# Common Fixed Point Theorems for Probabilistic Nearly Densifying Mappings

Authors: Aeshah Hassan Zakri; Sumitra Dalal; Sunny Chauhan; Jelena Vujaković
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

The aim of this paper is to prove some coincidence and common fixed point theorems for probabilistic nearly densifying mappings in complete Menger spaces. Our results improve the results of Chamola et al. (1991), Dimri and Pant (2002), and Pant et al. (2004) and extend the results of Khan and Liu (1997) in the framework of probabilistic settings.


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## Body

## 1. Introduction and Preliminaries

Banach contraction mapping principle is one of the most interesting and useful tools in applied mathematics. In recent years many generalizations of Banach contraction mapping principle have appeared. The notion of probabilistic metric spaces (in short PM-spaces) is a probabilistic generalization of metric spaces which are appropriate to carry out the study of those situations wherein distances are measured in the sense of distribution functions rather than nonnegative real numbers. The study of PM-spaces was initiated by Menger [1]. Since then, Schweizer and Sklar [2] enriched this concept and provided a new impetus by proving some fundamental results on this theme. The first result on fixed point theory in PM-spaces was given by Sehgal and Bharucha-Reid [3] wherein the notion of probabilistic contraction was introduced as a generalization of the classical Banach fixed point principle in terms of probabilistic settings. Some recent fixed point results can be studied in [4–7].Kuratowski [8] introduced the notion of measure of noncompactness of a bounded subset of a metric space. Further, this study was carried on by Furi and Vignoli [9]. They introduced the notion of densifying (also called condensing) mapping in terms of Kuratowski’s measure of noncompactness and obtained some fixed point theorems. Following Furi and Vignoli [9], a number of mathematicians worked on densifying mappings and proved some metrical fixed point theorem (cf. [10–14]). As a generalization of Kuratowski’s measure of noncompactness, Bocsan and Constantin [15] introduced the notion of Kuratowski’s measure of noncompactness in PM-spaces. Subsequently, Bocşan [16] studied the notion of probabilistic densifying mappings. Later, Hadžić [17], Tan [18], Chamola et al. [19], Dimri and Pant [20], Pant et al. [21], Pant et al. [22], and Singh and Pant [23] proved some results for such mappings. In [24], Ganguly et al. introduced the notion of probabilistic nearly densifying mappings and proved some interesting results in this setting.The aim of this paper is to prove some coincidence and common fixed point theorems for certain classes of nearly densifying mappings in complete Menger spaces. First, we give some topological definitions and terminology defined in [8, 15–17].Definition 1.
A semigroupG is said to be left reversible if for any r,s∈G there exist a,b∈G such that ra=sb.It is easy to see that the notion of left reversibility is equivalent to the statement that any two right ideals ofG have nonempty intersection.Definition 2.
LetG be a family of self-mappings in X. A subset Y of X is called G-invariant if gY⊆Y for all g∈G.Definition 3.
LetG* be the semigroup generated by G under composition *. Clearly, G*⊇gn:n≥0 for any g∈G and G*u=u∪gu:g∈G* for u∈X.We restate the notion of probabilistic diameter for the sake of quick reference.Definition 4.
LetA be a nonempty subset of X. A function DA· defined by(1)DAx=sup⁡y<x⁡inf⁡u,v∈A⁡Fu,vyis called probabilistic diameter of A. A is said to be bounded if(2)sup⁡x∈R⁡DAx=1.The following definition is due to Bocsan and Constantin [15].Definition 5.
For a probabilistic bounded subsetA of X, αA(x) defined by αAx=sup⁡⁡ε≥0:∃afinitecoverAofAsuchthatDSx≥εforallS∈A is called Kuratowski’s function.The following properties of Kuratowski’s functions are proved in [8]:(a)
αA∈I, the set of distribution functions;(b)
αAx≥DA(x);(c)
ifϕ≠A⊂B⊂X, then αAx≥αBx;(d)
αA∪Bx=min⁡⁡αAx,αBx;(e)
letA- be the closure of A in the ε,λ-topology on X; then(3)αA-x=αAx;(f)
A is probabilistic precompact (totally bounded) if αA=H,whereH denotes the specific distribution function defined by(4)Hx=0,x≤0;1,x>0.Definition 6.
LetX,F be a PM-space. A continuous mapping f of X into X is called a probabilistic densifying mapping if and only if, for every subset A of X, αA<H implies αf(A)>αA.Definition 7.
A self-mappingf:X→X is probabilistic nearly densifying if αf(A)>αA, whenever αA<H, A⊂H, and A is f-invariant.Definition 8.
Supposeϕ:0,∞→[0,∞) is an upper semicontinuous function with ϕ(0)=0 and ϕ(t)<t for all t>0.


