# On Existence of Fixed Points for Multivalued Generalizedwb-Contractive Mappings and Applications

Authors: Abdul Latif; Reem Fahad Al Subaie; Monairah Omar Alansari
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we present some new results on the existence of fixed points for multivalued generalizedwb-contractive mappings in the frame work of metric type spaces. Consequently, presented results unify and generalize several known metric fixed-point results. In support of our main results, examples are provided to show that the results are genuine generalization of the known corresponding results of metric fixed-point theory.


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## Body

## 1. Introduction

The concept of a metric space plays a vital role in the development of metric fixed-point theory and nonlinear functional analysis and also in various scientific branches. In the literature, this notion of metric space has been extended in several directions by reducing or modifying the metric axioms. Czerwik [1, 2] introduced and studied the concept of b-metric space (metric type space), where the triangle inequality replaced with the weaker condition. In fact, the basic idea of b-metric was given by Bakhtin [3]. It has been observed that the class of metric type spaces is effectively larger than the class of metric spaces [1]. In literature, a number of metric fixed-point results for single-valued and multivalued mappings have been shown; see, for example, [4–11] and references therein.Using a concept of the Hausdorff–Pompieu metric, Nadler [12] introduced a notion of multivalued contraction mappings and proved splendid result in metric fixed-point theory for such mappings, known as Nadler contraction principle. Due to its importance, matric fixed-point theory of multivalued contractions has been further developed in various directions by a number of authors. A real generalization of the Nadler contraction principle is obtained by Mizoguchi and Takahashi [13]. Without using the idea of the Hausdorff–Pompieu metric, a number of authors obtained interesting fixed-point results and improved various results of metric fixed-point theory including the results of Nadler and Mizoguchi–Takahashi and others. See, for example, [14–17] and references therein.In [18] Kada et al. introduced a concept of generalized distance, namely, w-distance on metric spaces, and improved some classical results by replacing the involved metric by a generalized distance. Based on this set up, a number of authors studied fixed-point results of mappings with respect to w-distance. Suzuki and Takahashi [19] introduced notions of single-valued and multivalued weakly contractive mappings and studied the existence of fixed points for such mappings. Consequently, they generalized the Banach contraction principle and Nadler contraction principle. For further fixed-point results with applications, see, for example, [20–24] and references therein. In [25], Hussain et al. defined w-distance on metric-type spaces called wt-distance (here, we call it wb-distance), and they proved fixed-point and common fixed-point results for single-valued mappings with respect to wb-distance. A number of articles with applications on this topic can also be found in [26–29] and references therein.


## 2. Preliminaries

Now, we recall some notations, concepts, and facts which are useful for our results.LetS,d be a metric space. Let 2S denote a collection of nonempty subsets of S, ClS denote a collection of nonempty closed subsets of S, CBS denote a collection of nonempty closed bounded subsets of S, and KS denote a collection of all nonempty compact subsets of S. An element u∈S is called a fixed point of a multivalued mapping J:S⟶2S if u∈Ju. We denote FixJ=u∈S:u∈Ju. A sequence un in S is called an orbit of J at u0∈S if un∈Jun−1, for all n≥1. A map f:S⟶ℝ is called J-orbitally lower semicontinuous at z∈S if, for any orbit un of J at u0∈S with un⟶z implies fz≤liminfn⟶∞fun. For a constant c∈0,1, we say a function ξc:[0,∞)⟶[0,c) is a strong MT-function if limsupr⟶s+ξcr<c, for all s∈[0,∞). In case c=1, the function ξ1 is denoted by ξ, known as MT-function. It has been observed that a function ξ is MT-function if and only if, for any strictly decreasing sequence un in 0,∞, we have 0≤supnξun<1. For more characterizations of MT-function, see [30].Using the concept of Hausdorff–Pompieu metric, Nadler [12] proved a multivalued version of the well-known Banach contraction principle.Theorem 1 (see [12]).
