# Multivalued Fixed Point Results for Two Families of Mappings in Modular-Like Metric Spaces with Applications

Authors: Tahair Rasham; Abdullah Shoaib; Choonkil Park; Manuel de la Sen; Hassen Aydi; Jung Rye Lee
Journal: Complexity
 (2020)

---

## Abstract

The aim of this research work is to find out some results in fixed point theory for a pair of families of multivalued mappings fulfilling a new type ofU-contractions in modular-like metric spaces. Some new results in graph theory for multigraph-dominated contractions in modular-like metric spaces are developed. An application has been presented to ensure the uniqueness and existence of a solution of families of nonlinear integral equations.


---

## Body

## 1. Introduction and Preliminaries

If the image of a pointx under two mappings is x itself, then x is called a common fixed point of those mappings. Theory of fixed point has a basic role in analysis (see [1–39]). Chistyakov [7] established the concept of modular metric spaces and showed briefly about modular convergence, convex modular, equivalent metrics, abstract convex cone, and metric semigroup. Padcharoen et al. [16] introduced the concept of α-type F-contractions in modular metric spaces and discussed some related results. Further results on these spaces in different directions can be seen in [6, 13, 14].In this paper, we establish some common fixed point theorems for two families of set-valued mappings satisfying a generalized contraction on a sequence only in a more generalized setting of modular-like metric spaces. New results can be established in dislocated metric spaces, ordered spaces, partial metric spaces, and metric spaces as a consequence of our findings. To support our results, some applications and examples are discussed. We give the following preliminary concepts which will be used in our results.Definition 1.
(see [16]). Let A≠ϕ. A function w:0,∞×A×A⟶0,∞ is called a modular-like metric on A if for all a,b,c∈A, l>0, and wla,b=wl,a,b, it satisfies(i)
wla,b=wlb,a for all l>0(ii)
wla,b=0 for all l>0 and then a=b(iii)
wl+ua,b≤wla,c+wuc,b for all l,u>0
If we replace (ii) bywla,b=0 for all l>0 if and only if a=b, then w becomes a modular metric on A. If we replace (ii) by wla,b=0 for some l>0 and then a=b, then w becomes a regular modular metric on A. For g∈A and ε>0, Bwlg,ε¯=p∈A:wlg,p≤ε is a closed ball in A,w. We will use m.l.m. space instead of modular-like metric space.Definition 2.
(see [16]). Let A,w be an m.l.m. space.(i)
E⊆A is known as w-complete if for any sequence ann∈ℕ in E and for some l>0, wlam,an⟶0 as m,n⟶∞ implies wlan,a⟶0 as n⟶∞ for some a∈E(ii)
The sequenceann∈ℕ in A is known as w-Cauchy for some l>0 if wlam,an⟶0 as m,n⟶∞(iii)
The sequenceann∈ℕ in A is known as w-convergent to a∈A for some l>0 if and only if wlan,a⟶0 as n⟶∞Definition 3.
LetA,w be an m.l.m. space and E⊆A. A member p0 which belongs to E is said to be a best approximation in E for g∈A if(1)wlg,E=infp∈Ewlg,p=wlg,p0.
If eachg∈A has a best approximation in E, then E is known as a proximinal set. PA is equal to the family of proximinal sets in A. Let A=ℝ+∪0 and wlg,p=1/lg+p for all l>0. Define a set E=4,6; then, for each y∈A,(2)wly,E=wly,4,6=infu∈4,6wly,u=wly,4.
Hence, 4 is the best approximation inE for each y∈A. Also, 4,6 is a proximinal set.Definition 4.
The set-valued mappingHwl:PA×PA⟶0,∞, defined by(3)HwlN,M=maxsupn∈Nwln,M,supm∈MwlN,m,is known as wl-Hausdorff metric. The pair PA,Hwl is called the wl-Hausdorff metric space. Let A=ℝ+∪0 and wlg,p=1/lg+p for all l>0. If P=5,6 and O=9,10, then HwlP,O=13/l.Definition 5.
LetX,w be a modular-like metric space. Then, we will say that w satisfies the ΔM-condition if it is the case that limn,m⟶∞wpxn,xm=0, for p=m−n implies limn,m⟶∞wlxn,xm=0m,n∈ℕ,m>n for some l>0.Definition 6.
(see [33]). Let A≠ϕ, ξ:A⟶PA be a set-valued mapping, B⊆A, and α:A×A⟶0,+∞. Then, ξ is called α∗-admissible on K if α∗ξa,ξc=infαu,v:u∈ξa,v∈ξc≥1 whenever αa,c≥1, for all a,c∈B.Definition 7.
LetA≠ϕ, ξ:A⟶PA be a set-valued mapping, M⊆A, and α:A×A⟶0,+∞. Then, ξ is called α∗-dominated on M if for all a∈M,α∗a,ξa=infαa,l:l∈ξa≥1.Definition 8.
(see [39]). Consider a metric space W,d. A function H:W⟶W is said A-contraction if for all c,k∈W, there exists τ>0 such that dHa,Hc>0 implies(4)τ+AdHa,Hc≤Ada,c,where A:ℝ+⟶ℝ is a mapping which satisfies the following:(F1) There existsk∈0,1 such that limσ⟶0+σkAσ=0(F2) For alla,c∈ℝ+ such that a<c, Aa<Ac, that is, A is strictly increasing(F3)limn⟶+∞σn=0 if and only if limn⟶+∞Aσn=−∞, for each sequence σnn=1∞ of positive numbers
The family of all mappings satisfying conditions (F1) to (F3) is denoted byϜ.Lemma 1.
LetA,w be an m.l.m. space. Let PA,Hwl be a Hausdorff wl-metric-like space. Then, for each a∈K and for all K,M∈PA, there exists ba∈M such that HwlK,M≥wla,ba.Proof.
IfHwlK,M=supa∈Kwla,M, then HwlK,M≥wla,M for each a∈K. As M is the proximinal set, for every a∈A, there exists at least one best approximation ba∈M which satisfies wla,M=wla,ba. Now, we have HwlK,M≥wla,ba. Now,(5)HwlK,M=suph∈MwlK,h≥supa∈Kwla,M≥wla,ba,for someba∈M.
Hence proved.Example 1.
(see [21]). Let A=ℝ. Define B:A×A⟶0,∞ by(6)Bb,t=1ifb>t,14ifb≯t.
DefineG,M:A⟶PA by(7)Gb=−4+b,−3+b,Mt=−2+t,−1+t.
Then,G and M are not α∗-admissible, but they are α∗-dominated.


