# Data Dependence, Strict Fixed Point Results, and Well-Posedness of Multivalued Weakly Picard Operators

Authors: Azhar Hussain; Ahsan Ali; Vahid Parvaneh; Hassen Aydi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce the notion ofs,r-contractive multivalued weakly Picard operators via simulation functions, named as Zs,r-contractions. We present some related fixed point theorems. We investigate data dependence and strict fixed point results. The well-posedness for such operators is also considered. Moreover, we generalize the results of Moţ and Petruşel. To show the usability of our results, we give some examples and an application to resolve a functional equation arising in dynamical systems.


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## Body

## 1. Introduction and Preliminaries

The fundamental theorem concerning the existence of a fixed point for a self-mapping on a metric spaceN,d is due to Banach [1].Definition 1.
A self-mapQ on N is called a contraction if there exists p˜∈0,1 such that(1)dQμ,Qω≤p˜dμ,ωholds for all μ,ω∈N.On the basis of the above definition, Banach [1] in 1922 stated the following well-known contraction principle.Theorem 1.
LetQ:N⟶N be a contraction mapping defined on a complete metric space N,d. Then, Q possesses a unique fixed point μ∗ in N. Furthermore, for any μ∈N, we have(2)limn⟶∞Qnμ=μ∗,with(3)dQnμ,μ∗≤p˜n1−p˜dμ,Qμ.In 1969, investigating the case for multivalued mappings, Nadler [2] proved that the multivalued contraction mapping possesses at least one fixed point. Before moving towards this new generalization, we recall that PN, ℭN, CBN, and KN the nonempty, closed, closed and bounded, and compact subsets of a metric space N,d, respectively. For ω∈N and U,V∈CBN, consider(4)DU,V=infdσ,λ:σ∈U,λ∈V,the Pompeiu–Hausdorff metric is defined by(5)ℋU,V≔maxsupu∈UDu,V,supv∈VDv,U,where Dω,V=infv∈Vdω,v.Lemma 1 (see [2]).
SupposeU⊆N and ξ>1. Then, for μ∈N, there is v∈V so that dμ,v≤ξDμ,V.Theorem 2 (see [2]).
LetQ:N⟶CBN be a multivalued contraction mapping on a complete metric space N,d. Then, Q possesses a fixed point.It is obvious that ifN,d is a complete metric space, then the pair CBN,ℋ and ℭN,ℋ is also complete (see e.g., [3–5]).Definition 2 (see [6]).
A mappingQ:N⟶CBN is called a multivalued weakly Picard (MWP) operator if, for all μ∈N and ω∈Qμ, there exists a sequence μn in N such that(i)
μ0=μ,μ1=ω(ii)
μn+1∈Qμn, for all n≥0(iii)
μn is convergent and its limit is a fixed point of QPopescu [7] defined the notion of s,r-contractive multivalued operators.Definition 3 (see [7]).
A multivalued operatorQ:N⟶CBN on a complete metric space N,d is called s,r-contractive if r∈0,1, s≥r, and μ,ω∈N such that(6)Dω,Qμ≤sdω,μ,impliesℋQμ,Qω≤rMTμ,ω,where(7)MTμ,ω=maxdμ,ω,Dμ,Qμ,Dω,Qω,Dμ,Qω+Dω,Qμ2.Popescu [7] showed that s,r-contractive multivalued operators are Picard, while in the single-valued case, it was shown that such operator possesses a unique fixed point. Later on, Kamran and Hussain [8] generalized the results of Popescu [7] to a weakly s,r-contractive multivalued operator. The set of fixed points of the mapping Q is defined as FixQ:=μ∈N:μ∈Qμ, while the set of strict fixed points is defined as SFixQ:=μ∈N:μ=Qμ. It is clear that SFixQ⊆FixQ.Definition 4.
(see [9, 10]). Let Y∈PN, where N,d is a metric space, and Q:N⟶ℭN be a multivalued operator. Then, the fixed point problem is well-posed for Q appropriate to D if(i)
FixQ=μ(ii)
For a sequenceμn in Y, Dμn,Qμn⟶0 as n⟶∞; then, dμn,μ⟶0 as n⟶∞Observe that a fixed point problem, which is well-posed forQ appropriate to D, is also well-posed for Q appropriate to ℋ. Moţ and Petruşel in [11] proved the results of strict fixed point sets, well-posedness, and also data dependence of the fixed point sets.An important class of functions was proposed by Khojasteh and Shukla [12] and was named as the set of simulation functions. Let ξ:0,∞×0,∞⟶ℝ verify the following conditions:ξ0ξ0,0=0.ξ1ξμ,ω<ω−μ, for all μ,ω>0.ξ2 If ωnandμn are sequences in 0,∞ such that limn⟶∞ωn=limn⟶∞μn>0, then(8)limn⟶∞supξμn,ωn<0.Suchξ is known as a simulation function proposed by Khojasteh et al. Utilizing such broad class of functions, they defined the notion of Z-contractions. We denote such class of functions by Δ.Definition 5 (see [12]).
A self-mapQ on N is said to be Z-contraction appropriate to ξ if the inequality(9)ξdQμ,Qω,dμ,ω≥0,∀μ,ω∈N,is fulfilled.On the basis of contraction mappings defined above, they gave a version of the contraction principle, which generalizes and unifies several existing fixed point results in the literature.After this work, studies involving simulation functions have been performed by various researchers (see [13–15] and references therein). Later on, Argoubi et al. [16] reshaped the notion of a simulation function by withdrawing the assertion ξ0. We denote such class of functions by Λ.Example 1 (see [16]).
Letξλ:0,∞×0,∞⟶ℝ be a function defined by(10)ξλω,μ=1,ifω,μ=0,0,λμ−ω,otherwise,where λ∈0,1. Then, ξλ∈Λ.Later on, the assertionζ3 of a simulation function was replaced with ζ3′ by Roldán-López-de-Hierro et al. [17].ζ3′ If μnandωn are sequences in 0,∞ such that limn⟶∞μn=limn⟶∞ωn>0 and μn<ωn, then(11)limn⟶∞supζμn,ωn<0.The class of simulation functionsζ fulfilling ζ1,ζ2, and ζ3′ is known as the class of simulation functions in the manner of Rolda´n−Lo´pez-de-Hierro, and we denote it by Σ. We need the following lemma.Lemma 2 (see [18]).
LetN,d be a metric space and let μn be a sequence in X such that(12)limn⟶∞dμn,μn+1=0.
Ifμn is not a Cauchy sequence in N, then there exist ε>0 and two sequences μmk and μnk of positive integers such that μnk>μmk>k and the following sequences tend to ε+ when k⟶∞:(13)dμmk,μnk,dμmk,μnk+1,dμmk−1,μnk,dμmk−1,μnk+1,dμmk+1,μnk+1.The purpose of this paper is to introduce the notion of weakly multivaluedZs,r-contractions and to prove some fixed point results. We also discuss examples to illustrate and elaborate these new concepts. After that, we present data dependence, strict fixed point set, and well-posedness results. Following these ideas, we generalize the Moţ and Petruşel result. Moreover, we present an application to functional equations arising in dynamical systems to show the usability of our results.