## 2. Main Results

First, we prove some fixed point theorems for probabilistic nearly densifying mappings in Menger spaces.Theorem 9.
LetP, Q, and R be three continuous and nearly densifying self-mappings on a complete Menger space X,F,* such that sup⁡⁡x*x=1 and R commutes with P and Q. If, for all x<1, u,v∈X, the following conditions are satisfied:(5)ϕ1Pu,Qv>min⁡⁡ϕ2Ru,Puϕ1Rv,Qvϕ2Ru,Rvϕ2Ru,Rv,ϕ2Ru,Pu,llllllllllϕ1Rv,Qv,ϕ2Ru,Puϕ1Rv,Qvϕ2Ru,RvlllllllllllllllllllllllllllllforRu≠Rv,Pu≠Qv;(6)ϕ2Qu,Pv>min⁡⁡ϕ1(Ru,Qu)ϕ2(Rv,Pv)ϕ1(Ru,Rv)ϕ1Ru,Rv,ϕ1Ru,Qu,ϕ2Rv,Pv,lllllllllϕ1(Ru,Qu)ϕ2(Rv,Pv)ϕ1(Ru,Rv),lllllllllllforRu≠Rv,Qu≠Pv,where ϕ1 and ϕ2 are real valued mappings from X×X to ς, the collection of all distribution functions, with either ϕ1 or ϕ2 being upper semicontinuous (u.s.c.) and ϕ1(u,u)=ϕ2(u,u)=1 for all u∈X.
Further, if, for someu0∈X, Gu0=PiQjRku0:i=0,1,2,…;j=0,1,2,…;k=0,1,2,… is bounded, then P and R or Q and R have a coincidence point.Proof.
Foru0∈X, let A=G(u0) and S=PQR.
ThenA=u0∪PA∪QA∪R(A).
IfαA<H, then(7)αA=αu0∪PA∪QA∪RA=min⁡⁡αPA,αQA,αR(A)>αA,a contradiction. It implies that A- is precompact.
LetB=⋂n=0∞PQRnA-.
Then it is easy to see thatSB=B and B is nonempty compact subset of A. By the continuity of P, Q, and R, it follows that PA-⊂A-, QA-⊂A-, and RA-⊂A-. Further, it is clear that PB-⊂B-, QB-⊂B-, and RB-⊂B-.
Note that(8)RB=⋂n=0∞RPQRnA-⊂⋂n=0∞PQRnRA-⊂B,B=PQRB=RPQB⊂RPB⊂RB,which implies R(B)=B or R2(B)=B.
Now, assume thatϕ1 is upper semicontinuous. Then the function T:B→I, defined by T(u)=ϕ1(Ru,Qu), is u.s.c. So T assumes its maximal value at some point p in B. Clearly, p∈R2(B), so there is a w∈B such that p=R2(w). Suppose that neither P and R nor Q and R have a coincidence point. Then(9)TPQw=ϕ1(RPQ(w),QPQw)=ϕ1PRQw,QPQwby5,>min⁡⁡ϕ2(R2Q(w),PRQ(w))ϕ1(RPQ(w),QPQ(w))ϕ2(R2Q(w),RPQ(w))ϕ2R2Qw,RPQw,llllllllllllllllϕ2R2Qw,PRQw,ϕ1RPQw,QPQw,lllllllllllllllϕ2(R2Q(w),PRQ(w))ϕ1(RPQ(w),QPQ(w))ϕ2(R2Q(w),RPQ(w))=ϕ2QR2w,PRQw,by6,>min⁡⁡ϕ1(RR2(w),QR2(w))ϕ2(R2Q(w),PRQ(w))ϕ1(RR2(w),R2Q(w))ϕ1RR2w,R2Qw,ϕ1RR2w,QR2w,lllllllllllllllllϕ2R2Qw,PRQw,llllllllllllllllϕ1(RR2(w),QR2(w))ϕ2(R2Q(w),PRQ(w))ϕ1(RR2(w),R2Q(w))=ϕ1RR2w,QR2w=ϕ1Rp,Qp=Tp,a contradiction to the selection of p. Hence, P and R or Q and R must have a coincidence point.
The same result holds good ifϕ2 is upper semicontinuous. This completes the proof of the theorem.Remark 10.
The above theorem extends the results of Khan and Liu [25, Theorem 3.1 and Corollary 3.3] to PM-spaces.Theorem 11.
LetX, P, Q, and R be as in Theorem 9. Further, let P, Q, and R satisfying (5) and (6) have a coincidence point w; then Rw is a unique common fixed point of P, Q, and R.Proof.
We havePw=Qw=Rw. By commutativity of R with P and Q, PRw=RPw=RRw and QRw=RQw=RRw, or PRw=RRw=QRw.