## 2. Main Results

Letℑ,w be an m.l.m. space and c0∈ℑ; let Sσ:σ∈Ω and Tβ:β∈Φ be two families of multifunctions from ℑ to Pℑ. Let c1∈Sac0 be an element such that wc0,Sac0=wc0,c1. Let c2∈Tbc1 be such that wc1,Tbc1=wc1,c2. Let c3∈Scc2 such that wc2,Scc2=wc2,c3. In this way, we get a sequence TβSσcn in ℑ, where c2n+1∈Sic2n, c2n+2∈Tjc2n+1, n∈ℕ, i∈Ω, and j∈Φ. Also, wc2n,Sic2n=wc2n,c2n+1 and wc2n+1,Tjc2n+1=wc2n+1,c2n+2. TβSσcn is said to be a sequence in ℑ generated by c0. If Sσ:σ∈Ω=Tβ:β∈Φ, then we denote Sσcn instead of TβSσcn.Theorem 1.
Letℑ,w be a complete m.l.m. space. Assume that w is regular and satisfies the ΔM-condition. Let c0∈ℑ, α:ℑ×ℑ⟶0,∞, Tβ:β∈Φ, and Sσ:σ∈Ω be the families of α∗-dominated set-valued functions on w. Suppose there exist τ>0 and U∈Ϝ such that(8)τ+UHw1Sσt,Tβg≤Umaxw1t,g,w1t,Sσt,w2t,Tβg2,w1t,Sσt.w1g,Tβg1+w1t,g,whenever t,g∈ℑ∩TβSσcn, αt,g≥1, σ∈Ω, β∈Φ, and Hw1Sσt,Tβg>0.
Then, the sequenceTβSσcn generated by c0 converges to c´∈ℑ, and for each n∈ℕ, αcn,cn+1≥1. Also, if c´ satisfies (8) and either αcn,c´≥1 or αc´,cn≥1 for all n∈ℕ∪0, then Sσ and Tβ have a common fixed point c´ in ℑ for all σ∈Ω and β∈Φ.Proof.
Consider a sequenceTβSσcn. Obviously, cn∈ℑ for each n∈ℕ. If j is odd, then j=2ı`+1 for some ı`∈ℕ. By the definition of α∗-dominated mappings, we have α∗c2ı`,Sσc2ı`≥1 and α∗c2ı`+1,Tβc2ı`+1≥1 for all σ∈Ω and β∈Φ. As α∗c2ı`,Sσc2ı`≥1, this implies infαc2ı`,b:b∈Sσc2ı`≥1. Also, c2ı`+1∈Sfc2ı` for some f∈Ω, so αc2ı`,c2ı`+1≥1. Also, c2ı`+2∈Tgc2ı`+1 for some g∈Φ. Now, by using Lemma 1, we have(9)τ+Uw1c2ı`+1,c2ı`+2≤τ+UHw1Sfc2ı`,Tgc2ı`+1≤Umaxw1c2ı`,c2ı`+1,w1c2ı`,Sfc2ı`,w2c2ı`,Tgc2ı`+12,w1c2ı`,Sfc2ı`.w1c2ı`+1,Tgc2ı`+11+w1c2ı`,c2ı`+1≤Umaxw1c2ı`,c2ı`+1,w1c2ı`,c2ı`+1,w1c2ı`,c2ı`+1+w1c2ı`+1,c2ı`+22,w1c2ı`,c2ı`+1.w1c2ı`+1,c2ı`+21+w1c2ı`,c2ı`+1≤Umaxw1c2ı`,c2ı`+1,w1c2ı`+1,c2ı`+2.
This implies(10)τ+Uw1c2ı`+1,c2ı`+2≤Umaxw1c2ı`,c2ı`+1,w1c2ı`+1,c2ı`+2.
Now, if(11)maxw1c2ı`,c2ı`+1,w1c2ı`+1,c2ı`+2=w1c2ı`+1,c2ı`+2,then from (10), we have(12)Uw1c2i+1,c2i+2≤Uw1c2i+1,c2i+2−τ,a contradiction. Therefore,(13)maxw1c2ı`,c2ı`+1,w1c2ı`+1,c2ı`+2=w1c2ı`,c2ı`+1,for all i∈ℕ∪0. Hence, from (10), we have(14)Uw1c2i+1,c2i+2≤Uw1c2i,c2i+1−τ.