## 2. Main Results

We begin with the following definition.Definition 6.
LetN,d be a metric space. A mapping Q:N⟶CBN is called a weakly multivalued Zs,r-contraction with respect to ζ if there are r∈0,1, s≥r, and μ,ω∈N so that(14)Dω,Qμ≤sdω,μimplies(15)ζℋQμ,Qω,rMμ,ω≥0,where r∈0,1, ℒ=L≥0, and(16)Nμ,ω=maxdμ,ω,Dμ,Qμ,Dω,Qω,12Dμ,Qω+Dω,Qμ+ℒmindμ,ω,dω,Qμ.Example 2.
LetN=0,1 and dμ,ω=μ−ω. Define Q:N⟶CBN by(17)Qμ=0,μ+1μ+3.
Chooses=0.8. For μ,ω∈N, μ≠ω, we have(18)Dμ,Qω<sdμ,ω,which implies(19)ζℋQμ,Qω,rMμ,ω=rMμ,ω−ℋQμ,Qω≥0,∵L≥0,r∈0,1.
Hence,Q is a weakly multivalued Z0.8,r-contraction.Theorem 3.
LetN,d be a complete metric space and Q:N⟶CBN be a weakly multivalued Zs,r-contraction. Then, Q is a MWP operator.Proof.
Letμ0∈N, μ1∈Qμ0, and t be a real number such that 0<r≤t≤s<1. Choose μ2∈Qμ1. Then,(20)Dμ2,Qμ1=0≤sdμ2,μ1.
From (15) and ζ1∈Z, we have(21)0≤ζℋQμ1,Qμ2,rMμ1,μ2<rMμ1,μ2−ℋQμ1,Qμ2,which implies(22)ℋQμ1,Qμ2<rMμ1,μ2,Dμ2,Qμ2≤ℋQμ1,Qμ2<rMμ1,μ2.
Hence,(23)Dμ2,Qμ2<rMμ1,μ2=rmaxdμ1,μ2,Dμ1,Qμ1,Dμ2,Qμ2,12Dμ1,Qμ2+Dμ2,Qμ1+Lmindμ1,μ2,dμ2,Qμ1implies(24)dμ2,μ3<rdμ1,μ2+Lmindμ1,μ2,0=rdμ1,μ2.
Similarly, we can get a sequenceμn in N such that μn+1∈Qμn and(25)dμn+1,μn+2<rdμn,μn+1<rndμ1,μ2≤tndμ1,μ2.
Sincen⟶∞,dμn+1,μn+2⟶0, this shows that μn is a Cauchy sequence. N is complete, so there exists u∈N such that(26)limn⟶∞μn=u.
We now show that there exists a subsequenceμnm of μn such that(27)Du,Qμnm≤sdu,μnm,∀m∈ℕ.
On the contrary, we assume that there is a positive integerN∈ℕ such that(28)Du,Qμn>sdu,μn,∀n≥N.
This implies(29)du,μn+1>sdu,μn,∀n≥N.
By induction, we obtain(30)du,μn+p>spdu,μn,∀n≥N,p≥1.
Recall that(31)dμn+p,μn≤dμn,μn+1+dμn+1,μn+2+⋯+dμn+p−1,μn+p<dμn,μn+11+t+t2+t3+⋯+tp−1≤1−tp1−tdμn,μn+1<11−tdμn,μn+1,∀n≥N,p≥1.
Takingp⟶∞, we get that(32)du,μn<11−tdμn,μn+1,∀n≥1.
Thus, we have(33)du,μn+p<11−tdμn+p,μn+p+1<tp1−tdμn,μn+1,∀n≥N,p≥1.
From (30) and (33), one writes(34)du,μn<t/sp1−tdμn,μn+1,∀n≥N.
By takingp⟶∞, we have du,μn=0, for all n≥N. It is a contradiction with respect to equation (30). Therefore, there is a subsequence μnm of μn such that(35)Du,Qμnm≤sdu,μnm,∀m∈ℕ,which implies(36)ζℋQu,Qμnm,rMu,μnm≥0.
Now, from (36) and using ζ1 together with ζ2, we have(37)0≤limn⟶∞supζℋQu,Qμnm,rMu,μnm<limn⟶∞suprMu,μnm−ℋQu,Qμnm=limn⟶∞suprmaxdu,μnm,Du,Qu,Dμnm,Qμnm,12Du,Qμnm+Dμnm,Qu+Lmindu,μnm,Dμnm,Qu−maxDQu,Qμnm,DQμnm,Qu=limn⟶∞suprmaxdu,μnm,Du,Qu,Dμnm,μnm+1,12Du,μnm+1+Dμnm,Qu+LminDu,μnm,Dμnm,Qu−maxDQu,μnm+1,Dμnm+1,Qu=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=suprmaxDu,Qu,Du,Qu2+Lmin0,Du,Qu−maxDQu,u,Du,Qu=suprdu,Qu+0−Du,Qu=r−1Du,Qu,where r∈0,1. So,(38)0≤r−1Du,Qu<0.
From this contradiction, we getDu,Qu=0, that is, u∈Qu. Hence, Q is a MWP operator.Example 3.