Now letR2w≠Rw; then by (5) and (6), we have(10)ϕ1R2w,Rw=ϕ1PRw,Qw>min⁡⁡ϕ2R2w,PRwϕ1Rw,Qwϕ2R2w,Rwϕ2R2w,Rw,ϕ2R2w,PRw,lllllllllllllllllϕ1Rw,Qw,ϕ2R2w,PRwϕ1Rw,Qwϕ2R2w,Rw=ϕ2R2w,PRw=ϕ2QRw,Pw>min⁡⁡ϕ1R2w,QRwϕ2Rw,Pwϕ1R2w,Rwϕ1R2w,Rw,ϕ1R2w,QRw,lllllllllllllllllϕ2Rw,Pw,ϕ1R2w,QRwϕ2Rw,Pwϕ1R2w,Rw=ϕ1(R2w,Rw),which is a contradiction. Hence, R2w=Rw. Thus, Rw is a fixed point of R. Thus, Rw=RRw=PRw=QRw. Therefore, Rw is a common fixed point of P, Q, and R.
The uniqueness ofRw as a common fixed point of P, Q, and R follows from (5) and (6).Theorem 12.
Letf and g be commuting, continuous, and nearly densifying self-mappings on a complete Menger space X satisfying(11)ϕgu,gv>min⁡⁡ϕfu,fv,ϕfu,gu,ϕfv,gvfor fu≠fv, gu≠gv, and u,v∈X, where ϕ:X×X→ζ is u.s.c. and ϕu,u,=1, u∈X. If, for some u0 in X, Gu0=figju0:i=0,1,2,…;j=0,1,2,… is bounded, then f and g have a unique common fixed point.Proof.
LetA=G(u0). Since f and g are commuting and continuous, we have fA-⊆A- and gA-⊆A- and then A=u0∪fA∪gA.
IfαA<H, then(12)αA=αu0∪fA∪gA=min⁡⁡αfA,αgA>αA,which is a contradiction. It implies that A- is precompact.
Now defineB=⋂n=0∞fgnA-. Since fgnA is a decreasing sequence of nonempty compact subset of A, it follows that B is nonempty set such that fB-⊂B-, gB-⊂B-.
Suppose thatu∈B; then u∈fgn+1A- for all n. Hence, there exists xn⊆fgnA-. Since fgnA- is compact and closed for all n, f and g are continuous and nearly densifying; therefore, there exists a point p∈fgnA- for all n so that fgp=u. Hence, u∈fB and u∈gB. Thus, we have(13)fB=B=gB.
Let us define a real valued functionψ on B by ψu=ϕfu,gu. It is u.s.c. and hence attains its maximum at some point p∈B. Then there exists a w∈B such that p=fw.
Suppose that there is no pointu in X such that fu=gu; then we have by (11)(14)ψgw=ϕfgw,ggw=ϕgfw,ggw>min⁡⁡ϕf2w,fgw,ϕf2w,gfw,ϕfgw,ggw=min⁡⁡ϕf2w,fgw,ϕfgw,ggw=ϕf2w,fgw=ϕfp,gp=ψp,which is a contradiction to the selection of p. Hence, there exists a w0∈B such that fw0=gw0 or f2w0=fgw0=gfw0.
Supposef2w0≠fw0; then we have(15)ϕf2w0,fw0=ϕgfw0,gw0>min⁡⁡ϕf2w0,fw0,ϕf2w0,gfw0,ϕfw0,gw0=ϕf2w0,fw0,which is a contradiction. Hence, f2w0=gfw0=fw0. Therefore, fw0 is common fixed point of f and g. Now we will prove the uniqueness of fw0. Let w be the other fixed point of f and g; then, by (11), we have(16)ϕw,fw0=ϕgw,fgw0=ϕgw,gfw0>min⁡⁡ϕfw,f2w0,ϕfw,gw,ϕf2w0,gfw0=ϕfw,f2w0=ϕw,fw0,acontradiction.Hence, fw0 is unique. This completes the proof of the theorem.Remark 13.
Theorems9, 11, and 12 improve the result of Chamola et al. [19], Dimri and Pant [20], Ganguly et al. [24], and Pant et al. [21] under more natural conditions.



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