Similarly, we have(15)Uw1c2i,c2i+1≤Uw1c2i−1,c2i−τ,for all i∈ℕ∪0. By (14) and (15), we have(16)Uw1c2i+1,c2i+2≤Uw1c2i−1,c2i−2τ.
Repeating these steps, we get(17)Uw1c2i+1,c2i+2≤Uw1c0,c1−2i+1τ.
Similarly, we have(18)Uw1c2i,c2i+1≤Uw1c0,c1−2iτ.
Inequalities (17) and (18) can jointly be written as(19)Uw1cn,cn+1≤Uw1c0,c1−nτ.
Taking the limit asn⟶∞ in (19), we have(20)limn⟶∞Uw1cn,cn+1=−∞.
SinceU∈Ϝ,(21)limn⟶∞w1cn,cn+1=0.
Applying the property (F1) ofϜ, there exists k∈0,1 such that(22)limn⟶∞w1cn,cn+1kUw1cn,cn+1=0.
By (19), for all n∈ℕ, we obtain(23)w1cn,cn+1kUw1cn,cn+1−Uw1c0,c1≤−w1cn,cn+1knτ≤0.
Considering (21) and (22) and letting n⟶∞ in (23), we have(24)limn⟶∞nw1cn,cn+1k=0.
Since (24) holds, there exists n1∈ℕ such that nw1cn,cn+1k≤1 for all n≥n1 or(25)w1cn,cn+1≤1n1/kfor alln≥n1.
Takep>0 and m=n+p>n>n1; then,(26)wpcn,cm≤w1cn,cn+1+w1cn+1,cn+2+⋯+w1cm,cm+1≤1n1/k+1n+11/k+⋯+1m1/k.
Applying the limit asn,m⟶∞ on both sides, we have(27)limn,m⟶∞wpcn,cm≤limn,m⟶∞1n1/k+limn,m⟶∞1n+11/k+⋯+limn,m⟶∞1m1/k.
Ask∈0,1, then 1/k>1, and so,(28)limn,m⟶∞1n1/k=limn,m⟶∞1n+11/k=limn,m⟶∞1m1/k=0.
Asw:0,∞×ℑ×ℑ⟶0,∞, then(29)limn,m⟶∞wpcn,cm=0.
Sincew satisfies the ΔM-condition, we have(30)limn,m⟶∞w1cn,cm=0.
Hence,TβSσcn is a Cauchy sequence in w. Since ℑ,w is a regular complete modular-like metric space, there exists c´∈ℑ such that TβSσcn⟶c´ as n⟶∞, and so,(31)limn⟶∞w1cn,c´=0.
Now, by Lemma1, we have(32)τ+Uw1c2n+1,Tβc´≤τ+UHw1Sec2n,Tβc´,for some β∈Φ and some e∈Ω. Now, there exists c2n+1∈Sec2n such that w1c2n,Sec2n=w1c2n,c2n+1. From the assumption, αcn,c´≥1. Assume that w1c´,Tβc´>0; then, there must be a positive real number p such that w1c2n+1,Tβc´>0 for n≥p. Now, Hw1Sec2n,Tβc´>0; then, by using inequality (8), we have(33)τ+Uw1c2n+1,Tβc´≤Umaxw1c2n,c´,w1c2n,c´,w1c2n,c2n+1+w1c2n+1,Tβc´2,w1c2n,Sec2n⋅w1f,Tβc´1+w1c2n,c´.
Lettingn⟶∞ and using (31), we get(34)τ+Uw1c´,Tβc´≤Uw1c´,Tβc´.
SinceU is strictly increasing, (32) implies(35)w1c´,Tβc´<w1c´,Tβc´.