LetN=1,2,3,4,5 and dμ,ω=μ−ω. Define(39)Qμ=5,ifμ∈5,2,3,ifμ∈2,3,1,4,otherwise.
Now, forμ,ω∈N, values of d, D, and ℋ for all possible pair of points are given in Table 1.
By choosings=0.5, we have(40)D1,Q2>sd1,2,D3,Q5>sd3,5,D1,Q5>sd1,5,D4,Q3>sd4,3,D2,Q1>sd2,1,D4,Q5>sd4,5,D2,Q5>sd2,5,D5,Q3>sd5,3,D3,Q1>sd3,1,D5,Q4>sd5,4,D3,Q4>sd3,4.
Furthermore, fors=0.5, we also have(41)D1,Q1=sd1,1,D5,Q5=sd5,5,D1,Q3=sd1,3,D1,Q4<sd1,4,D2,Q2=sd2,2,D2,Q3<sd4,5,D3,Q3=sd3,3,D3,Q2<sd5,3,D4,Q2=sd4,2,D4,Q1<sd5,4,D4,Q4=sd4,4,D5,Q1<sd5,1.
Now, choosingL=1 and r=0.4, we have(42)ζℋQ1,Q1,rM1,1=rM1,1−ℋQ1,Q1=0.40+10−0=0,ζℋQ1,Q3,rM1,3=rM1,3−ℋQ1,Q3=0.42+11−1>0,ζℋQ2,Q2,rM2,2=rM2,2−ℋQ2,Q2=0.40+10−0=0,ζℋQ3,Q3,rM3,3=rM3,3−ℋQ3,Q3=0.40+10−0=0,ζℋQ4,Q2,rM4,2=rM4,2−ℋQ4,Q2=0.42+11−1>0,ζℋQ4,Q4,rM4,4=rM4,4−ℋQ4,Q4=0.40+10−0=0,ζℋQ5,Q5,rM5,5=rM5,5−ℋQ5,Q5=0.40+10−0=0,ζℋQ1,Q4,rM1,4=rM1,4−ℋQ1,Q4=0.43+10−0>0,ζℋQ2,Q3,rM2,3=rM2,3−ℋQ2,Q3=0.41+10−0>0,ζℋQ3,Q2,rM3,2=rM3,2−ℋQ3,Q2=0.41+10−0>0,ζℋQ4,Q1,rM4,1=rM4,1−ℋQ4,Q1=0.43+10−0>0,ζℋQ5,Q1,rM5,1=rM5,1−ℋQ5,Q1=0.41+14−4>0.
Hence,Q is a weakly multivalued Z0.5,0.3-contraction. Thus, by Theorem 3, Q is a MWP operator.Table 1
Values ofd, D, and ℋ.
μ,ωdμ,ωDμ,QμDω,QμℋQμ,Qω(1, 1)0000(1, 2)1011(1, 3)2011(1, 4)3000(1, 5)4014(2, 1)1011(2, 2)0000(2, 3)1000(2, 4)2011(2, 5)3023(3, 1)2011(3, 2)1000(3, 3)0000(3, 4)1011(3, 5)2023(4, 1)3000(4, 2)2011(4, 3)1011(4, 4)0000(4, 5)1014(5, 1)4044(5, 2)3033(5, 3)2023(5, 4)1014(5, 5)0000Remark 1.
The weakly multivaluedZs,r-contraction is the generalization of Popescu notion of s,r-contractions. By using Definition 3 for some μ,ω∈N and s=0.5, we have(43)D1,Q3=sd1,3,which implies(44)rM1,3=0.8<ℋQ1,Q3=1,(45)D4,Q2=sd4,2,which implies(46)rM4,2=0.8<ℋQ4,Q2=1,(47)D5,Q1<sd5,1,which implies(48)rM5,1=0.4<ℋQ5,Q1=4.
Hence, from equations (44)–(48), it is clear that the notion of s,r-contraction defined by Popescu [7] is failed for Example 3.Theorem 4.
LetN,d be a complete metric space and Q:N⟶N be a weakly single-valued Zs,r-contraction operator. Then, Q possesses a fixed point. Moreover, if s≥1 and L+r<1, then Q possesses a unique fixed point.Proof.
From Theorem3, Q possesses a fixed point. Suppose that s≥1 and L+r<1. Assume that Q possesses two distinct fixed points μ and ω. Then,(49)Dω,Qμ=dμ,ω≤sdμ,ω⇒ζℋQμ,Qω,rMμ,ω≥0.
From (49) and by using ζ1, we have(50)ζℋQμ,Qω,rMμ,ω<rMμ,ω−ℋQμ,Qω=r+Ldμ,ω−dμ,ω=r+L−1dμ,ω.
We know thatr+L−1<0. Therefore, we get ζℋQμ,Qω,rMμ,ω<0, which is a contradiction to our assumption. Thus, Q possesses a unique fixed point.Theorem 5.
LetQ be a weakly multivalued Zs,r-contraction from N into CBN. Assume that there exist r,s∈0,1 such that(51)11+rDμ,Qμ≤dμ,ω≤11−sDμ,Qμ,which implies(52)ζℋQμ,Qω,rMμ,ω≥0.