LetS,d be a complete metric space and let J:S⟶CBS be a multivalued contraction mapping (that is, for a fixed constant h∈0,1 and for each u,v∈S, HJu,Jv≤hdu,v, where H is the Hausdorff–Pompieu metric on CBS). Then, FixJ≠∅.
This result known as Nadler contraction principle, which has been generalized in various directions. The first real generalization of Theorem1 is obtained by Mizoguchi and Takahashi [13].Theorem 2 (see [13]).
LetS,d be a complete metric space and let J:S⟶CBS be a multivalued mapping. Assume that there exists MT-function ξ such that, for each u,v∈S,(1)HJu,Jv≤ξdu,vdu,v.
Then,FixJ≠∅.
On the contrary, without using the concept of the Hausdorff–Pompieu metric, Feng and Liu [16] generalized Theorem 1 as follows.Theorem 3 (see [16]).
LetS,d be a complete metric space and let J:S⟶ClS be a multivalued mapping. Suppose that a real-valued function g on S, gu=du,Ju, is lower semicontinuous. Then, FixJ≠∅ provided there exist constants c,h∈0,1,h<c, such that, for any u∈S, there is v∈Ju satisfying(2)cdu,v≤gu,gv≤hdu,v.Later, Klim and Wardowski [17] generalized Theorem 3 as follows.Theorem 4 (see [17]).
LetS,d be a complete metric space and let J:S⟶ClS be a multivalued mapping such that a real-valued function g on S, gu=du,Ju, is lower semicontinuous. Then, FixJ≠∅ provided that there exists a strong MT-function ξc such that, for any u∈S, there is v∈Ju satisfying(3)cdu,v≤gu,gv≤ξcdu,vdu,v.UsingMT-functions, Klim and Wardowski [17] also proved fixed-point result for compact valued mappings of metric spaces as follows.Theorem 5 (see [17]).
LetS,d be a complete metric space and let J:S⟶KS be a multivalued mapping such that a real-valued function g on S, gu=du,Ju, is lower semicontinuous. Then, FixJ≠∅ provided that there exists MT-function ξ such that, for any u∈S, there is v∈Ju satisfying(4)du,v=gu,gv≤ξdu,vdu,v.It is worth mentioning that Theorem4 generalizes Theorem 1 and Theorem 3 but does not generalize Theorem 2 because the strong-MT-function ξc in Theorem 4 is stronger than the MT-function ξ used in Theorem 2 as c<1.However, some more general interesting fixed-point results in this direction obtained by Ćirić [14, 15] unify and generalize the corresponding abovementioned results.In [18], Kada et al. introduced the concept of w-distance as follows.LetS,d be a metric space. A function p:S×S⟶[0,∞) is called w-distance on S if it satisfies the following, for each u,v,t∈S:(1)
pu,t≤pu,v+pv,t(2)
A functionp(u,⋅):S⟶[0,∞) is lower semicontinuous(3)
For anyϵ>0, there exists δ>0 such that pt,u≤δ and pt,v≤δ imply du,v≤ϵUsing the concept ofw-distances, they improved a number of known important results of metric fixed-point theory. Note that, in general, for u,v∈S,pu,v≠pv,u and not either of the implications pu,v=0⇔u=v necessarily hold. Clearly, the metric d is a w-distance on S. Let W,. be a normed space. Then, the functions p1,p2:W×W⟶[0,∞) defined by p1u,v=v and p2u,v=u+v, for all u,v∈W, are w-distances [18]. For more examples and properties of the w-distance, see [18, 19, 24]. Using the concept of w-distance, Suzuki and Takahashi [19] introduced single-valued and multivalued weakly contractive mappings and then improved the Banach contraction principle and Nadler contraction principle. For further general fixed-point results in this direction, see [20, 21, 23, 24] and references therein.Czerwik [1, 2] introduced a concept of b-metric space as follows.LetS be a nonempty set. Let Δ:S×S⟶[0,∞) be a function which satisfies the following, for all u,v,t∈S:(1)
