# Fixed Point Theorems forα,k,θ,φ-Contraction Multivalued Mappings in b-Metric Space

Authors: Youssef Errai; El Miloudi Marhrani; Mohamed Aamri
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

We present the concept ofα,k,θ,φ-contractive multivalued mappings in b-metric spaces and prove some fixed point results for these mappings in this study. Our results expand and refine some of the literature’s findings in fixed point theory.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory is one of the most active research areas of mathematics. The findings are used to solve problems in a wide variety of disciplines, including transportation theory, economics, and biomathematics. The development of functional analysis methods assisted in the broadening of the well-known Banach contraction theory (see [1–21] and references therein). One of them is the extension of the principle of contraction to cover applications and multivalued applications.Samet et al. [20] presented the definition of α-admissible mapping in complete metric spaces and partially ordered metric spaces in 2012 and proved several fixed point theorems under the generalized contraction in both. Following that, Hasanzade et al. [22] introduced the idea of α∗-admissible mapping, which is the multivalued edition of α-admissible single-valued mapping established in [20], and they provided a fixed point result for multivalued mappings in complete metric spaces satisfying many generalized contractive conditions.In 2013, Mohammadi et al. [16] developed the notion of an α∗-admissible mapping to the class of α-admissible mappings.In 2014, Jleli and Samet [11] expanded the contractive condition by considering the following function Θ:Definition 1 ([11]).
LetΘ denote the set of all functions θ:0,∞⟶1,∞ fulfilling the following criteria:
(Θ1): θ is nondecreasing;
(Θ2): from every sequence zn⊆ℝ+, limn⟶∞θzn=1 if and only if limn⟶∞zn=0;
(Θ3): there exists 0<r<1 and l∈0,∞ such that limz⟶0+θz−1/zr=l;
(Θ4): every θ∈Θ is continuous.Definition 2 ([11]).
LetΩ,d be a metric space. A mapping F:Ω⟶Ω is called a θ-contraction if θ:0,∞⟶1,∞ fulfills Θ1−Θ3 and there exists a constant k∈0,1 such that for all x,y∈Ω,
(1)dFx,Fy≠0⟹θdFx,Fy≤θdx,yk.Several researchers extended (1) in various ways and proved fixed point theorems for a single and multivalued contractive mappings (see [7, 12, 17, 18, 23]).Any Banach contraction is aθ-contraction, although this is not so for the inverse ([7]).Definition 3 ([24, 25]).
Lets≥1 be a given real number and Ω be a nonempty set. A function d:Ω×Ω⟶ℝ+ is a b-metric if the following conditions are fulfilled for all x,y,z∈Ω.
(bM1): dx,x=0 if and only if x=y;
(bM2): dx,y=dy,x;
(bM3): dx,z≤sdx,y+dy,z.
The pairΩ,d is called a b-metric space.
It is worth noting thatb-metric space is a broader category than metric spaces.The following notations are used in this paper:(2)Mx,y=maxdx,y,dx,fx,dy,fy,dx,fy+dy,fx2s.Qx,y=mindx,fx,dx,fy,dy,fx, and Ψ is the set of all self mapping on 0,+∞such that ψ is a continuous monotone and nondecreasing, and ψt=0⟺t=0.Definition 4 ([26]).
LetΩ,d be a b-metric space. X⊂Ω, and consider a sequence un in Ω. Then,
(i)
un is b-convergent if there exists u∈Ω such that dun,u⟶0 as n⟶∞
In this case, we writelimn⟶∞un=u.
(ii)
un is b-Cauchy sequence if dun,um⟶0 as n,m⟶∞(iii)
X is closed if and only if for each sequence un of points in X with dun,u⟶0 as n⟶∞, we have u∈X.Remark 5 ([26]).
The following statements hold in ab-metric space Ω,d.
(1)
Ab-convergent sequence has a unique limit(2)
Eachb-convergent sequence is a b-Cauchy sequence(3)
In general, ab-metric is not continuous(4)
d does not generally induce a topology on Ω.ConsiderΩ;d;s be a b-metric space and CLΩ the class of nonempty closed subsets of Ω. On CLΩ, consider H.;. be the generalized Pompeiu-Hausdorff b-metric (see [27]); i.e., for all G;R∈CLΩ,
(3)HG;R=maxsupa∈Gda,R,supb∈RdG,b,ifthemaximumexists,∞,otherwise,where da;R=infda;b:b∈R.ForG;R∈CLΩ, we put δG;R=supa∈Gda;R; we note that δG;R≤HG;R.Definition 6 ([26, 28]).