Then,Q is a Picard operator.Proof.
Without loss of generality chooseQ. Take a real number t such that 0<r<t<s. Let μ1∈N and μ2∈Qμ1 such that dμ1,μ2≤1−t/1−sDμ1,Qμ1.
Then,(53)11+rDμ1,Qμ1≤Dμ1,Qμ1≤dμ1,μ2≤11−sDμ1,Qμ1.
By hypothesis, we have(54)0≤ζℋQμ1,Qμ2,rMμ1,μ2<rMμ1,μ2−ℋQμ1,Qμ2.
So,(55)ℋQμ1,Qμ2<rMμ1,μ2.
Following similar steps as in Theorem3, we can easily obtain that dμ2,μ3<rdμ1,μ2. Therefore, a sequence μn can be constructed in N such that μn+1∈Qμn and dμn+1,Qμn+2<tdμn,μn+2 for all n∈N(56)∑n=1∞dμn,μn+1<∑n=1∞tn−1dμ1,μ2<∞.
That is,μn is a Cauchy sequence. Completeness of N yields that there is u∈N so that μn converges to u.
Since(57)dμn+p,μn≤dμn,μn+1+dμn+1,μn+2+⋯+dμn+p−1,μn+p<dμn,μn+11+t+t2+t3+⋯+tp−1≤1−tp1−tdμn,μn+1,∀n≥N,p≥1.
By takingp⟶∞, we obtain(58)du,μn<11−tdμn,μn+1,∀n≥1.
We have(59)dμn,μn+1≤1−t1−sDμn,Qμn,∀n∈ℕ,du,μn+1<11−sDμn,Qμn,∀n∈ℕ.
Suppose now that there isN>0, such that(60)du,μn<11+rDμn,Qμn,∀n≥0.
Therefore,(61)dμn,μn+1≤du,μn+du,μn+1<11+rDμn,Qμn+Dμn+1,Qμn+1<11+rdμn,μn+1+rdμn,μn+1implies(62)dμn,μn+1<Dμn,Qμn,a contradiction. So, there exists a subsequence μnm of μn such that(63)du,μnm≥11+rDμnm,Qμnm,∀m≥N.
Since(64)du,μnm≤11−sDμnm,Qμnm,∀n≥1,thus, we have(65)11+rDu,Qμnm≤Du,Qμnm≤du,μnm≤11−sDu,Qμnm.
This implies that(66)ζℋQu,Qμnm,rMu,μnm≥0.
From (66), ζ1, and ζ2, one obtains(67)0≤limn⟶∞supζℋQu,Qμnm,rMu,μnm<limn⟶∞suprMu,μnm−ℋQu,Qμnm=limn⟶∞suprmaxdu,μnm,Du,Qu,Dμnm,Qμnm,12Du,Qμnm+Dμnm,Qu+Lmindu,μnm,Dμnm,Qu−maxDQu,Qμnm,DQμnm,Qu=limn⟶∞suprmaxdu,μnm,Du,Qu,Dμnm,μnm+1,12Du,μnm+1+Dμnm,Qu+Lmindu,μnm,Dμm,Qu−maxDQu,μnm+1,Dμnm+1,Qu=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=suprmaxdu,u,Du,Qu2+Lmin0,Du,Qu−maxDQu,u,Du,Qu=suprdu,Qu+0−Du,Qu=r−1Du,Qu,where r∈0,1. Hence,(68)0≤r−1Du,Qu<0.
This contradiction shows thatDu,Qu=0, that is, u∈Qu. Hence, Q is a Picard operator.Corollary 1.
LetQ be a weakly single-valued Zs,r-contraction mapping from N into N. Assume that there exist r,s∈0,1 so that(69)11+rDμ,Qμ≤dμ,ω≤11−sDμ,Qμimplies(70)ζℋQμ,Qω,rMμ,ω≥0.
Then, there existsu∈N such that u=Qu.Proof.
One can easily show that, for everyμ1∈N, the sequence μn defined by μn+1=Qμn satisfies the relationship dμn+1,μn+2<rdμn,μn+1 as done in Theorem 3. Thus, the sequence μn is Cauchy, so there is u∈N such that limn⟶∞μn=u. Following Theorem 5, we can show that du,μn≤1/1−rdμn,μn+1 for all n≥1 and there exists a subsequence μnm of μn such that du,μnm≥1/1+rdμnm,μnm+1 for all m≥N. Therefore, we obtain that(71)0≤limn⟶∞supζℋQu,Qμnm,rMu,μnm<limn⟶∞suprMu,μnm−ℋQu,Qμnm=suprmaxdu,u,Du,Qu,du,u,12du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=r−1Du,Qu<0,∵r∈0,1.
We deduce thatDu,Qu=0. Thus, we get u=Qu.


