# Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions

Authors: Iram Iqbal; Nawab Hussain; Hamed H. Al-Sulami; Shanza Hassan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature.


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## Body

## 1. Introduction and Preliminaries

LetZ,d be a metric space (in short MS). The set of all nonempty subsets of Z is denoted by PZ, the set of all nonempty closed subsets of Z is denoted by CLZ, the set of all nonempty closed and bounded subsets of Z is denoted by CBZ, and the set of all nonempty compact subsets of X is denoted by KZ. It is obvious that CBZ includes KZ. For U,V∈CBZ, define H:CBZ×CBZ⟶0,∞ by
(1)HU,V=maxsupu∈UDu,V,supv∈VDv,U,whereDu,V=infdu,v:v∈V. Such a function H is called the Pompei-Hausdorff metric induced by d, for more details, see, e.g., [1].Lemma 1 [2].
LetZ,d be a MS and A,B∈CLZ with HA,B>0. Then, for each h>1 and for each a∈A, there exists b=ba∈B such that da,b<hHA,B.IfΩ:Z⟶PZ is a multivalued operator, then an element ϖ∈Z is called a fixed point for Ω if x∈Ωϖ. The symbol fix Ω=ϖ∈Z:x∈Ωϖ denotes the fixed point set of Ω. On the other hand, a strict fixed point for Ω is an element ϖ∈Z with the property x=Ωϖ. The set of all strict fixed points of Ω is denoted by SFix Ω.Banach’s contraction principle [3] is the most fundamental result in metric fixed point theory. Since then, many authors have extended and generalized Banach’s contraction principle in many ways. Extensions of Banach’s contraction principle have spawned a wealth of literature. (see [13, 29]). One of an attractive and important generalization is given by Wardowski in [10]. He introduced a new type of contraction called F-contraction and proved a new fixed point theorem concerning F-contraction.Definition 2 [10].
LetZ,d be a MS. A mapping Ω:Z⟶Z is said to be F-contraction if there exists τ>0 such that
(2)dΩϖ,Ωω>0impliesτ+FdΩϖ,Ωω≤Fdϖ,ω,
for allx,y∈X, where F:0,∞⟶ℝ is a function satisfying
(Ϝ1) F is strictly increasing
(Ϝ2) For all sequence tn⊆0,∞, limn⟶∞tn=0, if and only if limn⟶∞Ftn=−∞
(Ϝ3) There exists 0<k<1 such that limk⟶0+tkFt=0We denote byΔϜ the collection of all functions F:0,∞⟶ℝ satisfying (Ϝ1), (Ϝ2), and (Ϝ3). Also, define
(3)Δ○∗=F∈ΔϜ∣FsatifiesϜ4,where(Ϝ4) FinfA=infFA for all A⊂0,∞ with infA>0.Theorem 3 [10].
LetZ,d be a complete MS and Ω:Z⟶Z be a F-contraction. Then, Ω has a unique fixed point ϖ∗∈Z and for every ϖ0∈Z, a picard sequence Tnϖ0n∈ℕ converges to ϖ∗.Further, Turinici [11] is replaced (Ϝ2) by the following condition: (Ϝ2′) limt⟶0+Ft=−∞.Note that, in general,F∈ΔϜ is not continuous. However, by Ϝ1 and the properties of the monotone functions, we have the following proposition.Proposition 4 [11].
LetF:0,∞⟶ℝ be a function satisfying Ϝ1 and Ϝ2, and then there exists a countable subset ΛF⊆0,1 such that
(4)Ft−0=Ft=Ft+0foreacht∈0,1\ΛF.Lemma 5 [11].
LetF:0,∞⟶ℝ be a function satisfying Ϝ1 and Ϝ2′. Then, for each sequence tn in 0,1,
(5)Ftn⟶−∞⇒tn⟶0.After this, many authors generalized theF-contraction in several ways (see [12–22] and references therein). In 2015, Klim and Wardowski [23] extended the concept of F-contractive mappings to the case of nonlinear F-contractions and proved fixed point theorems via the dynamic processes. In 2017, Wardowski [24] omitted one of the conditions of F-contraction and introduced nonlinear F-contraction).Definition 6 [24].
A mappingΩ:Z⟶Z is said to be a φ,F-contraction (or nonlinear F-contraction), if there exists F∈F and a function φ:0,∞⟶0,∞ satisfying
(H1)liminfs⟶t+φs>0,forallt≥0.
(H2)φdϖ,ω+FdΩϖ,Ωω≤Fdϖ,ω,forallϖ,ω∈ZsuchthatΩϖ≠ΩωTheorem 7 [24].
LetZ,d be a complete MS and let Ω:Z⟶Z be a φ,F-contraction. Then, Ω has a unique fixed point in Z.Very recently, Iqbal and Rizwan [25] considered a rich class of functions and generalized Definition 6 to obtain some new fixed point theorems for nonlinear F-contractions involving generalized distance. On unifying the concept of Wardowski [10], Nadler [9] and Altun et al. [26] gave the concept of multivalued F-contraction as follows.Definition 8 [26].
LetZ,d be a complete MS and Ω:Z⟶CBZ be a mapping. Then, Ω is a multivalued F-contraction, if there exists τ>0 and F∈ΔF such that for all ϖ,ω∈Z,
(6)HΩϖ,Ωω>0⇒τ+FHΩϖ,Ωω≤Fdϖ,ω.Theorem 9 [26].
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued F-contraction, and then Ω has a fixed point in Z.Afterwards, Olgun et al. [27] proved the nonlinear case of Theorem 9 as follows.Theorem 10 [27].
LetZ,d be a complete MS and Ω:Z⟶KZ, if there exists F∈ΔϜ and φ:0,1⟶0,1 satisfying
(7)liminfs⟶t+φs>0forallϖ,ω∈Z,φdϖ,ω+FHΩϖ,Ωω≤Fdϖ,ω.
Then,Ω has a fixed point in Z.For more directions for nonlinearF-contractions, consult [28, 29] and references there in. Next, we denote by P the set of all continuous mappings ρ:0,∞5⟶0,∞ satisfying the following conditions:(ρ1) ρ1,1,1,2,0∈0,1;(ρ2) ρ is subhomogeneous; that is, for all ϖ1,ϖ2,ϖ3,ϖ4,ϖ5∈0,∞5 and α≥0, we have ραϖ1,αϖ2,αϖ3,αϖ4,αϖ5≤αρϖ1,ϖ2,ϖ3,ϖ4,ϖ5(ρ3) ρ is nondecreasing function; that is, for ϖi,ωi∈ℝ+, ϖi≤ωi, i=1,⋯,5, we have ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5≤ρω1,ω2,ω3,ω4,ω5Ifϖi,ωi∈ℝ+, ϖi<ωi, i=1,⋯,4, then ρϖ1,ϖ2,ϖ3,ϖ4,0<ρω1,ω2,ω3,ω4,0 and ρϖ1,ϖ2,ϖ3,0,ϖ5<ρω1,ω2,ω3,0,ω5.Also, define(8)ℙ=ρ∈Pρ1,0,0,1,1∈0,1.Note thatℙ⊆P.Example 1.