This is not true. So, our assumption is wrong. Hence,w1c´,Tβc´=0 or c´∈Tβc´ for each β∈Φ. Similarly, by proceeding with Lemma 1 and inequality (8), we can prove that w1c´,Sσc´=0 or c´∈Sσc´ for all σ∈Ω. Hence, c´ is a common fixed point of both the mappings Sσ and Tβ in ℑ for all σ∈Ω and β∈Φ.Example 2.
Letℑ=ℝ+∪0. Take w2q,g=q+g and w1q,g=1/2q+g for all q,g∈ℑ. Suppose that Sσ,Tβ:ℑ×ℑ⟶Pℑ are two families of multivalued mappings defined by(36)Smv=v3m,2v3mifv∈ℑ,where m=1,2,3,…, and(37)Tnv=v4n,3v4nifv∈ℑ,where n=1,2,3,…. Suppose that v0=1 and w1v0,S1v0=w11,S11=w11,1/3. So, v1=1/3. Now, w1v1,T1v1=w11/3,T11/3=w11/3,1/12. So, v2=1/12. Now, w1v2,S2v2=w11/12,S21/12=w11/12,1/72. So, v3=1/72. Continuing in this way, we have TnSmvn=1,1/3,1/12,1/72,…. Consider the mapping α:ℑ×ℑ⟶0,∞ defined by(38)αr,t=1,ifr>t,12,otherwise.
Now, ifv,y∈ℑ∩TβSσvn with αv,y≥1, we have(39)Hw1Smv,Tny=maxsupa∈Smxw1a,Tny,supb∈Tnyw1Smv,b=maxw12v3m,y4n,3y4n,w1v3m,2v3m,3y4n=maxw12v3m,y4n,w1v3m,3y4n=12max2v3m+y4n,v3m+3y4n,Wx,y=max12v+y,v+v3m,v+y4n,v+v/3m⋅y+y/4n1+v+y.Case (i).
Ifmax2v/3m+y/4n,y/3m+3y/4n=2v/3m+y/4n and τ=ln1.2, then we have(40)8vn+3my≤10mnx+10mny,48vn+18my≤60mnx+60mny,652v3m+y4n≤x+y,ln1.2+ln2v3m+y4n≤lnx+y.
This implies that(41)τ+UHw1Smv,Tny≤UWx,y.Case (ii).
Ifmax2v/3m+y/4n,y/3m+3y/4n=v/3m+3y/4n and τ=ln1.2, then we have(42)4vn+9my≤10mnx+10mny,24vn+54my≤60mnx+60mny,65v3m+3y4n≤x+y,ln1.2+lnv3m+3y4n≤lnx+y.
This implies that(43)τ+UHw1Smv,Tny≤UWx,y.
Hence, all the requirements of Theorem1 are satisfied. So, the families Sm and Tn have a common fixed point.
If we takeSσ:σ∈Ω=Tβ:β∈Φ in Theorem 1, then we obtain the above result.Corollary 1.
Letℑ,w be a complete m.l.m. space. Assume that w is regular and satisfies the ΔM -condition. Let c0∈ℑ, α:ℑ×ℑ⟶0,∞, and Sσ:σ∈Ω be a family of α∗-dominated set-valued functions on ℑ. If there exist τ>0 and U∈Ϝ such that(44)τ+UHwlSσt,Sβg≤Umaxw1t,g,w1t,Sσt,w2t,Sβg,w1t,Sσt⋅w1g,Sβg1+w1t,g,whenever t,g∈ℑ∩Sσcn, αt,g≥1, σ,β∈Ω, and Hw1Sσt,Sβg>0, then the sequence ℑSσcn generated by c0 converges to c´∈ℑ, and for each n∈ℕ, αcn,cn+1≥1. Also, if c´ satisfies (44) and either αcn,c´≥1 or αc´,cn≥1 for every n∈ℕ∪0, then Sσ:σ∈Ω has a common fixed point c´ in ℑ.