LetΩ,d and Y,δ be two b-metric spaces.
(D1): The space Ω,d is b-complete if every b-Cauchy sequence in Ωb-converges.
(D2): A function f:Ω⟶Y is b-continuous at a point u∈Ω if it is b-sequentially continuous at u; that is, whenever un is b-convergent to u, fun is b-convergent to fu.Definition 7 ([16, 22]).
LetΩ be a nonempty set, α:Ω×Ω⟶0,∞ be a given mapping, and NΩ the class of all nonempty subsets of Ω.
(1)
If the following condition applies, a mappingf:Ω⟶NΩ is called α∗-admissible: for x,y∈Ω, αx,y≥1⟹α∗fx,fy≥1, where α∗fx,fy≔infαa,b:a∈fx,b∈fy(2)
For eachx∈Ω and y∈fx with αx,y≥1, we have αy,z≥1; for all z∈fy, a mapping f:Ω⟶NΩ is said to be α-admissibleHussain et al. [8] in metric spaces used the concept of α-complete. We expand and apply it in b-metric spaces here.Definition 8.
LetΩ,d be a b-metric space and α:Ω×Ω⟶0,∞ be a mapping. The b-metric space Ω is said to be α-complete if and only if every b-Cauchy sequence un in Ω with αun,un+1≥1, for all n∈ℕ, b-converges in Ω.In 2015, Kutbi and Sintunavarat ([13]) introduced the notion of the α-continuity for multivalued mappings in metric spaces. We expand and apply it in b-metric spaces here.Definition 9.
LetΩ,d be a b-metric space and T:Ω⟶CLΩ and α:Ω×Ω⟶0,∞ be two given mappings. We say T is an α-continuous multivalued mapping on CLΩ,H if
(i)
for each sequenceun with un⟶u∈Ω as n⟶∞, and(ii)
αun,un+1≥1, for all n∈ℕ, we have Tun⟶Tu as n⟶∞Remark 10.
(1)
It is easy to see thatα∗-admissibility implies α-admissibility. But the converse may not be true as shown in example 15 of [15](2)
IfΩ is b-complete b-metric space, then Ω is also α-complete b-metric space. But the converse is not true (see Example 1)(3)
Note that theb-continuity implies the α-continuity. In general, the converse is not true (see in Example 2)Example 1.
LetΩ=0,∞,dx,y=x−y2 for all x,y∈Ω, and Λ be a closed subset of Ω. Define α:Ω×Ω⟶0,∞ by the following:
(4)αx,y=x−y2+1,ifx,y∈Λ,0,otherwise.
Clearly,Ω,d is not a b-complete b-metric space, but it is an α-complete b-metric space.Example 2.
LetΩ=0,∞ and dx,y=x−y2 for all x,y∈Ω; let T:Ω⟶CLΩ and α:Ω×Ω⟶0,∞ defined by the following:
(5)Tx=x3,ifx∈0,1,cosπx+3,ifx∈1,∞,αx,y=exy,ifx,y∈0,1,0,otherwise.
Clearly,T is not a b-continuous multivalued mapping on CLΩ,H. Indeed, if zn in Ω defined by zn=1+1/n for all n≥1, we see that zn=1+1/n⟶d1 but Tzn=cosπ+π/n+3⟶H2≠1=T1; however, T is a α-continuous multivalued mapping on CLΩ,H. Indeed, if zn⟶dz as n⟶∞ with αzn,zn+1≥1 for all n∈ℕ, then z,zn∈0,1 for all n∈ℕ and so Tzn=zn3⟶Hz3=Tz as n⟶∞.Lemma 11 [29].
LetΩ;d;s be a b-metric space, and let R∈CLΩ. If ς∈Ω and dς;R<η, then there exists ν∈R such that dς;ν<η.Lemma 12 [1].
LetΩ;d;s be a b-metric space, and let R∈CLΩ and ς∈Ω. Then, we have
(6)dς;R=0ifandonlyifς∈R¯=R,where R¯ denotes the closure of R.Lemma 13 ([30]).
Letun be a sequence in a b-metric space Ω,d,s with s≥1 such that
(7)dun,un+1≤λdun−1,un,for some λ∈0,1, and each n∈ℕ. Then, un is a b-Cauchy sequence in Ω,d,s.Lemma 14 [31].
IfΩ,d is a b-metric space with s≥1 and un and yn are b-convergent to u,y, respectively, we have
(8)1s2du,y≤liminfn⟶∞dun,yn≤limsupn⟶∞dun,yn≤s2du,y.