Defineρ1:0,∞5⟶0,∞ by
(9)ρ1ϖ1,ϖ2,ϖ3,ϖ4,ϖ5=ϖ1+ξx5,where ξ∈0,1. Then, ρ1∈P. Since ρ11,0,0,1,1=1+ξ>1, so ρ1∉ℙ. Also, note that ρ11,1,1,0,2=1+2ξ>1, so ρ1∉R [30].Example 2.
Defineρ2:0,∞5⟶0,∞ by
(10)ρ2ϖ1,ϖ2,ϖ3,ϖ4,ϖ5=gmaxϖ1,12ϖ2+ϖ3,12ϖ4+ϖ5,where g∈0,1. Then, ρ2∈ℙ.Example 3.
Defineρ3:0,∞5⟶0,∞ by
(11)ρ3ϖ1,ϖ2,ϖ3,ϖ4,ϖ5=aϖ1+bϖ2+ϖ3+cϖ4+ϖ5,where a+2b+2c<1. Then, ρ3∈P.Now, we prove the following Lemma.Lemma 11.
Ifρ∈P and u,v∈0,∞ are such that
(12)u≤maxρv,v,u,v+u,0,ρv,v,u,0,v+u,ρv,u,v,v+u,0,ρv,u,v,0,v+u,then u≤v.Proof.
Without loss of generality, we can suppose thatu≤ρv,v,u,v+u,0. If v<u, then
(13)u≤ρv,v,u,v+u,0<ρu,u,u,2u,0≤uρ1,1,1,2,0≤u,a contradiction. Thus, u≤v.Now, consider following examples.Example 4.
LetF:0,∞⟶ℝ be a function defined by
(14)Ft=−1tforallt∈0,∞.
Then,F satisfies Ϝ1, Ϝ2′, and F that is continuous but does not satisfies Ϝ3.Example 5.
LetF:0,∞⟶ℝ be a function defined by
(15)Ft=−1tift∈0,10otherwise.
Then,F satisfies Ϝ1 and Ϝ2′ but F is not continuous.Example 6.
[31] Let F:0,∞⟶ℝbe a function defined by
(16)Ft=−1t+tℓ,where t denotes the integral part of Ω and ℓ∈0,1/a, a>1. Then, F satisfies Ϝ1, Ϝ2′, and Ϝ3 but F is not continuous.Examples4–6 clearly show that there exist some functions F:0,∞⟶ℝ which does not satisfy the condition of continuity, Ϝ1, Ϝ2, and Ϝ3 at a time. By getting inspiration from this, in this paper, we prove fixed point results for contractive conditions involving functions F, not necessarily continuous and belongs to ΔϜ by taking support of a continuous function from P. Our results generalize many results appearing recently in the literature including Altun et al. [32], Olgun et al., [27] Sgroi and Vetro [33], Vetro [34], Wardowski [24], and Wardowski and Dung [35].For convenience, we setΦ, the collection of all functions χ:0,∞⟶0,∞ satisfying
(17)lims⟶t+infχs>0forallt≥0.Theorem 12.
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued mapping. Assume that there exists χ∈Φ, a nondecreasing real valued function F1 on 0,∞ and a real valued function F2on0,∞ satisfying condition Ϝ2′ and Ϝ3 such that the following conditions hold:
(N1)F1c≤F2c for all c>0
(N2) For allϖ,ω∈Z and ρ∈P, HΩϖ,Ωω>0 implies
(18)χdϖ,ω+F2HΩϖ,Ωω≤F1ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.
Then, fixΩ is nonempty.Proof.
Letϖ0∈Z be an arbitrary point and ϖ1∈Ωϖ0. Assume that ϖ1∈Ωϖ1; otherwise, ϖ1 is a fixed point of Ω, and the proof is complete. Then, Dϖ1,Ωϖ1>0 and consequently HΩϖ0,Ωϖ1>0. Compactness of Ωϖ1 ensures the existence of ϖ2∈Ωϖ1, such that dϖ1,ϖ2=Dϖ1,Ωϖ1. From (N1) and (N2), we get
(19)F1dϖ1,ϖ2=F1Dϖ1,Ωϖ1≤F1HΩϖ0,Ωϖ1≤F2HΩϖ0,Ωϖ1≤F1ρdϖ0,ϖ1,Dϖ0,Ωϖ0,Dϖ1,Ωϖ1,Dx0,Ωϖ1,Dϖ1,Ωϖ0−χdϖ0,ϖ1<F1ρdϖ0,ϖ1,dx0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ2,0.SinceF1 is an nondecreasing function, (19) with ρ3 implies that
(20)dϖ1,ϖ2<ρdϖ0,ϖ1,dx0,ϖ1,dϖ1,ϖ2,dx0,ϖ1,0≤ρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ1+dϖ1,ϖ2,0.By using Lemma11, (20) implies
(21)dϖ1,ϖ2<dϖ0,ϖ1.Next, arguing as previous, we getϖ3∈Ωϖ2, such that dϖ2,ϖ3=Dϖ2,Ωϖ2 with Dx2,Ωϖ2>0. Also, by using Lemma 11, from (N1) and (N2), we obtain
(22)dϖ2,ϖ3<dϖ1,ϖ2.Continuing in the same manner, we get a sequenceϖn⊂Z such that ϖn+1∈Ωϖn satisfying dϖn,ϖn+1=Dϖn,Ωϖn with Dϖn,Ωϖn>0 and
(23)dϖn,ϖn+1<dϖn−1,ϖn,for alln∈ℕ. (23) implies that dϖn,ϖn+1n∈ℕ is a decreasing sequence of positive real numbers. Hence, from (N1) and (N2), we get