Δu,v=0 if and only if u=v(2)
Δu,v=Δv,u(3)
Δu,v≤bΔu,t+Δt,v, for some b≥1Then,Δ is called a b-metric on S and S,Δ is known as b-metric space (also known a metric-type space [8, 25]). In the sequel, we also call it metric-type space. Obviously, for b=1, we obtain a standard metric on S. In fact, the class of metric-type spaces is effectively larger than the class of metric spaces. Indeed, let S=ℝ be endowed with a mapping Δ:S×S⟶ℝ+ defined by Δu,v=u−v2, for each u,v∈S. Then, S,Δ is a metric-type space with b=2, but it is not a metric space [3]. For more examples of metric-type spaces, see [1, 31]. It is worth to point out that, unlike the case of standard metric, the b-metric Δ is not necessarily continuous due to the modified triangle inequality. In general, Δ is not continuous in each variable [5]. However, a metric-type space can be endowed with a topology induced by its convergence [5] and almost all the concepts and results which are valid for metric spaces can be extended to the framework of metric type spaces. In fact, for metric-type spaces, the notions of convergence, Cauchy sequence, and completeness and continuity can be defined similarly as in metric spaces, see [4, 8, 25]. Let us recall few such useful notions and facts in the framework of metric-type spaces.LetS,Δ be a metric-type space and let un be a sequence in S. Then,(1)
un converges in S if there exists u∈S such that limn⟶∞Δun,u=0(2)
un is a Cauchy sequence in S if limn,m⟶∞Δun,um=0(3)
S,Δ is complete if every Cauchy sequence in S is convergent in S(4)
A real-valued functionf on S is b-lower semicontinuous at a point u∈S if fu≤liminfn⟶∞bfun whenever un⟶uRecently, fixed-point theory for metric-type spaces studied and developed by many authors, for example, see [5, 7, 28, 32] and references therein.Motivated by the work of Kada et al. [18] and Hussain et al. [25] introduced wt-distance (here, we call it wb-distance) in the setting of metric-type space as follows.LetS,Δ be a metric-type space with constant b≥1. Then, a function pb:S×S⟶[0,∞) is called a wb-distance on S if, for any u,v,t∈S, the following hold:(1)
pbu,t≤bpbu,v+pbv,t(2)
pb(u,⋅):S⟶[0,∞) is b-lower semicontinuous (i.e., if, for any sequence vn in S, vn⟶v∈S, then pbu,v≤liminfn⟶∞bpbu,vn)(3)
For anyϵ>0, there exists δ>0 such that pbt,u≤δ and pbt,v≤δ imply Δu,v≤ϵRemark 1.
Note that, forb=1, each wb-distance is a w-distance. In general, wb-distance is not symmetric, see [25]. In fact, the class of wb-distances is much larger than the class of w-distance, see [28]. Every b-metric is wb-distance [25], but the converse is not true, see [28].Example 1.
(see [25]). Let S=ℝ and Δu,v=u−v2; then,(1)
The functionpb:S×S⟶[0,∞) defined by pbu,v=u2+v2 for every u,v∈S is a wb-distance on S(2)
The functionpb:S×S⟶[0,∞) defined by pbu,v=v2 for every u,v∈S is a wb-distance on S
The following result is an analogue of Lemma1 of [18], stated and used in [25, 26].Lemma 1 (see [25]).
LetS,Δ be a metric-type space with constant b≥1 and let pb be a wb-distance on S. Let un and vn be sequences in S; let αn and βn be sequences in 0,∞ converging to zero. Then, the following hold, for each u,v,t∈S.(i)
Ifpbun,v≤αn and pbun,t≤βn, for any n∈ℕ, then v=t. In particular, if pbu,v=0 and pbu,t=0, then v=t.(ii)
Ifpbun,vn≤αn and pbun,t≤βn, for any n∈ℕ, then Δvn,t⟶0.(iii)
Ifpbun,um≤αn, for any n,m∈ℕ with m>n, then un is a Cauchy sequence.(iv)
Ifpbv,un≤αn, for any n∈ℕ, then un is a Cauchy sequence.Lemma 2 (see [33]).