In particular, ifu=y, then we have limn⟶∞dun,yn=0. Moreover, for each z∈Ω, we have
(9)1sdu,z≤liminfn⟶∞dun,z≤limsupn⟶∞dun,z≤sdu,z.We will prove some new results of the fixed point inα-complete b-metric spaces, we will add a new kind of contraction for multivalued mappings called α,k,θ,φ-contraction multivalued mappings, and we give some examples to illustrate the main results of this paper.


## 2. Results

Definition 15.
LetΩ,d,s be a b-metric space. A mapping f:Ω⟶CLΩ is called weak α,k,θ,φ-contractive multivalued mapping if there exist α:Ω×Ω⟶0,∞,θ∈Θ,k∈0,1,L≥0,φ∈Ψ and constant λ∈0,1 such that
(10)αx,y≥1⟹θsHfx,fy≤λθMx,yk−φMx,y+LQx,y,for all x,y∈Ω.Example 3.
LetΩ=12/10,∞, and take the b-metric dx,y=x−y2 for all x,y∈Ω. Define f:Ω⟶CLΩ, φ:0,∞⟶0,∞, α:Ω×Ω⟶0,∞, and θ:0,∞⟶1,∞ by the following:
(11)fx=1210,1210+x4,φx=12x,αx,y=ex−y,ifx∈1210+y,∞,0,otherwise,and θt=etet. Note that for all t≥12/10, one has
(12)et/8et/8≤310et/2et−710t.
Now, for allx,y∈Ω such that x∈12/10+y,∞, we have Hfx,fy=x−y2/16.
Then,(13)θsHfx,fy=ex−y2/8ex−y2/8=edx,y/8edx,y/8≤eMx,y/8eMx,y/8≤310eMx,y/2eMx,y−710Mx,y=310θMx,y1/2−φMx,y=λθMx,yk−φMx,y.
And thus,(14)θsHfx,fy≤λθMx,yk−φMx,y+LQx,y.
Thus,f is weak α,k,θ,φ-contractive multivalued mapping, with λ=3/10 and k=1/2.The following is our primary result:Theorem 16.
LetΩ,d,s be a α-complete b-metric space and f:Ω⟶CLΩ is a weak α,k,θ,φ-contractive multivalued mapping. Assume that the following conditions hold:
(i)
f is a α-admissible(ii)
There existsx0,x1∈Ω such that x1∈fx0 and αx0,x1≥1(iii)
f isα-continuous
Then,f has a fixed point. Moreover, if u and v are fixed points of f such that αu,v≥1, then u=v.Proof.
Ifx0=x1 or x1∈fx1, then x1 is a fixed point of f. Assume that x0≠x1 and x1∈fx1.
Sincef is weak α,k,θ,φ-contractive multivalued mapping, we obtain
(15)1<θdx1,fx1≤θsHfx0,fx1≤λθMx0,x1k−φMx0,x1+LQx0,x1≤θMx0,x1k+LQx0,x1,with
(16)Mx0,x1=maxdx0,x1,dx0,fx0,dx1,fx1,dx0,fx1+dx1,fx02s≤maxdx0,x1,dx1,fx1,Qx0,x1=mindx1,fx1,dx0,fx1,dx1,fx0=0.
Assume thatMx0,x1=dx1,fx1, and from (15), we get
(17)1<θdx1,fx1≤θsHfx0,fx1≤θdx1,fx1k,which is a contradiction; it follows that Mx0,x1=dx0,x1, and (15) becomes
(18)1<θdx1,fx1≤θsHfx0,fx1≤θdx0,x1k.
By Lemma11, there exists x2∈fx1 such that dx1,x2≤dx0,x1, and by the monotonicity of θ, we obtain
(19)1<θdx1,x2≤θdx0,x1k.
Sincef is a α-admissible and αx0,x1≥1, we have αx1,x2≥1.
Ifx2∈fx2, this ends the proof; if x2∉fx2, we have dx2,fx2>0.
From (10), we have
(20)1<θdx2,fx2≤θsHfx1,fx2≤λθMx1,x2k−φMx1,x2+LQx1,x2≤λθMx1,x2k+LQx1,x2≤θMx1,x2k+LQx1,x2.
As above, we obtain(21)1<θdx2,fx2≤θsHfx1,fx2≤θdx1,x2k,and then, there exists x3∈fx2 such that
(22)1<θdx2,x3≤θdx1,x2k≤θdx0,x1k2.
We will assume that there exists the sequencexn in X such that xn≠xn+1∈fxn,αxn,xn+1≥1, using an inductive method.