### 2.1. Data Dependence of the Fixed Point Set

In this section, we study data dependence of the fixed point set for weakly multivaluedZs,r-contractions.Theorem 6.
LetN,d be a metric space and Q1 and Q2 be two multivalued operators. Assume that(1)
Qi is a weakly Zs,ri-contraction for each i∈1,2(2)
There exists a real numberλ>0 such that ℋQ1μ,Q2μ≤λ, ∀μ∈N
Then,(1)
FixQi∈ℭN for i∈1,2.(2)
Q1 and Q2 are weakly multivalued operators and(72)ℋFixQ1,FixQ2≤λ1−maxr1,r2.Proof.
From Theorem3, FixQi is nonempty for i∈1,2. First of all, we will show that the set of fixed point of a weakly multivalued operator Q is closed. Let μn be a sequence in FixQ such that μn⟶u as n⟶∞. One obtains(73)Du,Qμn≤du,μn.
This implies that(74)ζℋQμn,Qu,rMμn,u≥0.
Now, from (74), ζ1, and ζ2, we have(75)0≤limn⟶∞supζℋQu,Qμn,rMu,μn<limn⟶∞suprMu,μn−ℋQu,Qμn=limn⟶∞suprmaxdu,μn,Du,Qu,Dμn,Qμn,12Du,Qμn+Dμn,Qu+Lmindu,μn,Dμn,Qu−maxDQu,Qμn,DQμn,Qu=limn⟶∞suprmaxdu,μn,Du,Qu,Dμn,μn+1,12Du,μn+1+Dμn,Qu+Lmindu,μn,Dμn,Qu−maxDQu,μn+1,Dμn+1,Qu=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=r−1Du,Qu<0,∵r∈0,1.
Thus, from this contradiction, we deduce thatDu,Qu=0. Since Qu∈ℭN, we have u∈Qu. Hence, u∈FixQ. Secondly, from Theorem 3, we get that a weakly multivalued Z1,r-contractive operator is a MWP operator. Let q>1 be a real number and μ0∈FixQ1 be arbitrary. Then, there exists μ1∈Q2μ0 such that dμ0,μ1≤ℋQ1μ0,Q2μ0. Next, for μ1∈Q2μ0, there is μ2∈Q2μ1 such that dμ1,μ2≤ℋQ2μ0,Q2μ1. Since μ1∈Q2μ0,Dμ1,Q2μ0=0≤dμ0,μ1. So, we have(76)dμ1,μ2≤qℋQ2μ0,Q2μ1≤qr2dμ0,μ1.
Therefore, in a similar way, we obtain that the sequence of successive approximations forQ2 starting from μ0 fulfills the following assertions:(77)μn+1∈Qμn,dμn,μn+1≤qr2ndμ0,μ1,∀n≥1.
Hence, for alln≥N and p≥1,(78)dμn+p,μn≤dμn,μn+1+dμn+1,μn+2+⋯+dμn+p−1,μn+p≤qr2ndμ0,μ1+qr2n+1dμ0,μ1+⋯+qr2n+p−1dμ0,μ1≤qr2n1−qr2dμ0,μ1.
Choosing now1<q<min1/r1,1/r2 and letting n⟶∞, we obtain that μn is a Cauchy sequence in N,d. Then, there exists u∈N such that μn⟶u as n⟶∞. We will prove that u is a fixed point for Q2. Suppose that there exists a positive number N such that Du,Qμn>du,μn∀n≥N. Then, du,μn+1>du,μn∀n≥N. Since μn⟶u as n⟶∞, we get a contradiction. Hence, there exists a subsequence μnm such that Du,Q2μnm≤du,μnm∀m∈ℕ. Thus,(79)ζℋQ2u,Q2μnm,r2Mu,μnm≥0.
Now, from (79), ζ1, and ζ2, we have(80)0≤limn⟶∞supζℋQ2u,Q2μn,r2Mu,μnm<limn⟶∞supr2Mu,μnm−ℋQ2u,Q2μnm=limn⟶∞supr2maxdu,μnm,Du,Q2u,Dμnm,Q2μnm,12Du,Q2μnm+Dμnm,Q2u+Lmindu,μnm,Dμnm,Q2u−maxDQ2u,Q2μnm,DQ2μnm,Q2u=limn⟶∞supr2maxdu,μnm,Du,Q2u,Dμnm,μnm+1,12Du,μnm+1+Dμnm,Q2u+Lmindu,μnm,Dμnm,Q2u−maxDQ2u,μnm+1,Dμnm+1,Q2u=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=suprmaxdu,u,Du,Qu2+Lmin0,Du,Qu−maxDQu,u,Du,Qu=suprDu,Qu+0−Du,Qu=r−1Du,Qu,where r2∈0,1. So,(81)0≤r2−1Du,Q2u<0.
It is a contradiction. So, we obtain thatDu,Q2u=0, that is, u∈Q2u. Hence, u∈FixQ2.
By takingp⟶∞ in (78), we have du,μn≤qr2n/1−qr2dμ0,μ1 for each n∈N. Then, for n:=0, we get dμ0,u≤1/1−qr2dμ0,μ1≤qλ/1−qr2. In a similar way, we get that, for each u0∈FixQ2, there exists μ∈FixQ1 such that du0,μ≤1/1−qr1du0,u1≤qλ/1−qr1. Hence,(82)ℋFixQ1,FixQ2≤qλ1−maxqr1,qr2.
Lettingq⟶1 completes the proof. Moreover, we get that Qi is a MWP operator for i∈1,2.


### 2.2. Strict Fixed Point and Well-Posedness

Now, we prove the well-posedness for a weakly multivaluedZs,r-contractive operator with s>1.Theorem 7.
LetN,d be a complete metric space and Q:N⟶ℭN be a multivalued operator. Assume that(i)
Q is a weakly multivalued Zs,r-contractive operator with s≥1(ii)
SFixQ≠∅
Then,(a)
FixQ=SFixQ=z(b)
The fixed point problem is well-posed appropriate toℋ if s>1Proof.
(a)
We will prove thatFixQ=z. We suppose that u,z∈FixQ with u≠z.Since(83)Du,Qz=du,z≤sdu,z,one obtains(84)ζℋQu,Qz,rMu,z≥0.By usingζ1, we have(85)0≤ζℋQu,Qz,rMu,z<rMu,z−ℋQu,Qz.Hence, we obtainℋQu,Qz<rMu,z. We know that DQz,u≤ℋQu,Qz. So,(86)DQz,u=dz,u≤ℋQu,Qz<rMu,z=rdu,z+Lmindu,z,du,Qz=rdu,z.Thus,dz,u<rdu,z, where r∈0,1. It is a contradiction. Hence, we deduce that u=z and FixQ=z.(b)
Letμn∈N,n∈N, be such that Dμn,Qμn⟶0 as n⟶∞. We will prove that dμn,u⟶0 as n⟶∞. Arguing by contradiction, we suppose that dμn,u does not converge to 0. Then, there exist ε>0 and a subsequence μnm such that dμnm,u≥ε∀m∈N. If there exists a subsequence μnm of μnm such that(87)Du,Qμnm≤du,μnm,it implies(88)ζℋQu,Qμnm,rMu,μnm≥0.By following similar calculations as in Theorem3, one gets du,μnm⟶0 as n⟶∞. Hence, we obtain ε≤dμnm,u⟶0 as n⟶∞, which is a contradiction to our supposition, that is, ε>0.Thus, there existsDu,μnm>sdu,μnm for all m≥m1. We know that Dμn,Qμn⟶0 as n⟶∞, so there exists Dμnm,Qμnm<s−1ε for all m≥m2. Thus,(89)s−1ε>Dμnm,Qμnm≥Dz,Qμnm−du,μnm>sdu,μnm≤s−1ε,for allm≥m2. It is impossible. Therefore, μn⟶u as n⟶∞.