(24)χdϖn,ϖn+1+F2HΩϖn,Ωϖn+1≤F1ρdϖn,ϖn+1,Dϖn,Ωϖn,Dϖn+1,Ωϖn+1,Dϖn,Ωϖn+1,Dϖn+1,Ωϖn≤F1ρdϖn,ϖn+1,dϖn,ϖn+1,dϖn+1,ϖn+2,dϖn,ϖn+1+dϖn+1,ϖn+2,0≤F1ρdϖn,ϖn+1,dϖn,ϖn+1,dϖn,ϖn+1,2dϖn,ϖn+1,0≤F1dϖn,ϖn+1ρ1,1,1,2,0≤F1dϖn,ϖn+1=F1Dϖn,Ωϖn≤F1HΩϖn−1,Ωϖn≤F2HΩϖn−1,Ωϖn.Thus, for alln∈ℕ,
(25)F2HΩϖn,Ωϖn+1≤F2HΩϖn−1,Ωϖn−χdϖn,ϖn+1.Sinceχ∈Φ, there exists h>0 and n0∈ℕ such that χdϖn,ϖn+1>h, for all n≥n0. From (25), we obtain
(26)F2HΩϖn,Ωϖn+1≤F2HΩϖn−1,Ωϖn−χdϖn,ϖn+1≤F2HΩϖn−2,Ωϖn−1−χdϖn−1,ϖn−χdϖn,ϖn+1⋮≤F2HΩϖ0,Ωϖ1−∑i=1nχdϖi,ϖi+1=F2HΩϖ0,Ωϖ1−∑i=1n0−1χdϖi,ϖi+1−∑i=n0nχdϖi,ϖi+1≤F2HΩϖ0,Ωϖ1−n−n0h,n≥n0.Takingn⟶∞ in (26), we get F2HΩϖn,Ωϖn+1⟶−∞ and by Ϝ2′, we have
(27)limn⟶∞HΩϖn,Ωϖn+1=0,which further implies that(28)limn⟶∞dϖn,ϖn+1=limn⟶∞Dϖn,Ωϖn≤limn⟶∞HΩϖn−1,Ωϖn=0.Now fromϜ3, there exists k∈0,1 such that
(29)limn⟶∞HΩϖn,Ωϖn+1kF2HΩϖn,Ωϖn+1=0.Then, from (26), for all n∈ℕ, we have
(30)HΩϖn,Ωϖn+1kF2HΩϖn,Ωϖn+1−HΩϖn,Ωϖn+1kF2HΩϖ0,Ωϖ1≤HΩϖn,Ωϖn+1kF2HΩϖ0,Ωϖ1−n−n0h−HΩϖn,Ωϖn+1kF2HΩϖ0,Ωϖ1=−HΩϖn,Ωϖn+1kn−n0h≤0.Taking limitn⟶∞, in (30) and using (27) and (29), we have
(31)limn⟶∞nHΩϖn,Ωϖn+1k=0.Observe that from (31), there exist sn1∈ℕ such that nHΩϖn,Ωϖn+1k≤1 for all n≥n1. Thus, for all n≥n1, we have
(32)HΩϖn,Ωϖn+1≤1n1/kforalln≥n1,which further implies that(33)dϖn,ϖn+1=Dϖn,Ωϖn≤HΩϖn−1,Ωϖn≤1n1/kforalln≥n1.Now, in order to show thatϖnn∈ℕ is Cauchy sequence, consider m,n∈ℕ such that m>n>n1. From (33), we get
(34)dϖm,ϖn≤∑i=nm−1dϖi,ϖi+1≤∑i=nm−11i1/k≤∑i=n∞1i1/k.As a result of the above and the series’ convergence,∑i=n∞1/i1/k, we receive that ϖnn∈ℕ is Cauchy sequence. Since Z is a complete space, so there exists ϖ∗∈Z such that
(35)limn⟶∞ϖn=ϖ∗.Now,(36)F1HΩϖ,Ωω≤F2HΩϖ,Ωω<ϕdϖ,ω+F2HΩϖ,Ωω≤F1ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.SinceF1 is nondecreasing function, we obtain for all ϖ,ω∈Z.
(37)HΩϖ,Ωω≤ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.We claim thatϖ∗ is fixed point of Z. On contrary, suppose that Dϖ∗,Ωϖ∗>0 and by equation (37), we have
(38)Dϖ∗,Ωϖ∗≤dϖ∗,ϖn+1+Dϖn+1,Ωϖ∗<dϖ∗,ϖn+1+HΩϖn,Ωϖ∗<dϖ∗,ϖn+1+ρdϖn,ϖ∗,Dϖn,Ωϖn,Dϖ∗,Ωϖ∗,Dϖn,Ωϖ∗,Dx∗,Ωϖn≤dϖ∗,ϖn+1+ρdϖn,ϖ∗,dϖn,ϖn+1,Dϖ∗,Ωϖ∗,dϖn,ϖ∗+Dϖ∗,Ωϖ∗,dϖ∗,ϖn+1.Passing to limit asn⟶∞ in the above inequality, we obtain
(39)Dϖ∗,Ωϖ∗<ρ0,0,Dϖ∗,Ωϖ∗,0+Dϖ∗,Ωϖ∗,0,which implies by Lemma1 that
(40)0<Dϖ∗,Ωϖ∗<0,which is a contradiction. Hence,Dϖ∗,Ωϖ∗=0. Since Ωϖ∗ is closed, therefore, ϖ∗∈Ωϖ∗.Remark 13.
By definingF1=F2=F and ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=ϖ1 in Theorem 12, we get back Theorem 2.3 of [27].Example 7.
ConsiderZ=un=nn+1/2:n∈ℕ, and then Z is complete MS with metric du,v=∣u−v∣. Define functions F1,F2:0,∞⟶ℝ by
(41)F1u=−1uifu∈0,1uifu∈1,∞
andF2u=lnu+u for all u∈0,∞, respectively. Then F1 is nondecreasing, F2 satisfy the conditions Ϝ2′ and Ϝ3 and F1u≤F2u for all u>0.Next, defineΩ:Z⟶KZ, ρ:0,∞5⟶0,∞ and χ:0,∞⟶0,∞ by
(42)Ωϖ=u1ifϖ=u1,u1,u2ifϖ=un,n≥2,ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=ϖ1+ξϖ5,ξ∈0,1and χt=1/t for all t∈0,∞, respectively. Then χ∈Φ and ρ∈P (see Example 1). Observe that
(43)m,n∈ℕ,HΩum,Ωun>0⇔m>2andn=1.Assume thatHΩϖ,Ωω>0, and then m>2 and n=1. From Figure 1, it is clear that
(44)2∣m2+m−2∣+ln∣m2−m−2∣2+∣m2−m−2∣2≤∣m2+m−2∣2,Figure 1
Graph of Inequality (44).HΩum,Ωu1=∣um−1−1∣ and Du1,Ωum=0. Which further implies that
(45)χdum,u1+F2HΩum,Ωu1=1∣um−u1∣+F2∣um−1−1∣=2∣m2+m−2∣+ln∣m2−m−2∣2+∣m2−m−2∣2≤∣m2+m−2∣2=dum,u1+ξDu1,Ωum=F1dum,u1+ξDu1,Ωum=F1ρdum,u1,Du1,Ωu1,Dum,Ωum,Dum,Ωu1,Du1,Ωum.All hypothesis of Theorem12 are satisfied and fix Ω=u1,u2.Observe the following in Example7:
(i)
F1 is not continuous at 1(ii)
F1≠F2(iii)
ρ∉R(iv)
ρ∉ℙ.Corollary 14.