LetA be a closed subset of a metric-type space S,Δ and pb be a wb-distance on S. Suppose that there exists z∈S such that pbz,z=0. Then, pbz,A=0⇔z∈A, where pbz,A=infpbz,w:w∈A.Recently, some interesting results appeared in metric fixed-point theory with respect towb-distance on metric-type spaces; for example, see [25–28] and references therein.LetA∈(0,+∞]. Consider a real-valued function ψ on 0,A satisfying the following conditions:(1)
ψ0=0 and ψr>0, for each r∈0,A.(2)
ψ is nondecreasing on 0,A.(3)
ψ is subadditive, that is,(5)ψr1+r2≤ψr1+ψr2,for allr1,r2∈0,A.Examples and properties of such functions can be found in [34].We define(6)Ω[0,A)=ψ:ψsatisfies1−(3).The real-valued functionψ plays an important role in metric fixed-point theory; for example, see [22, 34, 35] and references therein. Among others, Latif and Abdou [22] proved some interesting fixed-point results for multivalued mapping with respect to w-distance. For example, the following results unify and extend a number of known metric fixed-point results.Theorem 6 (see [22]).
LetS,d be a complete metric space with a w-distance p. Let J:S⟶ClS be a multivalued mapping such that a real-valued function g on S, gu=pu,Ju, is lower semicontinuous. Assume that there exist c∈0,1 and η∈Ω[0,A) such that, for any u∈S, there is v∈Ju satisfying(7)cηpu,v≤ηgu,ηgv≤ξcpu,vηpu,v,where ξc is a strong MT-function. Then, there exists w0∈S such that gw0=0. Moreover, if pw0,w0=0, then w0∈Jw0.Theorem 7 (see [22]).
LetS,d be a complete metric space with a w-distance p. Let J:S⟶ClS be a multivalued mapping such that a real-valued function g on S, gu=pu,Ju, is lower semicontinuous. Assume that there exists η∈Ω[0,A) such that, for any u∈S, there is v∈Ju satisfying(8)ηpu,v=ηgu,ηgv≤ξpu,vηpu,v,where ξ is MT-function. Then, there exists w0∈S such that gw0=0. Moreover, if pw0,w0=0, then w0∈Jw0.The aim of this paper is to present some more general results on the existence of fixed points related to multivalued generalizedwb-contractive mappings defined on metric-type spaces. In particular, such mappings involve the function ψ∘pb, where ψ∈Ω[0,A) and the function pb is a wb-distance on a metric-type space. Consequently, our results unify and generalize the corresponding several known metric fixed-point results.


## 3. Results

Now, we present our first main result on the existence of fixed points for multivalued mapping with respect towb-distance, which improve and generalize a number of known fixed-point results including Theorem 6.Throughout this section,S,Δ is a complete metric-type space and pb is a wb-distance on S.Theorem 8.
LetJ:S⟶ClS be a multivalued mapping. Assume that there exist a strong MT-function ξc and a function ψ∈Ω[0,A) such that, for any u∈S, there is v∈Ju satisfying(9)cψpbu,v≤ψqu,ψqv≤ξcpbu,vψpbu,v,where q is a real-valued function on S defined by qu=pbu,Ju. Then,(1)
For anyu0∈S, there exist an orbit un of J at u0 and z0∈S such that limn⟶∞un=z0.(2)
pbz0,Jz0=0 if and only if the function q is J-orbitally b-lower semicontinuous at z0. Moreover, if pbz0,z0=0, then z0∈FixJ.Proof.
Letu0 be an arbitrary but fixed element of S. Then, there exists u1∈Ju0 such that(10)cψpbu0,u1≤ψqu0,(11)ψqu1≤ξcpbu0,u1ψpbu0,u1,(12)ξcpbu0,u1<c.
Thus, we have(13)ψqu0−ψqu1≥c−ξcpbu0,u1ψpbu0,u1>0.