### 2.3. An Extension of Moţ–Petruşel Theorem

Following Reich [19], Moţ and Petruşel initiated the following concepts.Definition 7.
LetN,d be a metric space and Y∈PN. Then, Q:Y⟶CBN is called an a,b,c-KSR multivalued operator if there are a,b,c∈R+ so that, for μ,ω∈Y, we have(90)1−b−c1+aDμ,Qμ≤dμ,ω⇒ℋQμ,Qω≤adμ,ω+bDμ,Qμ+cDω,Qω.Theorem 8.
LetN,d be a complete metric space and Q:N⟶CBN be an a,b,c-KSR multivalued operator. Then, FixQ≠∅. Moreover, Q is a MWP operator.Now, we prove the generalization of Theorem8 in the case b+c>0, by using a weakly multivalued Zs,r-contractive operator.Theorem 9.
LetN,d be a complete metric space and Q:N⟶CBN. Assume that there exist a,b,c∈R+ with a+b+c∈0,1,b+c>0, so that, for all μ,ω∈N, we have(91)Dω,Qμ≤a+b+c1−b−cdμ,ω,which implies(92)ζℋQμ,Qω,a+b+cMμ,ω≥0.
Then,FixQ≠∅.Proof.
Letμ0∈N and μ1∈Qμ0. Choose r1 be a real number such that(93)0<a+b+c<r1≤a+b+c1−b−c<1.
One can chooseμ2∈Qμ1, then(94)Dμ2,Qμ1=0≤a+b+c1−b−cdμ2,μ1,which implies(95)0≤ζℋQμ1,Qμ2,a+b+cMμ1,μ2<a+b+cMμ1,μ2−ℋQμ1,Qμ2.
That is,(96)ℋQμ1,Qμ2<a+b+cMμ1,μ2.
Therefore,(97)Dμ2,Qμ2≤ℋQμ1,Qμ2<a+b+cMμ1,μ2=a+b+cdμ1,μ2.
Thus, we obtain(98)dμ2,μ3<a+b+cdμ1,μ2.
From this way, we can easily construct the sequenceμn in N such that μn+1∈Qμn and(99)dμn+1,μn+2<a+b+cdμn,μn+1<a+b+cn−1dμ1,μ2≤r1n−1dμ1,μ2.
Asn⟶∞, one obtains(100)dμn+1,μn+2⟶0.
This shows thatμn is a Cauchy sequence. Since N is complete, there is u∈N so that limn⟶∞μn=u. Now, we suppose that there exists a subsequence μnm of μn such that(101)Du,Qμnm≤a+b+c1−b−cdu,μnm,for all m∈ℕ. Assume on the contrary that there is a positive integer N∈ℕ such that(102)Du,Qμn>a+b+c1−b−cdu,μn,∀n∈ℕ.
Then, following similar procedure of Theorem3, we can show that this contrary assumption leads to the contradiction. Therefore, there exists a subsequence μnm of μn such that(103)Du,Qμnm≤a+b+c1−b−cdu,μnm,for all m∈ℕ. This implies that(104)ζℋQu,Qμnm,a+b+cMu,μnm≥0.
By solving this inequality, using the definition of a simulation function, we will obtainDu,Qu=0, that is, u∈Qu. Hence, FixQ≠∅.