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued mapping. Assume that there exists χ∈Φ, a nondecreasing real valued function F1on0,∞ and a real valued function F2on0,∞ satisfying condition Ϝ2′ and Ϝ3 such that (N1) and the following condition holds:
HΩϖ,Ωω>0 implies χdϖ,ω+F2HΩϖ,Ωω≤F1Mϖ,ω for all ϖ,ω∈Z, where
(46)Mϖ,ω=maxdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dx,Ωω+Dω,Ωϖ2.
Then, fixΩ is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(47)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=maxϖ1,ϖ2,ϖ3,ϖ4+ϖ52.
Then,ρ∈P and result follow from Theorem 12.Remark 15.
Corollary14 generalizes and improves Theorem 2.4 of [35]. In fact, by taking F1=F2 and by defining Ωϖ=ϖ for all ϖ,ω∈Z and χt=τ>0 for all t∈0,∞ in Corollary 14, then we find Theorem 2.4 of [35]. Corollary 14 shows that condition Ϝ2 can be replaced by Ϝ2′ and the strictness of the monotonicity of F is not necessary.Corollary 16.
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued mapping. Assume that there exist χ∈Φ, a non decreasing real valued function F1on0,∞ and a real valued function F2 on 0,∞ satisfying condition Ϝ2′ and Ϝ3 such that (N1) and the following condition holds:
HΩϖ,Ωω>0 implies χdϖ,ω+F2HΩϖ,Ωω≤F1Nϖ,ω for all ϖ,ω∈Z, where
(48)Nϖ,ω=adϖ,ω+bDϖ,Ωϖ+cDω,Ωω+eDϖ,Ωω+Dω,Ωϖ,
a,b,c,e≥0 and a+b+c+2e<1. Then Fix Ω is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(49)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=aϖ1+bϖ2+cϖ3+eϖ4+ϖ5,where a,b,c,e>0 and a+b+c+2e<1. Then ρ∈P and result follows from Theorem 12.Now, in the next Theorem, we replace the conditionϜ3 of F2 by continuity of F1 in hypothesis of Theorem 12 and obtain another fixed point result.Theorem 17.
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued mapping. Assume that there exists χ∈Φ, a continuous, nondecreasing real-valued function F1 on 0,∞ and a real valued function F2 on 0,∞ satisfying condition Ϝ2′ such that following conditions hold:
(N1)F1c≤F2c for all c>0
(N2)HΩϖ,Ωω>0 implies
(50)χdϖ,ω+F2HΩϖ,Ωω≤F1ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ
for allϖ,ω∈Z and ρ∈ℙ.
Then, fixΩ is nonempty.Proof.
Letϖ0∈Z be an arbitrary point and ϖ1∈Ωϖ0. Then, as in proof of Theorem 12, we get a sequence ϖn⊂Z such that ϖn+1∈Ωϖn satisfying dϖn,ϖn+1=Dϖn,Ωϖn with Dϖn,Ωϖn>0,
(51)dϖn,ϖn+1<dϖn−1,ϖn,foralln∈ℕ,(52)F2HΩϖn,Ωϖn+1≤F2HΩϖ0,Ωϖ1−n−n0h,n≥n0.
Takingn⟶∞ in (52), we get F2HΩϖn,Ωϖn+1⟶−∞ and by Ϝ2′, we have
(53)limn⟶∞HΩϖn,Ωϖn+1=0,which further implies that
(54)limn⟶∞dϖn,ϖn+1=limn⟶∞Dϖn,Ωϖn≤limn⟶∞HΩϖn−1,Ωϖn=0.
Next, we claim that(55)limn,m⟶∞dϖn,ϖm=0.
If (55) is not true, then there exists δ>0 such that for all r≥0, there exists mk>nk>r(56)dϖn,ϖm>δ.
Also, there existsr0∈ℕ such that
(57)λr0=dϖn−1,ϖn<δforalln≥r0.
Consider two subsequencesϖnk and ϖmk of ϖn satisfying
(58)r0≤nk≤mk+1anddϖmk,ϖnk>δforallk>0.
Observe that(59)dϖmk−1,ϖnk≤δforallk,where mk is chosen as minimal index for which (59) is satisfied. Also, note that because of (58) and (59), the case nk+1≤nk is impossible. Thus, nk+2≤mk for all k. It implies
(60)nk+1<mk<mk+1forallk.
Using triangle inequality and by (58) and (59), we have
(61)δ<dϖmk,ϖnk≤dϖmk,ϖmk−1+dϖmk−1,ϖnk≤λmk+δ.
Letting limitk⟶∞ in (61) and using (53), we get
(62)limk⟶∞dϖmk,ϖnk=δ.
Now, by using (53) and (62), we obtain
(63)limk⟶∞dϖmk+1,ϖnk+1=δ.
Then, fromN1, N2, and monotonicity of F1, we get
(64)χdϖmk,ϖnk+F1dϖmk+1,ϖnk+1=χdϖmk,ϖnk+F1Dϖmk+1,Ωϖnk≤χdϖmk,ϖnk+F1HΩϖmk,Ωϖnk≤χdϖmk,ϖnk+F2HΩϖmk,Ωϖnk≤F1ρdϖmk,ϖnk,Dϖmk,Ωϖmk,Dϖnk,Ωϖnk,Dϖmk,Ωϖnk,Dϖnk,Ωϖmk≤F1ρdϖmk,ϖnk,dϖmk,ϖmk+1,dϖnk,ϖnk+1,dϖnk+1,ϖnk+dϖnk,ϖmk,dϖnk,ϖnk+1+dϖnk+1,ϖmk+1.
SinceF1 is continuous, so by passing the limit k⟶∞, using equations (62) and (63), we have
(65)limk⟶∞χdϖmk,ϖnk+F1δ≤F1ρδ,0,0,δ,δ≤F1δρ1,0,0,1,1.
Now, sinceρ∈ℙ, we have ρ1,0,0,1,1∈0,1; so, (65) implies
(66)lims⟶δ+infϕs≤0,which is a contradiction to (17). Hence, (55) holds, which implies that ϖn is a Cauchy sequence. Completeness of Z ensures the existence of ϖ∗∈Z such that
(67)limn⟶∞ϖn=ϖ∗.
By following the same steps as in the proof of Theorem12, we get ϖ∗∈Ωϖ∗. This completes the proof.Corollary 18.