## 3. Applications in Graph Theory

Jachymski [11] developed a relation between fixed point theory and graph theory by the induction of graphic contractions. Hussain et al. [9] established some results for the new type of contractions endowed with a graph and also showed an application. Further useful results on the graph can be seen in [34, 35, 40].Definition 9.
LetA be a nonempty set and Y=VY,LY be a graph with VY=A. A mapping F from A to PA is known as multigraph-dominated on A if a,b∈LY, whenever a∈A and b∈Fa.Theorem 2.
Letℑ,w be a complete m.l.m. space endowed with a graph Y, c0∈ℑ, and the following hold:(i)
Tβ:β∈Φ and Sσ:σ∈Ω are sets of multigraph-dominated functions on ℑ∩TβSσcn.(ii)
There existτ>0 and U∈Ϝ such that(45)τ+UHw1Sσt,Tβy≤Umaxw1t,y,w1t,Sσt,w2t,Tβy2,w1t,Sσt⋅w1y,Tβy1+w1t,y,whenever t,y∈ℑ∩TβSσcn, t,y∈LY, σ∈Ω, β∈Φ, and Hw1Sσt,Tβy>0.
Assume thatℑ is regular and satisfies the ΔM-condition. Then, TβSσcn is a sequence in ℑ, cn,cn+1∈LY, and TβSσcn⟶g∗. Also, if g∗ satisfies (45) and cn,g∗∈LY or g∗,cn∈LY for each n∈ℕ∪0, then Sσ and Tβ have common fixed point g∗ in ℑ for all σ∈Ω and β∈Φ.Proof.
Defineα:ℑ×ℑ⟶0,∞ by αt,y=1 if t∈ℑ and t,y∈LY. Otherwise, αt,y=0. By the definition of the graph dominated on ℑ, we have t,y∈LY for all y∈Sσt and t,y∈LY for each y∈Tβt. So, αt,y=1 for all y∈Sσt and αt,y=1 for every y∈Tβt. This means that infαt,y:y∈Sσt=1 and infαt,y:y∈Tβt=1. Hence, α∗t,Sσt=1 and α∗t,Tβt=1 for every t∈ℑ. So, the families of mappings are α∗-dominated on ℑ. Furthermore, inequality (45) can be expressed as(46)τ+UHw1Sσt,Tβy≤Umaxw1t,y,w1t,Sσt,w2t,Tβy2,w1t,Sσt⋅w1y,Tβy1+w1t,y,whenever t,y∈ℑ∩TβSσcn, αt,y≥1, and Hw1Sσt,Tβy>0. Also, (ii) holds. Then, by Theorem 1, we have TβSσcn is a sequence in ℑ, and TβSσcn⟶g∗∈ℑ. Now, cn,g∗∈ℑ, and either cn,g∗∈LY or g∗,cn∈LY implies that either αcn,g∗≥1 or αg∗,cn≥1. So, all the requirements of Theorem 1 are satisfied. Hence, from Theorem 1, Sσ and Tβ have a common fixed point g∗ in ℑ, and w1g∗,g∗=0.


