∈L.Now, the properties which the mappingπ has are(4)0≤πe≤1,for eache∈L,πγt≤γr−1πt,for eachγ∈0,1,and eache∈L.Now, the gauge mapping which is defined onϱ is written as(5)0≤πe≤1,for eache∈L.ForQ-order of convergence with at least order r≥1, sequence vb converges to η. If a constant c exists which is greater then zero such that dvb+1,η≤cdvb,ηr for sufficiently large b, then there are the different cases which depend on r; but, in that case, r=1 and b≤1. Now, the type of convergence linearly and quadratically depends on the order r, respectively, 1 and 2.Assume a sequencevn whose converges with order R with constant η and r≥1. Let us take a real number sequence αb whose order of convergence is Q and that converge to zero with r≥1 such that dvb,η≤αb.


## 3. Main Result

Definition 2.
LetV,d be a metric space and define a mapping T:V⟶CBV such that HTv,Tw be a Hausdorff metric defined on CBV; then,(6)dTv,w≤HTv,Tw,HTv,Tw≤ϱdv,w,where the gauge mapping is ϱ.Lemma 2.
LetV be a nonempty and arbitrary set and define a mapping T:C⊂V⟶CBV such that a mapping which has the starting condition of T with gauge mapping of order r is ϱ on the interval L which is D:C⟶ℝ+. Now, for T, each starting point v0∈C and each b≥0. We have(7)Dvb+1≤ϱDvb,Dvb≤Dv0λSbr.Proof.
Here,vb+1∈D and v0 is the starting point of T. As vb+1∈Tvb, using Definition 2 and Lemma 2, we obtain(8)Dvb+1=dvb+1,Tvb+1≤HTvb,Tvb+1≤ϱDvb.
Thus, we obtain(9)Dvb+1≤ϱDvb.
Now prove(10)Dvb≤Ev0λsbr.
Asϱ is nondecreasing on L, we have(11)Dvb≤ϱDvb−1≤ϱϱDvb−2≤⋯≤ϱbDv0,where ϱ is thw gauge function and λ=ϱDv0sbr. Then,(12)Dvb=ϱbDv0≤Dv0ϱDv0sbr=Dv0λsbr.Theorem 1.
Assume a metric spaceV,d and define a mapping T:C⊂V⟶CBV. The initial condition of T is D:C⟶ℝ+ and gauge mapping ϱ:L⟶L fulfills the condition ϱe≤e for all e∈L. Let η be in C. Assume that(13)HTv,Tη≤γDxdv,η,∀v∈CwithDη∈L,dTv,η≤HTv,Tη,where γ is a nondecreasing mapping on L such that γe∈0,1,∀e∈L. Then, T has a unique fixed point η at which we define E=v∈C:Dv∈L. Furthermore, for every starting point v0 of T, Picard iteration vb+1∈Tvb remains in V and converges Q-linearly to η with error which we can estimate as(14)dvb+1,η≤gdvn,η,dvn,η≤gbdv0,η,where g=γDv0.Proof.
Substitutev=η and obtain(15)dTη,η≤HTη,Tη≤γDηdη,η,where dη,η=0.
Thus,(16)dTη,η=0,this implies thatη∈Tη.
Now, we prove thatη is unique. Suppose that contrary μ is another fixed point such that μ∈Tμ and μ≠η. Then,(17)dη,μ≤HTη,Tμ≤γDηdη,μ<dη,μ.
This implies that(18)dη,μ<dη,μ,which is a contradiction such that(19)η=μ,given v0 is the initial point of Picard iteration. We obtain(20)dvb+1,η≤HTvb,Tη≤γDvbdvb,η,as Dvb≤Dv0. Then, we obtain γEvb≤γEv0. Now, following from the above inequality, we obtain(21)dvb+1,η≤HTvb,Tη≤γDvbdvb,η,and this implies that(22)dvb+1,η≤γDvbdvb,η≤γDv0dvb,η,dvb+1,η≤γDv0dvb+1,η,given g=γDv0. So(23)dvb+1,η≤gdvb,η,and dvb,η≤gdvb−1,η≤ggdvb−2,η,…,gbdv0,η. Therefore,(24)dvb,η≤gbdv0,η.This proof is complete and it is necessary to note that 0≤g≤1.Remark 1.