LetZ,d be a complete MS and Ω:Z⟶KZ be a multivalued mapping. Assume that there exists χ∈Φ, a continuous, nondecreasing real-valued function F1 on 0,∞ and a real valued function F2 on 0,∞ satisfying condition Ϝ2′ such that (N1) and the following condition holds:
(68)HΩϖ,Ωω>0impliesχdϖ,ω+F2HΩϖ,Ωω≤F1℘1dϖ,ω+℘2Dϖ,Ωϖ+℘3Dω,Ωω+℘4Dϖ,Ωω+℘5Dω,Ωϖforallϖ,ω∈Z,where ℘i≥0, ℘1+℘2+℘3+2℘4=1 and ℘1+℘3+℘4≤1. Then, fix Ω is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(69)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=℘1ϖ1+℘2ϖ2+℘3ϖ3+℘4ϖ4+℘5ϖ5,where ℘i≥0, ℘1+℘2+℘3+2℘4=1 and ℘1+℘3+℘4≤1. Then, ρ∈ℙ and result follow from Theorem 17.Remark 19.
Corollary18 improves Theorem 1 of [35]. In fact, by taking F1=F2 and by defining Ωϖ=ϖfor all ϖ,ω∈Z in Corollary 18, then we are back to Theorem 1 of [34]. In Corollary 18, condition Ϝ2 is weakened to the condition Ϝ2′.Next, we considerΩϖ that are closed subsets of Z instead of compact subsets for all Z and obtain the following theorems.Theorem 20.
LetZ,d be a complete MS and Ω:Z⟶CX be a multivalued mapping. Assume that there exists χ∈Φ, F∈ΔϜ∗ and a real-valued function L on 0,∞ such that following holds:
(G1)Fc≤Lc for all c>0
(G2)HΩϖ,Ωω>0 implies
(70)χdϖ,ω+LHΩϖ,Ωω≤Fρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ,
for allϖ,ω∈Z and ρ∈P.
Then, fix (Ω) is nonempty.Proof.
Letx0∈Z be an arbitrary point and ϖ1∈Ωϖ0. Assume that ϖ1∈Ωϖ1; otherwise, ϖ1 is a fixed point of Ω, and the proof is complete. Then, since Ωϖ1 is closed, Dϖ1,Ωϖ1>0 and consequently, HΩϖ0,Ωϖ1>0. Due to Ϝ4, we obtain
(71)FDϖ1,Ωϖ1=infz∈Ωϖ1Fdϖ1,z.Then, (71) with (G1) and (G2) gives
(72)infz∈Ωϖ1Fdϖ1,z=FDϖ1,Ωϖ1≤FHΩϖ0,Ωϖ1≤LHΩϖ0,Ωϖ1≤Fρdϖ0,ϖ1,Dϖ0,Ωϖ0,Dϖ1,Ωϖ1,Dϖ0,Ωϖ1,Dϖ1,Ωϖ0−χdϖ0,ϖ1<Fρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ2,0.Thus, there existsx2∈Ωϖ1 such that
(73)Fdϖ1,ϖ2<Fρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ1,0.SinceF is an nondecreasing function, (73) with ρ3 implies that
(74)dϖ1,ϖ2<ρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ1,0≤ρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,ϖ2,dϖ0,ϖ1+dϖ1,ϖ2,0.By using Lemma11, (74) implies
(75)dϖ1,ϖ2<dϖ0,ϖ1.Next, arguing as previous, we getϖ3∈Ωϖ2 with Dϖ2,Ωϖ2>0. Also, by using Lemma 11, from (G1) and (G2), we obtain
(76)dϖ2,ϖ3<dϖ1,ϖ2Continuing in the same manner, we get a sequenceϖn⊂Z such that ϖn+1∈Ωϖn with Dϖn,Ωϖn+1>0 and
(77)dϖn,ϖn+1<dϖn−1,ϖn,for all n∈ℕ. (77) implies that dϖn,ϖn+1n∈ℕ is a decreasing sequence of positive real numbers. Hence, from F4, (G1), and (G2), we get
(78)infz∈ΩϖnFdϖn,z=FDϖn,Ωϖn≤FHΩϖn−1,Ωϖn≤LHΩϖn−1,Ωϖn≤Fρdϖn−1,ϖn,Dϖn−1,Ωϖn−1,Dϖn,Ωϖn,Dϖn−1,Ωϖn,Dϖn,Ωϖn−1−χdϖn−1,ϖn≤Fρdϖn−1,ϖn,dϖn−1,ϖn,dϖn,ϖn+1,dϖn−1,ϖn+dϖn,ϖn+1,0−χdϖn−1,ϖn≤Fρdϖn−1,ϖn,dϖn−1,ϖn,dϖn−1,ϖn,2dϖn−1,ϖn,0−χdϖn−1,ϖn≤Fdϖn−1,ϖnρ1,1,1,2,0−χdϖn−1,ϖn≤Fdϖn−1,ϖn−χdϖn−1,ϖn.Thus, for alln∈ℕ,
(79)infz∈ΩϖnFdϖn,z≤Fdϖn−1,ϖn−χdϖn−1,ϖn.Thus, from (79), there exists ϖn+1∈Ωϖn such that
(80)Fdϖn,ϖn+1≤Fdϖn−1,ϖn−χdϖn−1,ϖn.Sinceχ∈Φ, there exists h>0 and n0∈ℕ such that χdϖn,ϖn+1<h, for all n≥n0. From (80), we obtain
(81)Fdϖn,ϖn+1≤Fdϖn−1,ϖn−χdϖn−1,ϖn≤Fϖn−2,ϖn−1−χdϖn−2,ϖn−1−χdϖn−1,ϖn⋮≤Fdϖ0,ϖ1−∑i=1n−1χdϖi−1,ϖi=Fdϖ0,ϖ1−∑i=1n0−1χdϖi−1,ϖi−∑i=n0n−1χdϖi−1,ϖi=Fdϖ0,ϖ1−n−n0h,n≥n0.Takingn⟶∞ in (81), we get Fdϖn−1,ϖn⟶−∞ and by Ϝ2′, we have
(82)limn⟶∞dϖn−1,ϖn=0,Now, fromϜ3, there exists k∈0,1 such that
(83)limn⟶∞dϖn−1,ϖnkFdϖn−1,ϖn=0.Then, from (81), for all n∈ℕ, we have
(84)dϖn−1,ϖnkFdϖn−1,ϖn−dϖn−1,ϖnkFdϖ0,ϖ1≤dϖn−1,ϖnkFdϖn−1,ϖn−n−n0h−dϖn−1,ϖnkFdϖ0,ϖ1=−dϖn−1,ϖnkn−n0h≤0.Taking limitn⟶∞, in (84) and using (82) and (83), we have
(85)limn⟶∞ndϖn−1,ϖnk=0.Observe that from (85), there exists n1∈ℕ such that ndϖn−1,ϖnk≤1 for all n≥n1. Thus, for all n≥n1, we have
(86)dϖn−1,ϖn≤1n1/kforalln≥n1,Now, in order to show thatxnn∈ℕ is Cauchy sequence, consider m,n∈ℕ such that m>n>n1. From (86), we get
(87)dϖm,ϖn≤∑i=nm−1dϖi,ϖi+1≤∑i=nm−11i1/k≤∑i=n∞1i1/k.As a result of the above and the series’ convergence,∑i=n∞1/i1/k, we receive that ϖnn∈ℕ is Cauchy sequence. Since Z is a complete space, so there exists ϖ∗∈Z such that
(88)limn⟶∞xn=ϖ∗.Now,(89)FHΩϖ,Ωω≤LHΩϖ,Ωω<ϕdϖ,ω+LHΩϖ,Ωω≤Fρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.SinceF is nondecreasing function, we obtain
(90)HΩϖ,Ωω≤ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ,forallϖ,ω∈Z.We claim thatϖ∗ is a fixed point of Z. On contrary, suppose that Dϖ∗,Ωϖ∗>0 and by equation (90), we have
(91)Dϖ∗,Ωϖ∗≤dϖ∗,ϖn+1+Dϖn+1,Ωϖ∗<dϖ∗,ϖn+1+HΩϖn,Ωϖ∗<dϖ∗,ϖn+1+ρdϖn,ϖ∗,Dϖn,Ωϖn,Dϖ∗,Ωϖ∗,Dϖn,Ωϖ∗,Dϖ∗,Ωϖn≤dϖ∗,ϖn+1+ρdϖn,ϖ∗,dϖn,ϖn+1,Dϖ∗,Ωϖ∗,Dϖn,ϖ∗+Dϖ∗,Ωϖ∗,dϖ∗,ϖn+1.Passing to limit asn⟶∞ in the above inequality, we obtain
(92)Dϖ∗,Ωϖ∗<ρ0,0,Dϖ∗,Ωϖ∗,0+Dϖ∗,Ωϖ∗,0,which implies by Lemma 1 that
(93)0<Dx∗,Ωϖ∗<0,which is a contradiction. Hence, Dϖ∗,Ωϖ∗=0. Since Ωϖ∗ is closed, therefore, ϖ∗∈Ωϖ∗.Corollary 21.