## 4. Results on Single-Valued Mappings

In this section, some consequences of our results related to single-valued mappings inm.l.m. spaces have been discussed. Let ℑ,w be an m.l.m. space, c0∈ℑ, and Sσ,Tβ:ℑ⟶ℑ be two families of mappings. Let c1=Sσc0, c2=Tβc1, and c3=Sσc2. Adopting this way, we make a sequence cn in ℑ so that c2n+1=Sσc2n and c2n+2=Tβc2n+1, where n=0,1,2,…. We represent this kind of iterative sequence by TβSσcn. We say that TβSσcn is a sequence in ℑ generated by c0. If Sσ:σ∈Ω=Tβ:β∈Φ, then we denote ℑSσcn instead of TβSσcn.Theorem 3.
Letℑ,w be a complete m.l.m. space. Assume that w is regular and satisfies the △M-condition. Let r>0, c0∈Bw1c0,r¯⊆ℑ, α:ℑ×ℑ⟶0,∞, and Sσ:σ∈Ω, Tβ:β∈Φ be two families of α-dominated mappings from ℑ to ℑ. Then, there exist τ>0 and U∈Ϝ such that(47)τ+UHw1Sσt,Tβg≤Umaxw1t,g,w1t,Sσt,w2t,Tβg2,w1t,Sσt⋅w1g,Tβg1+w1t,g,whenever t,g∈ℑ∩TβSσcn, αt,g≥1, σ∈Ω, β∈Φ, and w1Sσt,Tβg>0.
Then,TβSσcn is a sequence in ℑ, αcn,cn+1≥1 for each n∈ℕ∪0 and TβSσcn⟶h∈ℑ. Also, if u satisfies (47) and either αcn,h≥1 or αh,cn≥1 for all n∈ℕ∪0, then Sσ and Tβ have a common fixed point h in ℑ for every σ∈Ω and β∈Φ.Proof.
The proof is similar to the proof of Theorem1.
If we takeSσ:σ∈Ω=Tβ:β∈Φ in Theorem 3, then we can get the following result.Corollary 2.
Letℑ,w be a complete m.l.m. space. Assume that w is regular and satisfies the ΔM-condition. Let c0∈ℑ, α:ℑ×ℑ⟶0,∞, and Sσ:σ∈Ω be a family of α-dominated mapping from ℑ to ℑ. Then, there exist τ>0 and U∈Ϝ such that(48)τ+UHw1Sσt,Sβg≤Umaxw1t,g,w1t,Sσt,w2t,Tβg2,w1t,Sσt⋅w1g,Sβg1+w1t,g,whenever t,g∈ℑ∩ℑSσcn, αt,g≥1, σ,β∈Ω, and w1Sσt,Sσg>0. Then, ℑSσcn is a sequence in ℑ, αcn,cn+1≥1 for each n∈ℕ∪0, and ℑSσcn⟶h∈ℑ. Also, if h satisfies (48) and either αcn,h≥1 or αh,cn≥1 for each n∈ℕ∪0, then h is the fixed point of Sσ in ℑ for every σ∈Ω.