Letη be a number in C. It is easily represented that following the assumption of theorem, the important point Dη∈L is fulfilled iff the gauge mapping has ϱ as the fixed point. Now, take η as the root of mapping C which can be selected from the theorem. So the fixed point of ϱ is 0.Remark 2.
Without using the measurement of the error, the above result will be true. If gamma is used as one of the nondecreasing mappings onL, then gamma is right continuous.Theorem 2.
Assume a metric spaceV,d and define a mapping T:C⊂V⟶CBV. The initial condition of T is D:C⟶ℝ+ and gauge mapping ϱ:L⟶L fulfills the condition ϱe≤e for all e∈L. Let η be in C. Assume that(25)HTv,Tη≤γDvdv,η,∀v∈CwithDη∈L,Tv,η≤HTv,Tη,where γ is a nondecreasing mapping on L such that eγe is a gauge mapping which is strict of order r on L. Furthermore, for e∈L:ωe=0, it implies that γe=0ω is a nondecreasing nonnegative function on J fulfilling(26)φe=eωe,∀e∈L.Then,η is fixed point of T which is unique with E=v∈C:Dv∈L. Furthermore, for every starting point v0 of T, the following conditions are fulfilled:(1)
The iterative sequencevb+1∈Tvn is in the set E and it will converge to η. If ωDv0<1, then the sequence vb+1∈Tvb converges with R-order r.(2)
For eachb≥0, the error estimate is(27)dvb,η≤ϕnμsbrdv0,η,whereμ=ωDv0, ϕ=ψDv0, and ψ is a nonnegative mapping of L such that γe=ωeψe for each e∈L.(3)
If there is another mapping which is continuousB:G⟶ℝ+ at η, then Dv≤Bvdv,η,∀v∈G. Hence, we use the sequence which is iterative vb+1=Tvb and converges to η with the order Q- with r as(28)dvb+1,η≤Bvdvb,ηr,∀b≥0,where Bv=Bvbr−1ηDvb⟶Bηr−1η0 is b⟶∞ and a mapping which is real, η:L⟶ℝ+ is continuous at 0, and γe=er−1ηe for all e∈L.Proof.
We have already proved that fixed point is unique. Now, we have to proveR-order of convergence. Given that γe=ωeψe for all e∈L, we select a arbitrary number b which is nonnegative and represented as τ:L⟶ℝ+ as follows:(29)τe=γeωe,ifωe>0,b,ifωe=0.
Now, by result, we observe that(30)0≤μ≤1,ω≥0,and 0≤ωμ<1. Then, (31)dvb+1,η≤HTvb,Tη≤γEvbdvb,η≤γEv0dvb,η≤dvb,η≤dvb,0.
Here,dvb,η=αb. According to abovementioned lemma, the function γ satisfies(32)γπe≤πr−1γe,for allπ∈0,1ande∈L.
Now, we shall prove the next part by induction. Substituteb=0 and obtain dv0,η=dv0,η. Assume that it fulfills for b. Now, we prove that for b+1,(33)dvb+1,η≤HTvb,Tη≤γDvbdvb,η≤γDvbωnμsbrdv0,η,γμe≤μr−1γt,∀μ∈0,1ande∈L,γvb≤Dv0μbsr,γDvb≤γDv0μsbr≤μr−1sbrγDv0=ωμ1+r−1sbr,Sbr=1+r+r2+r3+⋯+rb−1,r−1Sbr=r+r2+⋯+rb−1−r−r2−⋯−rb−1=−1+rn,γDvb=ωμrb,dvb+1,η≤ωn+1μb+1Srdv0,η.
Now, we have to prove thatdvb+1,η≤cbdvb,ηr,∀b≥0.
We already know that(34)dTv,η≤HTv,Tη≤γDvdv,η,γe=er−1μe,Dv≤Cvdv,η,dvb+1,η≤HTvb,Tη≤γDvbdvb,η=Dvbr−1μDvbdvb,η≤Cvbr−1dvb,ηr−1μDvbdvb,η=Cvbr−1μDvbdvb,ηr=Cbdvb,ηr.Corollary 1.