## 2.1. Data Dependence of the Fixed Point Set

In this section, we study data dependence of the fixed point set for weakly multivaluedZs,r-contractions.Theorem 6.
LetN,d be a metric space and Q1 and Q2 be two multivalued operators. Assume that(1)
Qi is a weakly Zs,ri-contraction for each i∈1,2(2)
There exists a real numberλ>0 such that ℋQ1μ,Q2μ≤λ, ∀μ∈N
Then,(1)
FixQi∈ℭN for i∈1,2.(2)
Q1 and Q2 are weakly multivalued operators and(72)ℋFixQ1,FixQ2≤λ1−maxr1,r2.Proof.
From Theorem3, FixQi is nonempty for i∈1,2. First of all, we will show that the set of fixed point of a weakly multivalued operator Q is closed. Let μn be a sequence in FixQ such that μn⟶u as n⟶∞. One obtains(73)Du,Qμn≤du,μn.
This implies that(74)ζℋQμn,Qu,rMμn,u≥0.
Now, from (74), ζ1, and ζ2, we have(75)0≤limn⟶∞supζℋQu,Qμn,rMu,μn<limn⟶∞suprMu,μn−ℋQu,Qμn=limn⟶∞suprmaxdu,μn,Du,Qu,Dμn,Qμn,12Du,Qμn+Dμn,Qu+Lmindu,μn,Dμn,Qu−maxDQu,Qμn,DQμn,Qu=limn⟶∞suprmaxdu,μn,Du,Qu,Dμn,μn+1,12Du,μn+1+Dμn,Qu+Lmindu,μn,Dμn,Qu−maxDQu,μn+1,Dμn+1,Qu=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=r−1Du,Qu<0,∵r∈0,1.
Thus, from this contradiction, we deduce thatDu,Qu=0. Since Qu∈ℭN, we have u∈Qu. Hence, u∈FixQ. Secondly, from Theorem 3, we get that a weakly multivalued Z1,r-contractive operator is a MWP operator. Let q>1 be a real number and μ0∈FixQ1 be arbitrary. Then, there exists μ1∈Q2μ0 such that dμ0,μ1≤ℋQ1μ0,Q2μ0. Next, for μ1∈Q2μ0, there is μ2∈Q2μ1 such that dμ1,μ2≤ℋQ2μ0,Q2μ1. Since μ1∈Q2μ0,Dμ1,Q2μ0=0≤dμ0,μ1. So, we have(76)dμ1,μ2≤qℋQ2μ0,Q2μ1≤qr2dμ0,μ1.
Therefore, in a similar way, we obtain that the sequence of successive approximations forQ2 starting from μ0 fulfills the following assertions:(77)μn+1∈Qμn,dμn,μn+1≤qr2ndμ0,μ1,∀n≥1.
Hence, for alln≥N and p≥1,(78)dμn+p,μn≤dμn,μn+1+dμn+1,μn+2+⋯+dμn+p−1,μn+p≤qr2ndμ0,μ1+qr2n+1dμ0,μ1+⋯+qr2n+p−1dμ0,μ1≤qr2n1−qr2dμ0,μ1.
Choosing now1<q<min1/r1,1/r2 and letting n⟶∞, we obtain that μn is a Cauchy sequence in N,d. Then, there exists u∈N such that μn⟶u as n⟶∞. We will prove that u is a fixed point for Q2. Suppose that there exists a positive number N such that Du,Qμn>du,μn∀n≥N. Then, du,μn+1>du,μn∀n≥N. Since μn⟶u as n⟶∞, we get a contradiction. Hence, there exists a subsequence μnm such that Du,Q2μnm≤du,μnm∀m∈ℕ. Thus,(79)ζℋQ2u,Q2μnm,r2Mu,μnm≥0.
Now, from (79), ζ1, and ζ2, we have(80)0≤limn⟶∞supζℋQ2u,Q2μn,r2Mu,μnm<limn⟶∞supr2Mu,μnm−ℋQ2u,Q2μnm=limn⟶∞supr2maxdu,μnm,Du,Q2u,Dμnm,Q2μnm,12Du,Q2μnm+Dμnm,Q2u+Lmindu,μnm,Dμnm,Q2u−maxDQ2u,Q2μnm,DQ2μnm,Q2u=limn⟶∞supr2maxdu,μnm,Du,Q2u,Dμnm,μnm+1,12Du,μnm+1+Dμnm,Q2u+Lmindu,μnm,Dμnm,Q2u−maxDQ2u,μnm+1,Dμnm+1,Q2u=suprmaxdu,u,Du,Qu,Du,u,12Du,u+Du,Qu+Lmindu,u,Du,Qu−maxDQu,u,Du,Qu=suprmaxdu,u,Du,Qu2+Lmin0,Du,Qu−maxDQu,u,Du,Qu=suprDu,Qu+0−Du,Qu=r−1Du,Qu,where r2∈0,1. So,(81)0≤r2−1Du,Q2u<0.
It is a contradiction. So, we obtain thatDu,Q2u=0, that is, u∈Q2u. Hence, u∈FixQ2.
By takingp⟶∞ in (78), we have du,μn≤qr2n/1−qr2dμ0,μ1 for each n∈N. Then, for n:=0, we get dμ0,u≤1/1−qr2dμ0,μ1≤qλ/1−qr2. In a similar way, we get that, for each u0∈FixQ2, there exists μ∈FixQ1 such that du0,μ≤1/1−qr1du0,u1≤qλ/1−qr1. Hence,(82)ℋFixQ1,FixQ2≤qλ1−maxqr1,qr2.
Lettingq⟶1 completes the proof. Moreover, we get that Qi is a MWP operator for i∈1,2.


## 2.2. Strict Fixed Point and Well-Posedness

Now, we prove the well-posedness for a weakly multivaluedZs,r-contractive operator with s>1.Theorem 7.
LetN,d be a complete metric space and Q:N⟶ℭN be a multivalued operator. Assume that(i)
Q is a weakly multivalued Zs,r-contractive operator with s≥1(ii)
SFixQ≠∅
Then,(a)
FixQ=SFixQ=z(b)
The fixed point problem is well-posed appropriate toℋ if s>1Proof.
(a)
We will prove thatFixQ=z. We suppose that u,z∈FixQ with u≠z.Since(83)Du,Qz=du,z≤sdu,z,one obtains(84)ζℋQu,Qz,rMu,z≥0.By usingζ1, we have(85)0≤ζℋQu,Qz,rMu,z<rMu,z−ℋQu,Qz.Hence, we obtainℋQu,Qz<rMu,z. We know that DQz,u≤ℋQu,Qz. So,(86)DQz,u=dz,u≤ℋQu,Qz<rMu,z=rdu,z+Lmindu,z,du,Qz=rdu,z.Thus,dz,u<rdu,z, where r∈0,1. It is a contradiction. Hence, we deduce that u=z and FixQ=z.(b)
Letμn∈N,n∈N, be such that Dμn,Qμn⟶0 as n⟶∞. We will prove that dμn,u⟶0 as n⟶∞. Arguing by contradiction, we suppose that dμn,u does not converge to 0. Then, there exist ε>0 and a subsequence μnm such that dμnm,u≥ε∀m∈N. If there exists a subsequence μnm of μnm such that(87)Du,Qμnm≤du,μnm,it implies(88)ζℋQu,Qμnm,rMu,μnm≥0.By following similar calculations as in Theorem3, one gets du,μnm⟶0 as n⟶∞. Hence, we obtain ε≤dμnm,u⟶0 as n⟶∞, which is a contradiction to our supposition, that is, ε>0.Thus, there existsDu,μnm>sdu,μnm for all m≥m1. We know that Dμn,Qμn⟶0 as n⟶∞, so there exists Dμnm,Qμnm<s−1ε for all m≥m2. Thus,(89)s−1ε>Dμnm,Qμnm≥Dz,Qμnm−du,μnm>sdu,μnm≤s−1ε,for allm≥m2. It is impossible. Therefore, μn⟶u as n⟶∞.