Similarly, foru1∈S, there exists u2∈Ju1 such that(14)cψpbu1,u2≤ψqu1,ψqu2≤ξcpbu1,u2ψpbu1,u2,ξcpbu1,u2<c,(15)ψqu1−ψqu2≥c−ξcpbu1,u2ψpbu1,u2>0.
Continuing this process, we obtain an orbitun of J at u0∈S such that un+1∈Jun satisfying(16)cψpbun,un+1≤ψqun,(17)ψqun+1≤ξcpbun,un+1ψpbun,un+1,(18)ξcpbun,un+1<c,(19)ψqun−ψqun+1≥c−ξcpbun,un+1ψpbun,un+1>0,which imply that(20)ψqun+1<ψqun,n∈ℕ∪0.
While from (11), (12), and (14), it follows that(21)ψpbu1,u2≤1cψqu1≤1cξcpbu0,u1ψpbu0,u1<ψpbu0,u1.
Thus, for eachn∈ℕ, we obtain(22)ψpbun,un+1<ψpbun−1,un.
By (20) and (22), we note that the sequences ψqun and ψpbun,un+1 are decreasing. Now, since ψ is nondecreasing, it follows that qun and pbun,un+1 are decreasing sequences and are bounded from below, thus convergent. Now, by the definition of the function ξc, there exists δ∈[0,c) such that(23)limsupn⟶∞ξcpbun,un+1=δ.
Thus, for anyc0∈δ,c with c0c−1∈0,b−1, there exists n0∈ℕ such that(24)ξcpbun,un+1<c0,for alln>n0,and thus, for all n>n0, we have(25)ξcpbun,un+1×⋯×ξcpbun0+1,un0+2<c0n−n0.
Note that, for alln>n0, we have(26)ψqun+1≤ξcpbun,un+1ψpbun,un+1≤1cξcpbun,un+1ψqun≤1c2ξcpbun,un+1ξcpbun−1,unψqun−1⋮≤1cnξcpbun,un+1×⋯×ξcpbu1,u2ψqu1=ξcpbun,un+1×⋯×ξcpbun0+1,un0+2cn−n0×ξcpbun0,un0+1×⋯×ξcpbu1,u2ψqu1cn0,and thus,(27)ψqun+1<c0cn−n0ξcpbun0,un0+1×⋯×ξcpbu1,u2ψqu1cn0.
Putλ=c0c−1 and since λ<1, we have limn⟶∞λn−n0=0, and hence, the decreasing sequence ψqun converges to 0. Thus, we have(28)limn⟶∞qun=0.
Now, we show thatun is a Cauchy sequence. From (16), (17), and (24), we note that, for all n>n0,(29)ψpbun,un+1≤1cψqun≤1cξcpbun−1,unψpbun−1,un<λψpbun−1,un<λ2ψpbun−2,un−1⋮<λnψpbu0,u1.
Thus, we have(30)ψpbun,un+1≤λnψpbu0,u1,n∈ℕ∪0.
Now, for anyn,m∈ℕ,m>n,(31)ψpbun,um≤bψpbun,un+1+ψpbun+1,um≤bψpbun,un+1+bbψpbun+1,un+2+ψpbun+2,um=bψpbun,un+1+b2ψpbun+1,un+2+b2ψpbun+2,um≤bψpbun,un+1+b2ψpbun+1,un+2+b2bψpbun+2,un+3+ψpbun+3,um=bψpbun,un+1+b2ψpbun+1,un+2+b3ψpbun+2,un+3+ψpbun+3,um⋮≤bψpbun,un+1+b2ψpbun+1,un+2+…+bm−n−1ψpbum−2,um−1+ψpbum−1,um≤bλnψpbu0,u1+b2λn+1ψpbu0,u1+…+bm−n−1λm−2ψpbu0,u1+bm−n−1λm−1ψpbu0,u1=bλn1+bλ+bλ2+⋯+bλm−n−2+bm−n−2λm−n−1ψpbu0,u1.