LetV,d be a metric space and define an operator T:C⊂V⟶CBV and η∈C. Assume that(35)HTv,η≤ϱdv,η,∀v∈Canddv,η∈L,where a gauge mapping which is strict with order r≥1 is ϱ. Then, the fixed point T which is unique is η contains the set E=v∈C:dv,η∈L. Furthermore, if T:V⟶V, then for every point v0∈L, the following conditions are fulfilled:(1)
The sequencevb+1∈Tvb remains in V and converge to η with Q- order r(2)
Now, we have to estimate for allb≥0 as(36)dvb,η≤μsbrdv0,η,whereμ=Dv0 and ω is a nondecreasing nonnegative mapping on L satisfying ϱe=eωe,∀e∈L(3)
∀ we have to estimate dvb+1,η≤φdvb,ηProof.
As we know thatdTv,η≤HTv,Tη, now substitute v=η and obtain(37)dTη,η≤HTη,Tη≤ϱdη,η<dη,η=0,where η∈Tη and it is unique. Picard sequence converges with Q-order as(38)dvb+1,η≤HTvb,Tη≤ϱdvb,η<dvb,η=Dvb≤Cbdvb,ηr,dvb+1,η≤Cbdvb,ηr.
Now, we prove the other part by induction. Supposeb=0; then, equality holds dv0,η=dv0,η. Suppose that it is true for b,(39)dvb,η≤μSbrdv0,η.
Forb+1,(40)dvb+1,η≤HTvb,Tη≤ϱdvb,η=dvb,ηωdvb,ω≤μSbrdvb,ηωDvb=μSbrdv0,ημrb=μSb+1rdv0,η.Theorem 3.
Assume a metric spaceV,d and define a mapping T:C⊂V⟶CBV. The initial condition of T is D:C⟶ℝ+ and gauge mapping ϱ:L⟶L fulfills the condition ϱe≤e for all e∈L. Let η be in C. Assume that(41)HTv,η≤γDvdv,η,∀v∈CwithDη∈L,where γ is a nondecreasing mapping on L such that eγe is a gauge mapping which is strict of order r on L. Furthermore, for e∈L:ωe=0, this implies that γe=0:(42)γϱe≤γee,∀e∈L.Now, for the fixed point of the mappingT, η contains the set E=v∈C:Dc∈L. Furthermore, for the starting point v0 of T, the following statements hold:(1)
Picard sequencevb+1∈Tvb remains in V and converges to η with order of convergence R-order r. It converges with Q-order r given that Dv≤Bvdv,η,∀v∈C, where B:C⟶ℝ+ is contentious at η.(2)
For eachb≥0, we estimate that(43)dvb,η≤gsbrdv0,η,where g=γDv0 and ω is nondecreasing and nonnegative on L satisfying ϱe=rωe.Proof.
First, we prove thatη is a fixed point which is unique:(44)dTv,η≤HTv,Tη,HTv,Tη≤dv,η.Now, substituting v=η above, then(45)dTη,η≤HTη,TηγDηdη,η,dTη,η≤0.This implies that η∈Tη. Now, for uniqueness, suppose μ to be another fixed point and η≠μ, i.e.,(46)dη,μ≤HTη,Tμ≤γDηdη,μ≤dη,μ,which implies that(47)dη,μ<dη,μ,which is a contradiction so η=μ. Now, the first part of the theorem is already proved in the previous theorem. Now, we have the second part:(48)γDvb≤grb.We prove it by mathematical induction. Take b=0; then, equality holds, i.e.,(49)γDv0=γDv0.
Assume that it is true forn. Now, we prove that for b+1,(50)γDvb+1≤γϱvb≤γDvbr≤grbr=grb+1,HTvb,Tη≤grbdvb,η.We prove this by mathematical induction; now, for b=0,(51)dv0,η=dv0,η.
Assume that it is true forb; now, we prove that for b+1,(52)dvb+1,η≤HTvb,Tη≤grbdvb,η,and also note that(53)dvb,η≤gsbrdv0,η.Now,(54)dvb+1,η≤HTvb,Tη≤grb+sbrdv0,η=grb+1dv0,η.
Hence,(55)dvb+1,η≤grb+1dv0,η.


## 4. Application

We apply the proposed multivalued contraction mappings in integral theory.Theorem 4.