And(23)1<θdxn+1,xn+2≤θdx0,x1kn,for all n∈ℕ. This shows that limn⟶∞θdxn,xn+1=1 and so limn⟶∞dxn,xn+1=0.
Prove thatxn is a b-Cauchy sequence.
Fors>1, we have
(24)1<θdxn+1,xn+2≤θsdxn+1,xn+2≤θsHfxn,fxn+1≤θdxn,xn+1k≤θdxn,xn+1,and using the monotonicity of θ, we shall have
(25)dxn+1,xn+2≤μdxn,xn+1,for all n∈ℕ, where μ=1/s∈0,1.
By Lemma13, we conclude that xn is a b-Cauchy sequence.
Fors=1, we can use similar arguments as in the proof of Theorem 2.1 of [9] to prove that there exists n1∈ℕ and r∈0,1 such that
(26)dxn,xn+1≤1n1/r,for all n≥n1. Then, for m>n>n1, we have
(27)dxn,xm≤∑i=nm−1dxi,xi+1≤∑i=nm−11i1/r.
Since0<r<1, ∑i=n∞1/i1/r converges. Therefore, dxn,xm⟶0 as m,n⟶∞. It follows that xn is a Cauchy sequence in Ω; since Ω is α-complete b-metric space, there exists x∈Ω such that
(28)limn⟶∞dxn,x=0.
Sincef is α-continuous, we have
(29)limn⟶∞Hfxn,fx=0,and hence,
(30)dx,fx≤sdx,xn+1+sdxn+1,fx≤sdxn+1,x+sHfxn,fx,
which implies thatdx,fx=0. And since fx is closed, we obtain x∈fx.
Assume thatf has two fixed points u and v such du,v≠0 and αu,v≥1.
We have(31)1≤θdu,v≤θsHfu,fv≤λθMu,vk−φMu,v+LQu,v≤λθMu,vk+LQu,v≤θMu,vk+LQu,v,
where(32)Mu,v=maxdu,v,du,fu,dv,fv,du,fv+dv,fu2s=du,v,Qu,v=mindv,fv,du,fv,dv,fu=0.
Then,(33)1<θdu,v≤θdu,vk,which is a contradiction since k∈0,1; thus, u=v, which ends the proof.☐Corollary 17.
LetΩ,d,s be a b-complete b-metric space and f:Ω⟶CLΩ is weak α,k,θ,φ-contractive multivalued mapping. Assume that the following conditions hold:
(i)
f is a α-admissible(ii)
There existx0 and x1∈fx0 such that αx0,x1≥1(iii)
f is a α-continuous
Then,f has a fixed point.Corollary 18.
LetΩ,d,s≥1 be a α-complete b-metric space and f:Ω⟶CLΩ is weak α,k,θ,φ-contractive multivalued mapping. Assume that the following conditions are true:
(i)
f is a α-admissible(ii)
There existx0 and x1∈fx0 such that αx0,x1≥1(iii)
f is a continuous
Then,f has a fixed point.Corollary 19.
LetΩ,d,s be a α-complete b-metric space and f:Ω⟶CLΩ is weak α,k,θ,φ-contractive multi-valued mapping. Assume that the following conditions are true:
(i)
f is a α∗-admissible(ii)
There existx0 and x1∈fx0 such that αx0,x1≥1(iii)
f is a continuous
Then,f has a fixed point.Theorem 20.
LetΩ,d,s be a α-complete b-metric space and f:Ω⟶CLΩ is a weak α,k,θ,φ-contractive multivalued mapping. Assume that the following conditions hold:
(i)
f is a α-admissible(ii)
There existsx0 and x1∈fx0 such that αx0,x1≥1(iii)
Ifxn is sequence in X such that αxn,xn+1≥1 for all n∈ℕ and xn⟶x∈X as n⟶∞, then αxn,x≥1 for all n∈ℕ
Then,f has a fixed point.Proof.
As in the proof of Theorem16, we obtain a b-Cauchy sequence xn such that αxn,xn+1≥1 for all n∈ℕ and limndxn,x=0. Assume that dx,fx>0.
Sincef is weak α,k,θ,φ-contractive multivalued mapping, we have
(34)αxn,x≥1⟹θsdxn+1,fx≤θsHfxn,fx≤λθMxn,xk−φMxn,x+LQxn,x≤θMxn,xk+LQxn,x,where
(35)Mxn,x=maxdxn,x,dxn,fxn,dx,fx,dxn,fx+dx,fxn2s=maxdxn,x,dxn,xn+1,dx,fx,dxn,fx+dx,xn+12s≤maxdxn,x,dxn,xn+1,dx,fx,dxn,x+dx,fx2+dx,xn+12s,Qxn,x=mindx,fx,dxn,fx,dx,fxn=mindx,fx,dxn,fx,dx,xn+1.