Sinceλ<b−1, thus, for m,n∈ℕ with m>n>n0, we obtain(32)ψpbun,um≤bλn1−bλψpbu0,u1.
Thus, sincebλn/1−bλ⟶0 as n⟶+∞, we have limn,m⟶+∞ψpbun,um=0, and thus,(33)limn,m⟶+∞pbun,um=0.
By Lemma1iii, un is Cauchy sequence in S. Since S is complete, un converges to some point z0∈S. Note that the sequence un is an orbit of J at u0∈S with un⟶z0. Now, suppose that the function q is J-orbitally b-lower semicontinuous at z0; then, using (28), we have(34)0≤qz0≤liminfn⟶∞bqun=0,and hence, qz0=pbz0,Jz0=0. Conversely, if pbz0,Jz0=qz0=0, then, clearly, the function q is J-orbitally b-lower semicontinuity at z0 because qz0=0≤liminfn⟶∞bqun. Furthermore, if pbz0,z0=0, then, since Jz0 is closed, it follows from Lemma 2 that z0∈Jz0.
If we consider in Theorem8, a constant mapping ξcs=τ and s∈0,∞, where τ∈0,c; then, we have the following result.Corollary 1.
LetJ:S⟶ClS be a multivalued mapping satisfying that, for any constants c∈0,1 and for each u∈S, there is v∈Lcu such that(35)ψqv≤τψpbu,v,where Lcu=v∈Ju:cψpbu,v≤ψqu and a real-valued function q on S defined by qu=pbu,Ju is b-lower semicontinuous. Then, there exists z0∈S such that qz0=0. Furthermore, if pbz0,z0=0, then z0∈FixJ.Remark 2.
(1)
Theorem8 generalizes the fixed-point result (Theorem 2.2 of [36]). Indeed, if we take ψs=s, for all s∈[0,A) and b=1 (i.e., wb-distance as a w-distance) in Theorem 8, then we get Theorem 2.2 of [36]. Consequently, Theorem 8 also extends Theorem 4, which contains Theorem 3.(2)
Theorem8 generalizes the fixed-point result (Theorem 2.1 of [22]) for wb-distance in the frame work of metric-type spaces.(3)
Theorem8 contains Theorem 2.1 of [35] as a special case.(4)
Corollary1 contains the fixed-point results (Corollary 2.2 of [22] and Theorem 3.3 of [37]).
Now, without usingb-lower semicontinuity of the function q, we present a fixed-point result for multivalued mappings which extends the fixed-point result of Theorem 2.4 of [36] and reduces to Theorem 2.4 of [22].Theorem 9.
Suppose that all the hypotheses of Theorem8 (except the b-lower semicontinuity of the function q) hold. Assume that(36)infψpbu,z+ψqu:u∈S>0,for every z∈S with z∉Jz. Then, FixJ≠∅.Proof.
Following the proof of Theorem8, there exists an orbit un of J at u0∈S, which turns as a Cauchy sequence in a complete space S. Then, there exists z0∈S such that the sequence un converges to z0. Thus, by the b-lower semicontinuity of the function pbun,⋅, and from the proof of Theorem 8, it follows that, for all n>n0,(37)ψpbun,z0≤liminfm⟶∞bψpbun,um≤b2λn1−bλψpbu0,u1,where λ=c0/c<1. Also, note that(38)qun=pbun,Jun≤pbun,un+1,for all n, and since the function ψ is nondecreasing, we have(39)ψqun≤ψpbun,un+1,and thus,(40)ψqun<λnψpbu0,u1.
Assume thatz0∉Jz0. Then, we have(41)0<infψpbu,z0+ψqu:u∈S≤infψpbun,z0+ψqun:n>n0<infb2λn1−bλψpbu0,u1+λnψpbu0,u1:n>n0=b2−bλ+11−bλψpbu0,u1infλn:n>n0=0,which is impossible, and hence, z0∈FixJ.