LetZ,d be a complete MS and Ω:Z⟶CZ be a multivalued mapping. Assume that there exists χ∈Φ, F∈ΔϜ∗ and a real-valued function L on 0,∞ such that (G1) and the following condition holds:
(94)HΩϖ,Ωω>0impliesχdϖ,ω+FHΩϖ,Ωω≤L℘1dϖ,ω+℘2Dϖ,Ωϖ+℘3Dω,Ωω+℘4Dϖ,Ωω+℘5Dω,Ωϖforallϖ,ω∈Z,where ℘i≥0 and ℘1+℘2+℘3+2℘4=1. Then, fix Ω is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(95)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=℘1ϖ1+℘2ϖ2+℘3ϖ3+℘4ϖ4+℘5ϖ5,where ℘i≥0 and ℘1+℘2+℘3+2℘4=1. Then. ρ∈P and result follow from Theorem 20.Corollary 22.
LetZ,d be a complete MS and Ω:Z⟶CZ be a multivalued mapping. Assume that there exists χ∈Φ, F∈ΔϜ∗ and a real-valued function L on 0,∞ such that (G1) and the following condition holds:
(96)HΩϖ,Ωω>0impliesχdϖ,ω+FHΩϖ,Ωω≤Ldϖ,ω+λDω,Ωϖforallϖ,ω∈Z,where λ≥0. Then, fix Ω is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(97)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=ϖ1+λϖ5,where λ≥0. Then, ρ∈P and result follow from Theorem 20.Remark 23.
Corollary21 generalizes Theorem 24 of [33]. Indeed, by considering L=F and by defining χt=2τ for all ϖ,ω∈Z, where τ>0 in Corollary 21, we obtain Theorem 32 of [33]. Also, by considering L=F and by defining χt=τ for all ϖ,ω∈Z, where τ>0 in Corollary 22, we get back the Theorem 24 of [32].Theorem 24.
LetZ,d be a complete MS and Ω:Z⟶CZ be a multivalued mapping. Assume that there exists χ∈Φ, a nondecreasing and continuous real-valued function F:0,∞⟶ℝ satisfying condition Ϝ2′ and a real-valued function L on 0,∞ such that following holds:
(G1)Fc≤Lc for all c>0
(G2)HΩϖ,Ωω>0 implies
(98)χdϖ,ω+LHΩϖ,Ωω≤Fρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ,
for allϖ,ω∈Z and ρ∈ℙ.
Then, fixΩ is nonempty.Proof.
Letϖ0∈Z be an arbitrary point and ϖ1∈Ωϖ0. Then, as in proof of Theorem 20, we get a sequence ϖn⊂Z such that ϖn+1∈Ωϖn with Dϖn,Ωϖn+1>0,
(99)dϖn,ϖn+1<dϖn−1,ϖn,foralln∈ℕ,(100)Fdϖn−1,ϖn≤Fdϖ0,ϖ1−n−n0h,n≥n0.
Takingn⟶∞ in (100), we get Fdϖn−1,ϖn⟶−∞ and by Ϝ2′, we have
(101)limn⟶∞dϖn−1,ϖn=0.
Next, we claim that(102)limn,m⟶∞dϖn,ϖm=0.
If (102) is not true, then there exists δ>0 such that for all r≥0, there exists mk>nk>r(103)dϖn,ϖm>δ.
Also, there existsr0∈ℕ such that
(104)λr0=dϖn−1,ϖn<δforalln≥r0.
Consider two subsequencesϖnk and ϖmk of xn; then, as is proof of Theorem 17, we get
(105)limk⟶∞dϖmk,ϖnk=δ,limk⟶∞dϖmk+1,ϖnk+1=δ.
Then, from (G1), (G2), and monotonicity ofF, we get
(106)χdϖmk,ϖnk+Fdϖmk+1,ϖnk+1=χdϖmk,ϖnk+FDϖmk+1,Ωϖnk≤χdϖmk,ϖnk+FHΩϖmk,Ωϖnk≤χdϖmk,ϖnk+LHΩϖmk,Ωϖnk≤Fρdϖmk,ϖnk,Dϖmk,Ωϖmk,Dϖnk,Ωϖnk,Dϖmk,Ωϖnk,Dϖnk,Ωϖmk≤Fρdϖmk,ϖnk,dϖmk,ϖmk+1,dϖnk,ϖnk+1,dϖnk+1,ϖnk+dϖnk,ϖmk,dϖnk,ϖnk+1+dϖnk+1,ϖmk+1.
SinceF is continuous, so by passing the limit k⟶∞, using equations (105) and (106), we have
(107)limk⟶∞χdϖmk,ϖnk+Fδ≤Fρδ,0,0,δ,δ≤Fδρ1,0,0,1,1.
Now, sinceρ∈ℙ, we have ρ1,0,0,1,1∈0,1; so, (107) implies
(108)lims⟶δ+infϕs≤0,which is a contradiction to (17). Hence, (102) holds, which implies that ϖn is Cauchy sequence. Completeness of Z ensures the existence of ϖ∗∈Z such that
(109)limn⟶∞ϖn=ϖ∗.