Consider the integral(56)ve∈G∫e0epτ,vτdτ+η,∈GVve+η,where Vve=∫e0epτ,vτdτ,G is a compact subset of real line ℝ, and p:ℝ×ℝ⟶ℝ satisfies the following conditions:(1)
p is continuous on the rectangle R=e,v:e−e0≤1/λ,v−η≤c/λ2, where λ=maxg∈Gg,0<c<a(2)
p is continuous and bounded as pe,v<1/λ2 for all e,v∈R:(57)pe,ve−pe,ηe≤γy|ve−ηe,where y=ve−η−1/λ with ve∈L and y∈I. Then, the interval I=e0−1/λ,e0+1/λ has the solution of integral.Proof.
LetCI denote those spaces which contain all those mappings which are continuous. And define metric space as dv,η=supve−ηe. Consider C˜ to be a closed subspace of the space CI which is defined as C˜:=v∈CI:dv,η≤1/λ. Define T:C˜⟶PC˜ as(58)Tvt=G∫e0etpτ,vτdτ+η=GPve+η.
Finding the solution of integral (56) becomes equivalent to the fixed point problem of (58). We represent that T is well defined and T is defined for every v∈C˜, Tv is a compact subset of C˜ for each v∈C˜. For s∈I and by definition of C˜, we obtain vs−η≤1/λ. Thus, s,vs∈R. Since p is continuous on R, way integral also exists. So T is defined for each v∈C˜. Now, we prove for each v∈C˜andTv is a compact subset of C˜. Let fe∈Tve. Then, fe=gKve+η for some g∈G and(59)fe−η=gPve=gPve≤λ∫e0ePτ,vτdτ≤λ∫e0ePτ,vτdτ<λ×1λ2=1λ,which prove that df,η≤1/λ; then, f∈C˜ since f∈Tv was arbitrary and so Tv⊂C˜ for each v∈C˜. Now, for compactness, first we prove that Tx is compact. Consider a sequence bn⊂Tv and bn=gnPvt+η for some gn∈G;n=1,2,3,…. Since G is compact, then there exists subsequence gni of gn such that gni⟶g˜. Let b=p˜Kve+η; then, we obtain(60)dbn,b=supe∈I|gni−g˜Pve≤gni−g˜supe∈IPve⟶0,asj⟶∞,and we note that Dv=dv,Tv, where Tve=G∫e0epτ,vτdτ+η and(61)dv,Tv=infw∈Tvdv,w=infdv,g∫e0epτ,vτdτ+η=infsupve−g∫e0epτ,vτdτ−η=infsupve−η−g∫e0epτ,vτdτ,where inf apply on g and λ>infg:(62)≥supve−η−λ∫e0epτ,vτdτ≥supve−η−1λ≥supw,∀w∈I=supe0+1λ≥supe,∀e∈I.
Now,γ is a nondecreasing function such that(63)γDv≥supγw,∀w∈I.
Furthermore, note that(64)HTv,Tη=HGKve+η,GPηe+η≤HGPve,GPηe,HGPve,GPηe=maxmaxa∈GPveda,GPηe,maxb∈GPηdb,GPve.Now,(65)maxa∈GPveda,GPηe=maxa∈GPvmaxb∈GPηda,b=maxg∈Gmaxg∗∈Gdgpe,v,g∗pe,η=maxg∈Gmaxg∗∈Gsupe∈Igpe,v−g∗pe,η≤maxg∈Gmaxg∗∈Gsupe∈Igpe,η−g∗pe,η+gpe,η−gpe,v≤maxg∈Gmaxg∗∈Ggsupe∈Igpe,v,g∗pe,η+g−g∗supe∈Ipe,η=maxg∈Ggsupe∈Ipe,η−pe,v=λsupe∈Ipe,η−pe,v.and we have(66)pe,η−pe,v≤∫e0epτ,ητ−pτ,vτdτ≤∫e0eγw/ητ−vτdτ≤supe∈Iγwηe−ve∫e0edτ≤γDvdv,ηe−e0≤γDv1λdv,η.Thus, we obtain(67)maxa∈GPvda,GPηe≤λγDv1λdv,η=γDvdv,η.Now, interchanging V and η, we obtain(68)maxb∈GPηdb,GPve≤γDηdη,v.Thus,(69)HTv,Tη≤γDvdv,η.



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