UsingMT-functions (instead of strong MT-functions), we present a fixed-point result for multivalued wb-contraction mappings which extends Theorem 2.5 of [22] and thus contains a number of known metric fixed-point results.Theorem 10.
LetJ:S⟶ClS be a multivalued mapping. Assume that there exist an MT-function ξ and a function ψ∈Ω[0,A) such that, for any u∈S, there is v∈Ju satisfying(42)ψpbu,v=ψqu,ψqv≤ξpbu,vψpbu,v,where q is a real-valued function on S defined by qu=pbu,Ju. Then,(1)
For anyu0∈S, there exist an orbit un of J at u0 and z0∈S such that limn⟶∞un=z0.(2)
pbz0,Jz0=0 if and only if the function q is J-orbitally b-lower semicontinuous at z0. Moreover, if pbz0,z0=0, then z0∈FixJ.Proof.
Letu0 be an arbitrary but fixed element of S. Then, there is u1∈Ju0 such that(43)ψpbu0,u1=ψqu0,ψqu1≤ξpbu0,u1ψpbu0,u1,ξpbu0,u1<1.
Following the proof of Theorem8, there exists a Cauchy sequence un in the complete space S such that un+1∈Jun (that is, un is an orbit of J at u0) satisfying(44)ψpbun,un+1=ψqun,ψqun+1≤ξpbun,un+1ψpbun,un+1,ξpbun,un+1<1.
Consequently, there existsz0∈S such that the sequence un converges to z0. Now, the rest of the proof follows as of Theorem 8.Remark 3.
(1)
Forb=1, Theorem 10 reduces to Theorem 2.5 of [22].(2)
If we takeb=1 and ψs=s, for each s∈[0,A) in Theorem 10, the we obtain Theorem 2.5 of [38].(3)
It turns out that Theorem10 also generalizes Theorem 7 of [14] and Theorem 2.4 of [35].
Using the same technique as in the proof of Theorem9, we get the following fixed-point result (in the absence of the b-lower semicontinuity of the function q), which contains fixed-point result (Theorem 2.7 of [22]) as a special case.Theorem 11.
Suppose that all the hypotheses of Theorem10 except the b-lower semicontinuity of the function q hold. Assume that(45)infψpbu,z+ψqu:u∈S>0,for every z∈S with z∉Jz. Then, FixJ≠∅.


## 4. Example

Now, we present examples which show that our main results, namely, Theorems8 and 10 are genuine generalizations of Theorem 2.1 of [22] and Theorem 2.5 of [22], respectively.Example 2.
LetS=0,1. Define Δu,v=u−v2, for all u,v∈S. Then, S is a metric-type space with b=2. Define a wb-distance function on S by pbu,v=v2, for all u,v∈S. Let J:S⟶ClS be defined by(46)Ju=12u2;u∈0,1532∪1532,1,0,1796,14;u=1532.
LetA∈[1,∞) and let c=1/2. Define a function ψ:[0,A)⟶ℝ by ψs=s1/2. Clearly, ψ∈Ω[0,A). Define ξc:[0,∞)⟶[0,c) as follows:(47)ξcs=34s1/2;s∈0,12,38;s∈12,∞.
Clearly,ξc is a strong MT-function. Also, note that(48)qu=pbu,Ju=14u4;u∈0,1532∪1532,1,0;u=1532.
Now, for eachu∈0,15/32)∪(15/32,1, we have Ju=1/2u2. Take v=1/2u2∈Ju; then, we have(49)pbu,v=qu=14u4.
Thus, foru∈0,1,u≠15/32, we have(50)cψpbu,v≤ψqu,(51)ψqv=ψpb12u2,1212u22=ψ164u8=18u4≤316u4=ξcpbu,vψpbu,v.
Now, letu=15/32; then, we have Ju=0,17/96,1/4. Clearly, there exists v=0∈Ju such that(52)cψpbu,v=0=ψqu,(53)ψqv=ψpb0,0=ξcpbu,vψpbu,v.