## 2.3. An Extension of Moţ–Petruşel Theorem

Following Reich [19], Moţ and Petruşel initiated the following concepts.Definition 7.
LetN,d be a metric space and Y∈PN. Then, Q:Y⟶CBN is called an a,b,c-KSR multivalued operator if there are a,b,c∈R+ so that, for μ,ω∈Y, we have(90)1−b−c1+aDμ,Qμ≤dμ,ω⇒ℋQμ,Qω≤adμ,ω+bDμ,Qμ+cDω,Qω.Theorem 8.
LetN,d be a complete metric space and Q:N⟶CBN be an a,b,c-KSR multivalued operator. Then, FixQ≠∅. Moreover, Q is a MWP operator.Now, we prove the generalization of Theorem8 in the case b+c>0, by using a weakly multivalued Zs,r-contractive operator.Theorem 9.
LetN,d be a complete metric space and Q:N⟶CBN. Assume that there exist a,b,c∈R+ with a+b+c∈0,1,b+c>0, so that, for all μ,ω∈N, we have(91)Dω,Qμ≤a+b+c1−b−cdμ,ω,which implies(92)ζℋQμ,Qω,a+b+cMμ,ω≥0.
Then,FixQ≠∅.Proof.
Letμ0∈N and μ1∈Qμ0. Choose r1 be a real number such that(93)0<a+b+c<r1≤a+b+c1−b−c<1.
One can chooseμ2∈Qμ1, then(94)Dμ2,Qμ1=0≤a+b+c1−b−cdμ2,μ1,which implies(95)0≤ζℋQμ1,Qμ2,a+b+cMμ1,μ2<a+b+cMμ1,μ2−ℋQμ1,Qμ2.
That is,(96)ℋQμ1,Qμ2<a+b+cMμ1,μ2.
Therefore,(97)Dμ2,Qμ2≤ℋQμ1,Qμ2<a+b+cMμ1,μ2=a+b+cdμ1,μ2.
Thus, we obtain(98)dμ2,μ3<a+b+cdμ1,μ2.
From this way, we can easily construct the sequenceμn in N such that μn+1∈Qμn and(99)dμn+1,μn+2<a+b+cdμn,μn+1<a+b+cn−1dμ1,μ2≤r1n−1dμ1,μ2.
Asn⟶∞, one obtains(100)dμn+1,μn+2⟶0.
This shows thatμn is a Cauchy sequence. Since N is complete, there is u∈N so that limn⟶∞μn=u. Now, we suppose that there exists a subsequence μnm of μn such that(101)Du,Qμnm≤a+b+c1−b−cdu,μnm,for all m∈ℕ. Assume on the contrary that there is a positive integer N∈ℕ such that(102)Du,Qμn>a+b+c1−b−cdu,μn,∀n∈ℕ.
Then, following similar procedure of Theorem3, we can show that this contrary assumption leads to the contradiction. Therefore, there exists a subsequence μnm of μn such that(103)Du,Qμnm≤a+b+c1−b−cdu,μnm,for all m∈ℕ. This implies that(104)ζℋQu,Qμnm,a+b+cMu,μnm≥0.
By solving this inequality, using the definition of a simulation function, we will obtainDu,Qu=0, that is, u∈Qu. Hence, FixQ≠∅.


## 3. An Application

Many authors [20–22] have studied the existence and uniqueness of a solution of functional equations arising in dynamic programming. We make use of Theorem 4 to explore the existence and uniqueness of a solution for a class of functional equations. From now on, U and V are Banach spaces and W⊂U,D⊂V, and ℜ is the set of real numbers. We denote by BW the set of all bounded real-valued functions on W. The set BW equipped with the metric,(105)dBh,m=supμ∈Whμ−mμ,h,m∈BW,is a complete metric space. Viewing W and D as the state and decision space, respectively, the problem of dynamic programming reduces to the problem of solving the functional equation Pμ=supω∈Dℋμ,ω,ptμ,ω, where t:W×D⟶W represents the transformation of the process and Pμ corresponds to the optimal return function with an initial functional equation:(106)Pμ=supω∈Dgμ,ω+Gμ,ω,ptμ,ω,μ∈W,where g:W×D⟶ℜ and G:W×D×ℜ⟶ℜ are bounded functions. Let Q be defined by Qhμ=supω∈Dgμ,ω+Gμ,ω,ptμ,ω, where h∈BW and μ∈W.Theorem 10.
Suppose that there existr∈0,1,s>r such that, for every μ,ω∈W×D,h,m∈BW and t∈W, the inequality(107)mt−Qht≤smt−htimplies(108)ζGμ,ω,ht−Gμ,ω,mt,rMht,mt≥0,where(109)Mht,mt=maxht−mt,mt−Qmt,|ht−Qht,ht−Qmt+mt−Qht2+ℒminht−mt,mt−Qht,with ℒ=L≥0. Then, the functional equation (106) possesses a bounded solution. Moreover, if s≥1 and L+r<1, then such a solution is unique.Proof.
LetQ be a self-map of BW. Let λ be an arbitrary real number and h,m∈BW. Pick μ∈W. We can choose ω1,ω2∈D so that(110)Qhμ<gμ,ω1+Gμ,ω1,ht1+λ,(111)Qmμ<gμ,ω2+Gμ,ω2,mt2+λ,where ti=tiμ,ωi, i∈1,2.
Using the definition ofQ, we obtain(112)Qhμ≥gμ,ω2+Gμ,ω2,ht2,(113)Qmμ≥gμ,ω1+Gμ,ω1,mt1.
If inequality (107) holds, then(114)ζGμ,ω,ht−Gμ,ω,mt,rMhμ,mμ≥0.
From (114) and ζ1, one writes(115)0≤ζGμ,ω,ht−Gμ,ω,mt,rMhμ,mμ<rMhμ,mμ−Gμ,ω,ht−Gμ,ω,mt.
This yields that(116)Gμ,ω,ht−Gμ,ω,mt<rMhμ,mμ.
In view of (111), (114), and (117), we have(117)Qhμ−Qmμ<Gμ,ω1,ht1−Gμ,ω1,mt1+λ<rMhμ,mμ+λ.
Similarly, from (112), (113), and (117), we have(118)Qmμ−Qhμ<rMhμ,mμ+λ.
Thus, from equations (117) and (118), the following,(119)Qhμ−Qmμ<rMhμ,mμ+λ,holds for all μ∈W and λ>0. Hence, we get that(120)dBm,Qh≤sdBm,himplies(121)ζdBQh,Qm,rMh,m≥0,where(122)rMh,m=rmaxh−m,m−Qm,h−Qh,h−Qm+m−Qh2+Lminh−m,m−Qh,whererℒ=L≥0.
Hence, all assertions of Theorem4 are fulfilled for the mapping Q; therefore, we get the required result.



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*Source: 5518647-2021-03-16.xml*