By following the same steps as in the proof of Theorem20, we get ϖ∗∈Ωϖ∗. This completes the proof.Corollary 25.
LetZ,d be a complete MS and Ω:Z⟶CZ be a multivalued mapping. Assume that there exists χ∈Φ, a nondecreasing and continuous real-valued function F:0,∞⟶ℝ satisfying condition Ϝ2′ and a real-valued function L on 0,∞ such that (G1) and the following condition hold:
(110)HΩϖ,Ωω>0impliesχdϖ,ω+FHΩϖ,Ωω≤L℘1dϖ,ω+℘2Dϖ,Ωϖ+℘3Dω,Ωω+℘4Dϖ,Ωω+℘5Dω,Ωϖforallϖ,ω∈Z,where ℘i≥0, ℘1+℘2+℘3+2℘4=1, and ℘1+℘3+℘4≤1. Then, fix Ω is nonempty.Proof.
Defineρ:0,∞5⟶0,∞ by
(111)ρϖ1,ϖ2,ϖ3,ϖ4,ϖ5=℘1ϖ1+℘2ϖ2+℘3ϖ3+℘4ϖ4+℘5ϖ5,where ℘i≥0, ℘1+℘2+℘3+2℘4=1, and ℘1+℘3+℘4≤1. Then, ρ∈ℙ and result follow from Theorem 24.If we restrictλ=0 in Corollary 22, then ρ defined in the proof of Corollary 22 also satisfies ρ1,0,0,1,1=1 and hence ρ∈ℙ. Consequently, from Theorem 24, we getCorollary 26.
LetZ,d be a complete MS and Ω:Z⟶CZ be a multivalued mapping. Assume that there exists χ∈Φ, a nondecreasing and continuous real-valued function F:0,∞⟶ℝ satisfying condition Ϝ2′ and a real-valued function L on 0,∞ such that (G1) and the following condition hold:
(112)HΩϖ,Ωω>0impliesχdϖ,ω+FHΩϖ,Ωω≤Ldϖ,ω+λDω,Ωϖforallϖ,ω∈Z,where λ≥0. Then, fix Ω is nonempty.


## 2. Data Dependence

LetZ, V be two nonempty sets and Ω:Z⟶PV. Denote by GΩ, the graph of the multivalued operator is Ω. A multivalued operator Ω:Z⟶PV is said to be closed if GΩ is a closed set in Z×V. A selection for Ω is a singlevalued operator Ω:Z⟶V such that tϖ∈Ωϖ, for each ϖ∈Z. Mo t and Petrusel in [36] discussed some basic problems including data dependence of the fixed point theory for a new type contractive multivalued operator. In [37], Rus et al. gave an important abstract notion as follows:Definition 27.
LetZ,d be a MS and Ω:Z⟶CLZ a multivalued operator. Then, Ω is a multivalued weakly Picard operator (briefly MWP operator) if for all ϖ∈Z and ω∈Ωϖ, there exists a sequence ϖnn∈ℕ such that
(i)
ϖ0=ϖ, ϖ1=ω(ii)
ϖn+1∈Ωϖn, for all n∈ℕ(iii)
The sequenceϖnn∈ℕ is convergent, and its limit is a fixed point of ΩA sequenceϖnn∈ℕ satisfying the conditions (i) and (ii) in Definition 27 is also called a sequence of successive approximations of Ω starting from ϖ0. Now, we present the main result of this section.Theorem 28.
LetZ,d be a MS and Ω1,Ω2:Z⟶KZ be two multivalued operators. Assume that there exists χ∈Φ, a nondecreasing real-valued function F1 on 0,∞ and a real valued function F2 on 0,∞ satisfying condition Ϝ2′ and Ϝ3 such that Ωi satisfies (N1) and (N2) for all i∈1,2:
(i)
There existsλ>0 such that HΩ1ϖ,Ω2ϖ≤λ, for all ϖ∈Z(ii)
Then(iii)
FixΩi∈CLZ and i∈1,2(iv)
Ω1 and Ω2 are MWP operators and(113)HFixΩ1,FixΩ2≤λ1−maxρ11,1,1,2,0,ρ21,1,1,2,0.Proof.
(a) By Theorem12, we have that fix Ωi≠∅, for i∈1,2. Next, we prove that the fixed point set of multivalued operators Ωi is closed for i∈1,2. For this, let wn be a sequence in fix Ωi such that wn⟶w as n⟶∞. Then,
(114)F1HΩϖ,Ωω≤F2HΩϖ,Ωω<ϕdϖ,ω+F2HΩϖ,Ωω≤F1ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.
SinceF1 is nondecreasing function, we obtain for all ϖ,ω∈Z(115)HΩϖ,Ωω≤ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.
Suppose thatDw,Ωw>0, then we have
(116)Dw,Ωw≤dw,wn+1+Dwn+1,Ωw<dw,wn+1+HΩwn,Ωw<dw,wn+1+ρdwn,w,Dwn,Ωwn,Dw,Ωw,Dwn,Ωw,Dw,Ωwn≤dw,wn+1+ρdwn,w,dwn,wn+1,Dw,Tw,dwn,w+Dw,Tw,dw,wn+1.
Passing to limit asn⟶∞ in the above inequality, we obtain
(117)Dw,Ωw<ρ0,0,Dw,Ωw,0+Dw,Ωw,0,which implies by Lemma 1 that
(118)0<Dw,Tw<0,which is a contradiction. Hence, Dw,Ωw=0. Since Ωw is closed, so w∈Ωw.
(b) From the proof of Theorem12, we immediately get that Ωi operators are MWP operators for i∈1,2. Now, we will show that HFixΩ1,FixΩ2≤λ/1−maxρ11,1,1,2,0,ρ21,1,1,2,0. For this purpose, Let q>1, ϖ0∈FixΩ1, be arbitrary. Then, there exists ϖ1∈Ω2ϖ0 such that dϖ0,ϖ1=Dϖ0,Ω2ϖ0 and dϖ0,ϖ1≤qHΩ1ϖ0,Ω2ϖ0. Next, for ϖ1∈Ω2ϖ0, there exists ϖ2∈Ω2ϖ1 such that dϖ0,ϖ1=Dϖ0,Ω2ϖ0 and dϖ1,ϖ2≤qHΩ2ϖ0,Ω2ϖ1. Then, by using (3.1), we get dϖ1,ϖ2≤dϖ0,ϖ1 and
(119)dϖ1,ϖ2≤qHΩ2ϖ0,Ω2ϖ1≤qρdϖ0,ϖ1,Dϖ0,Ω2ϖ0,Dϖ1,Ω2ϖ1,Dϖ0,Ω2ϖ1,Dϖ1,Ω2ϖ0≤qρdϖ0,ϖ1,dϖ0,ϖ1,dϖ1,x2,dϖ0,ϖ1+dϖ1,x2,0≤qρdϖ0,ϖ1,dϖ0,ϖ1,dϖ0,ϖ1,2dϖ0,ϖ1,0≤qρ1,1,1,2,0dϖ0,ϖ1.