Note that, for eachu∈0,1, all the conditions of Theorem 8 are satisfied, and hence, it follows that FixJ≠∅. Note that FixJ=0.
Clearly,pb is not a metric d, even not a w-distance p on S, and thus, J does not satisfy the hypotheses of Theorem 2.1 of [22]. Note that the mapping J also does not satisfy the hypotheses of Theorem 2.5 of [22].Example 3.
LetS=0∪1/n:n∈ℕ. Denote Λ=0∪1/2n:n∈ℕ. Clearly, Λ⊆S. Let Δ:S×S⟶[0,∞) be defined by(54)Δu,v=0;ifu=v,2;ifu≠v∈0,1,u−v;ifu≠v∈Λ,4;otherwise.
Then,S is a metric-type space with b=8/3 (see [28]). Define a wb-distance pb:S×S⟶[0,∞) by(55)pbu,v=0;ifu=v,2;ifu≠v∈0,1,max3u−v,2v−u;ifu≠v∈Λ,4;otherwise.
LetJ:S⟶ClS be defined by(56)Ju=111u;ifu∈Λ,0,13;otherwise.
LetA∈[1,∞). Define a function ψ:[0,A)⟶ℝ by ψs=s1/2. Clearly, ψ∈Ω[0,A). Define ξ:[0,∞)⟶[0,1) as follows:(57)ξs=14s;ifs∈Λ,12;otherwise.
Clearly,ξ is MT-function. We need to examine the following cases:Case I: supposeu∈Λ∖0; we have Ju=1/11u and so(58)qu=pbu,Ju=max3u−111u,2111u−u=3011u.Takev=1/11u∈Ju; then, clearly, v∈Λ, and we have(59)pbu,v=qu=3011u.Thus, foru∈Λ∖0, we have(60)ψpbu,v=ψqu,(61)ψqv=ψpb111u,1112u=ψ30112u=3011u1/2≤123011u1/2=ξpbu,vψpbu,v.Case II: supposeu=0; then, we have Ju=1/11u=0. Clearly, there exists v=0∈Ju such that(62)ψpbu,v=0=ψqu,(63)ψqv=ψpb0,0=0=ξpbu,vψpbu,v.Case III: supposeu=1; then, we have Ju=0,1/3. Clearly, there exists v=0∈Ju such that(64)ψpbu,v=2=ψqu,(65)ψqv=ψpb0,0=0≤12=ξpbu,vψpbu,v.Case IV: supposeu=1/3; then, we have Ju=0,1/3. Clearly, there exists v=1/3∈Ju such that(66)ψpbu,v=0=ψqu,(67)ψqv=ψpb13,13=0=ξpbu,vψpbu,v.Case V: supposeu∈S∖Λ∪1,1/3; then, we have Ju=0,1/3. Clearly, there exists v=1/3∈Ju such that(68)ψpbu,v=2=ψqu,(69)ψqv=ψpb13,13=0≤1=ξpbu,vψpbu,v.
Note that, for eachu∈S, all the conditions of Theorem 10 are satisfied, and hence, it follows that FixJ≠∅. Note that FixJ=0.
Not that thewb-distance pb is not a metric d, even not a w-distance p on S, and thus, J do not satisfy the hypotheses of Theorem 2.5 of [22].


## 5. Conclusion

Among others, Feng-Liu [16], Klim and Wardowski [17], and Ćirić [14] studied the existence of fixed points for multivalued contractive-type mappings without using the Hausdorff–Pompieu metric, and consequently, they generalized some classical known fixed-point results including Theorems 1 and 2. In this study, we established some general fixed-point results for multivalued generalized contractive mappings on metric-type spaces (instead of normal metric spaces) with respect to wb-distances. Presented results generalize and improve a number of known fixed-point results, including the corresponding fixed-point results which are stated in Section 2. In support of our main fixed-point theorems, examples are also provided. Note that the family of metric-type spaces is effectively larger than one of metric spaces, and hence, our theorems are more general, different from the classical results.



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