Ifn⟶∞, we obtain that
(36)limsupn⟶∞Mxn,x=dx,fx,limsupn⟶∞Qxn,x=0.
Sinceθ∈Θ, we obtain that
(37)1<θdx,fx=θds1sdx,fx≤θlimsupn⟶∞sdxn+1,fx≤θlimsupn⟶∞sHfxn,fx≤θlimsupn⟶∞Mxn,xk−φliminfn⟶∞Mxn,x+Llimsupn⟶∞Qxn,x=θlimsupn⟶∞Mxn,xk+Llimsupn⟶∞Qxn,x=θdx,fxk,which is a contradiction since k∈0,1. Thus, x is a fixed point of f.☐


## 3. Examples

Example 4.
LetΩ=0,∞, and take the b-metric dx,y=x−y2 for all x,y∈Ω. Define f:Ω⟶CLΩ, φ:Ω⟶Ω, and α:Ω×Ω⟶0,∞ by the following:
(38)fx=1,1+x4,αx,y=exy,ifx∈1+y,∞,0,otherwise,φx=710x,and θt=etet. Note that for all t≥1,
(39)et/8et/8≤4761000et/2et−710t.
Now, for allx,y∈X and x∈1+y,∞, we have Hfx,fy=x−y2/16.
Then,(40)θsHfx,fy=ex−y2/8ex−y2/8=edx,y/8edx,y/8≤eMx,y/8eMx,y/8≤4761000eMx,y/2eMx,y−710Mx,y=4761000θMx,y1/2−φMx,y=λθMx,yk−φMx,y.
Then,(41)θsHfx,fy≤λθMx,yk−φMx,y+LQx,y.
Thus,f is weak α,k,θ,φ-contractive multivalued mapping, with λ=476/1000 and k=1/2.
So, all conditions of Theorem16 are fulfilled, which implies that f has a fixed point, which is x0=1,1742 (in particular if L=0).Example 5.
LetΩ=0,∞, and take the b-metric dx,y=x−y2 for all x,y∈Ω. Define f:Ω⟶CLΩ, φ:Ω⟶Ω, α:Ω×Ω⟶0,∞, and θ:0,∞⟶1,∞ by the following:
(42)fx=0,x4,φx=12x,αx,y=ex−y,ifx≥y+1810,0,otherwise,
andθt=etet. Note that for all t≥18/10, one has
(43)et/8et/8≤et/2et−12t.
Now, for allx,y∈Ω and x≥y+18/10, we have Hfx,fy=x−y2/16.
Then,(44)θsHfx,fy=ex−y2/8ex−y2/8=edx,y/8edx,y/8≤eMx,y/8eMx,y/8≤eMx,y/2eMx,y−12Mx,y=θMx,y1/2−φMx,y=λθMx,yk−φMx,y.
Then,(45)θsHfx,fy≤λθMx,yk−φMx,y+LQx,y.
Thus,f is weak α,k,θ,φ-contractive multivalued mapping, with λ=1 and k=1/2.
So, all conditions of Theorem16 are verified, which implies that f has a fixed point, which is x0=0.7586 (in particular if L=0).Example 6.
LetΩ=0,∞, and take the b-metric dx,y=x−y2 for all x,y∈Ω. Define f:Ω⟶CLΩ, φ:Ω⟶Ω, α:Ω×Ω⟶0,∞, and θ:0,∞⟶1,∞ by the following:
(46)fx=0,x4,φx=17x2,αx,y=x−y,ifx≥2+y,0,otherwise,and θt=etet. Note that for all t≥2, one has
(47)et/8et/8≤710etet/3−17t2.
Now, for allx,y∈Ω and x≥2+y, we have Hfx,fy=x−y2/16.
Then,(48)θsHfx,fy=ex−y2/8ex−y2/8=edx,y/8edx,y/8≤eMx,y8eMx,y/8≤710eMx,yeMx,y/3−17Mx,y2=710θMx,y1/3−φMx,y=λθMx,yk−φMx,y.
Then,(49)θsHfx,fy≤λθMx,yk−φMx,y+LQx,y.
Thus,f is weak α,k,θ,φ-contractive multivalued mapping, with λ=7/10 and k=1/3.
So, all conditions of Theorem16 are verified, which implies that f has a fixed point, which is x0=2,69 (in particular if L=0).



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