Inductively, we will obtain a sequence of successive approximations forΩ2 starting from ϖ0, satisfying the following:
(120)dϖn,ϖn+1≤qρ11,1,1,2,0ndϖ0,ϖ1,n∈ℕ,which further implies for each n∈ℕ,
(121)dϖn,ϖn+m≤qρ11,1,1,2,0n1−qρ11,1,1,2,0dϖ0,ϖ1.
Lettingn⟶∞, we get that ϖnn∈ℕ is Cauchy sequence in Z,d, and so it converges to an element u∈Z. As in the proof of Theorem 12, we get that u∈FixΩ2. From (121), letting m⟶∞ to get
(122)dϖn,u≤qρ11,1,1,2,0n1−qρ11,1,1,2,0dϖ0,ϖ1,foreachn∈ℕ.
Puttingn=0, we get that
(123)dϖ0,u≤11−qρ1o1,1,1,2,0dϖ0,ϖ1≤qλ1−qρ11,1,1,2,0.
By interchanging the roles ofΩ1 and Ω2, we obtain that for each u0∈FixΩ2, there exists x∈FixΩ1 such that
(124)du0,ϖ≤11−qρ21,1,1,0,2du0,u1≤qλ1−qρ21,1,1,0,2.
Hence,HFixΩ1,FixΩ2≤qλ/1−maxqρ11,1,1,2,0,qρ21,1,1,2,0, and letting q⟶1, we get the conclusion.


## 3. Strict Fixed Points and Well Posedness

Firstly, we define the notions of well posedness of a fixed point problem.Definition 29 [38, 39].
LetZ,d be a MS, V∈PZ, and Ω:V⟶CZ be a multivalued operator. Then, the fixed point problem is well posed for Ω with respect to D if
(a1) FixΩ=ϖ∗;
(b1) If ϖn∈V, n∈ℕ, and Dϖn,Ωϖn⟶0 as n⟶∞, then ϖn⟶dϖ∗∈FixΩ as n⟶∞Definition 30 [38, 39].
LetZ,d be a MS, V∈PZ, and Ω:V⟶CZ be a multivalued operator. Then, the fixed point problem is well posed for Ω with respect to H if
(a2) SFixΩ=ϖ∗
(b2) If ϖn∈V, n∈ℕ, and Hϖn,Ωϖn⟶0 as n⟶∞, then ϖn⟶dϖ∗∈SFixΩ as n⟶∞Remark 31.
Note that if the fixed point problem is well posed forΩ with respect to D and FixΩ=SFixΩ, then the fixed point problem is well posed for Ω with respect to H.Theorem 32.
LetZ,d be a MS and Ω:Z⟶KZ be a multivalued operators. Assume that
(1)
There existχ∈Φ, a continuous, nondecreasing real-valued function F1 on 0,∞ and a real-valued function F2 on 0,∞ satisfying condition Ϝ2′ such that (N1) and (N2) hold for ρ∈P with ρ1,0,0,1,1∈0,1,(125)SFixΩ≠∅.
Then,(a)
FixΩ=SFixΩ=ϖ∗;(b)
The fixed point problem is well posed forΩ with respect to HProof.
a By Theorem 17, we have that fix Ω≠∅. Next, We will prove that FixΩ=ϖ∗. From (N1) and (N2), we get that
(126)F1HΩϖ,Ωω≤F2HΩϖ,Ωω<ϕdϖ,ω+F2HΩϖ,Ωω≤F1ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.
SinceF1 is nondecreasing function, we obtain for all ϖ,ω∈Z,
(127)HΩϖ,Ωω≤ρdϖ,ω,Dϖ,Ωϖ,Dω,Ωω,Dϖ,Ωω,Dω,Ωϖ.
Letu∈FixΩ, with u≠ϖ∗; then Dϖ∗,Ωu>0, and we have
(128)Dϖ∗,Ωu=HΩϖ∗,Ωu≤ρdϖ∗,u,Dϖ∗,Ωϖ∗,Du,Ωu,Dϖ∗,Ωu,Du,Ωϖ∗≤ρdϖ∗,u,0,0,dϖ∗,u,du,ϖ∗≤dϖ∗,uρ1,0,0,1,1.
Sinceρ1,0,0,1,1∈0,1, above inequality implies that
(129)dϖ∗,u=Dϖ∗,Ωu<dϖ∗,u,which is a contradiction. Hence, dϖ∗,u=0; so, ϖ∗=u.
(b) Letϖn∈V, n∈ℕ, such that
(130)limn⟶∞Dϖn,Ωϖn=0.
We claim that(131)limn⟶∞dϖn,ϖ∗=0,where ϖ∗∈FixΩ. If (131) is not true, there exists ε>0 such that for each n∈ℕ, we have that
(132)dϖn,ϖ∗>ε.
On the other hand, from (130), there exists nε∈ℕ\0 such that Dϖn,Ωϖn<ε for each n>nε. Hence, for each n>nε, we get
(133)dϖn,ϖ∗=Dϖn,Ωϖ∗=Dϖn,Ωϖn+HΩϖn,Ωϖ∗≤Dϖn,Ωϖn+ρdϖn,ϖ∗,Dϖn,Ωϖn,Dϖ∗,Ωϖ∗,Dϖn,Ωϖ∗,Dϖ∗,Ωϖn≤Dϖn,Ωϖn+ρdϖn,ϖ∗,Dϖn,Ωϖn,dϖ∗,ϖ∗,dϖn,ϖ∗,dϖ∗,ϖn+Dϖn,Ωϖn.
Sinceρ1,0,0,1,1∈0,1, so by passing the limit n⟶∞, we obtain dxn,ϖ∗⟶0 as n⟶∞, a contradiction. Consequently, proof is complete by Remark 31.


## 4. Conclusion

In the theory of set-valued dynamic systems, fixed points and strict fixed points of multivalued operators are essential notions. A rest point of the dynamic system can be read as a fixed point for the multivalued mapΩ, whereas a strict fixed point for Ω can be viewed as the system’s endpoint. We have made a contribution in this approach by establishing some basic problems in multivalued fixed point and strict fixed point theory. We have proved several existence and data dependence results for multivalued nonlinear mappings satisfying a new class of contractive conditions via auxiliary functions. The obtained outcomes are backed up by a nontrivial example. The findings add to and expand on some of the most recent results in the literature.



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*Source: 2200903-2021-08-24.xml*