## 5. Application in Integral Equations

In this section, we discuss the application of our work in integral equations. First of all, we present our main result without a closed ball for self-mappings and withoutα∗-dominated functions and then apply it to attain an application in integral equations. We also discuss the uniqueness.Theorem 4.
Letℑ,w be a complete m.l.m. space. Assume that w is regular and satisfies the ΔM-condition. Let c0∈ℑ and Sσ:σ∈Ω and Tβ:β∈Φ be the families of self-mappings. Then, there exist τ>0 and a strictly increasing function U:ℝ+⟶ℝ such that(49)τ+UHw1Sσt,Tβg≤Umaxw1t,g,w1t,Sσt,w2t,Tβg2,w1t,Sσt⋅w1g,Tβg1+w1t,g,whenever t,g∈TβSσcn, σ∈Ω, β∈Φ, and w1Sσt,Tβg>0. Then, TβSσcn⟶h∈ℑ. Also, if inequality (49) holds for t,g∈h, then Sσ and Tβ have a unique common fixed point h in ℑ for all σ∈Ω and β∈Φ.Proof.
The proof is similar to the proof of Theorem1. We prove only uniqueness. Let Sσ and Tβ have another common fixed point v. Suppose w1Sσu,Tβv>0. Then,(50)τ+Uw1Sσu,Tβv≤Umaxw1u,v,w1u,Sσu,w2u,Tβv2,w1u,Sσu⋅w1v,Tβv1+w1u,v.
This implies that(51)w1u,v<μ1w1u,v<w1u,v,which is not true. So, w1Sσu,Tβv=0. Hence, u=v.
LetW=C0,1,ℝ+ be the set of all continuous functions on 0,1. Consider the families of integral equations(52)uk=∫0kHσk,h,uhdh+∈,(53)ck=∫0kGβk,h,chdh+∈,for all k∈0,1, σ∈Ω, β∈Φ, and Hσ,Gβ are the functions from 0,1×0,1×W to ℝ. For c∈C0,1,ℝ+, define the supremum norm as cτ=supk∈0,1cke−ηk, where η>0 is arbitrarily taken. Define(54)w1c,p=12supk∈0,1ck+pke−τk=12c+pτ,for all c,p∈C0,1,ℝ+; with these settings, C0,1,ℝ+,dτ becomes a complete m.l.m. space.
Now, we prove the following theorem to ensure the uniqueness and existence of a solution of families of nonlinear integral equations (52) and (53).Theorem 5.
Assume the following constraints are satisfied:(i)
Hσ,σ∈Ω and Gβ,β∈Φ are two families of mappings from 0,1×0,1×C0,1,ℝ+ to ℝ.(ii)
Define(55)Sσuk=∫0kHσk,h,uhdh+∈,Tβck=∫0kGβk,h,chdh+∈.
Suppose there existsτ>0 such that(56)Hσk,h,u+Gβk,h,c≤τEσ,βu,cτEσ,βu,c+1,for all k,h∈0,1 and u,c∈C0,1,ℝ+, where(57)Eσ,βu,c=max12u+cτ,u+Sσuτ,u+Sσuτ+c+Tβcτ2,u+Sσuτ.c+Tβcτ1+u+cτ.
Then, integral equations (52) and (53) have a unique solution.Proof.
By assumption (ii),(58)Sσu+Tβc=∫0kHσk,h,u+Gβk,h,cdh≤∫0kτEσ,βu,cτEσ,βu,c+1eτhdh≤τEσ,βu,cτEσ,βu,c+1∫0keτhdh≤Eσ,βu,cτEσ,βu,c+1eτk.
This implies(59)Sσu+Tβce−τk≤Eσ,βu,cτEσ,βu,c+1Sσu+Tβcτ≤Eσ,βu,cτEσ,βu,c+1τEσ,βu,c+1Eσ,βu,c≤1Sσu+Tβcττ+1Eσ,βu,c≤1Sσu+Tβcτ,which further implies(60)τ−1Sσuk+Tβckτ≤−1Eσ,βu,c.
So, all the requirements of Theorem5 are satisfied for Uf=−1/f; f>0 and w1f,c=1/2f+cτ. Hence, two families of integral equations given in (52) and (53) have a unique common solution.


## 6. Conclusion

In this article, we have achieved some new results for set-valued mappings belonging to two families which satisfy generalized rational-type Wardowski’s contraction. Dominated mappings are applied to find out the fixed point results. Applications in the subject of integral equations and graph theory are presented. Moreover, we investigate our results in new generalized modular-like metric spaces. Many consequences of our results in dislocated metric spaces, metric spaces, and partial metric spaces (even with a partial order) can be established easily.



---

*Source: 2690452-2020-11-16.xml*