Four mappings and generalized contractions

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R E S E A R C H

Open Access

Four mappings and generalized

contractions

Lech Pasicki

*

*_{Correspondence:}

[email protected]

Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Kraków, 30-059, Poland

Abstract

In the paper general theorems on common ﬁxed point for four mappings are presented. The results are compact and they extend and unify the respective part of the ﬁxed point theory.

MSC: 47H10; 54H25

Keywords: generalized contraction; dislocated metric; weakly compatible mappings; ﬁxed point

1 Introduction

The present paper was inspired by the advanced and sophisticated article of Liuet al.[]. Our assumptions are weaker, as the comparison function is much more general, and we do not assume the spaces under consideration to be metric. In addition the general con-traction condition is compact and abstract. Also the proofs are relatively simple.

2 Theorems

Let us recall (see []) thatis the family of all mappingsϕ: [,∞)→[,∞) such that

ϕ(α) <α, α> , and0 consists of mappingsϕ∈such thatϕ() = . In turn,Pis the family of all mappingsϕ: [,∞)→[,∞) for which every sequence (an)n∈Nsuch that an+≤ϕ(an),n∈N, converges to zero. It is well known ([], Proposition ) thatP⊂. In turn,P(see []) consists of all mappings of for which every sequence (an)n∈N such that  <an+≤ϕ(an),n∈Nconverges to zero. It is known ([], Corollary .) that

Pconsists of all mappingsϕ∈satisfying

for eachα> ,ϕ(·)≤αon some interval (α,α+). ()

Clearly,P⊂Pholds and consequently, all members ofPsatisfy ().

Lemma . If aϕ∈satisﬁes(),thenϕ∈P.

Proof Let (an)n∈Nbe a sequence such thatan+≤ϕ(an),n∈Nfor aϕ∈. Then we have

an+≤ϕ(an)≤an, n∈N.

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Therefore, (an)n∈Nis nonincreasing and it converges, say toα. Supposeα> . Then from

() it follows that there exists an interval (α,α+) on whichϕ(·)≤α. For largenallan belong to this interval, as

 <α≤ϕ(an+)≤an+≤an,

andϕ∈yieldα≤ϕ(an+) <an+. Now, we haveα<an+≤ϕ(an)≤α, a contradiction. Consequently,α= ,i.e.ϕ∈P.

Corollary . Pconsists of all mappingsϕ∈satisfying().

The notion of dislocated metric space presented below is due to Hitzler and Seda []. LetXbe a nonempty set, andp:X×X→[,∞) a mapping satisfying

p(x,y) =  yieldsx=y, x,y∈X, (a)

p(x,y) =p(y,x), x,y∈X, (b)

p(x,z)≤p(x,y) +p(y,z), x,y,z∈X. (c)

Thenpis called adislocated metric(brieﬂy ad-metric), and (X,p) is called adislocated metric space(brieﬂy ad-metric space). The topology of (X,p) is generated by balls.

Many authors applied sophisticated contraction conditions. To present a general idea let us consider a mappingh:X_→_[,_∞_{) satisfying the following requirements for each}

α> :

(a=dorb=c) yieldsh(a,b,c,d)≤maxp(a,b),p(c,a),p(d,b), (a)

p(a,b)→αandp(c,a),p(d,b)→yieldslim suph(a,b,c,d)≤α, (b) ifp(d,b)≥αandp(a,b),p(c,a) are small, thenh(a,b,c,d)≤p(d,b), (c) ifp(c,a)≥αandp(a,b),p(d,b) are small, thenh(a,b,c,d)≤p(c,a), (d)

h(a,b,a,b)≤p(a,b). (e)

In order to present an example let us recall the notion of partial metric due to Matthews ([], Deﬁnition .).

Apartial metricis a mappingp:X×X→[,∞) such that

x=yiﬀp(x,x) =p(x,y) =p(y,y), x,y∈X, (a)

p(x,x)≤p(x,y), x,y∈X, (b)

p(x,y) =p(y,x), x,y∈X, (c)

p(x,z)≤p(x,y) +p(y,z) –p(y,y), x,y,z∈X. (d)

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Example . Letpbe a partial metric onXand let

h(a,b,c,d) =p(a,d) +p(b,c)/. ()

Then fora=dwe have

p(a,d) +p(b,c)/≤p(a,d) +p(b,d) +p(d,c) –p(d,d)/ =p(b,d) +p(a,c)/≤maxp(c,a),p(d,b). Similarly, forb=cwe obtain

p(a,d) +p(b,c)/≤p(a,b) +p(b,d) –p(b,b) +p(b,c)/ =p(a,b) +p(b,d)/≤maxp(a,b),p(d,b). Consequently, (a) holds. From

p(a,d) +p(b,c)/

≤p(a,b) +p(b,d) –p(b,b) +p(b,a) +p(a,c) –p(a,a)/ =p(a,b) +p(d,b) –p(b,b) +p(c,a) –p(a,a)/

we obtain (b), (c), (d), and (e).

The notion of a -complete d-metric space (or a set) was presented in [], Deﬁnition . (condition (.)). Let us note that iflimn→∞p(y,xn) =limn→∞p(x,xn) =  holds, then from p(x,y)≤p(x,xn) +p(y,xn) it follows thatx=y. In addition,p(x,x)≤p(x,xn) means that p(x,x) = . Therefore, condition (.) of [] is equivalent to

for each sequence (xn)n∈NinXwith lim

m,n→∞p(xn,xm) = 

there exists a uniquex∈Xfor which lim

n→∞p(x,xn) =p(x,x) = .

()

The idea of -completeness for partial metric spaces is due to Romaguera ([], Deﬁ-nition .). A partial metric space (X,p) is called -complete if any -Cauchy sequence (xn)n∈NinX(i.e.such thatlimm,n→∞p(xn,xm) = ), converges (in the topology of (X,p)) to a pointx∈Xfor whichp(x,x) = .

It is well known (e.g. []) that x ∈ limn→∞xn in a partial metric space (X,p) iﬀ

limn→∞p(x,xn) =p(x,x). In addition, as it was noticed before, each partial metric is a d-metric. Hence we obtain the following.

Corollary . Any partial metric space(X,p)is-complete iff(X,p)treated as a d-metric space is-complete,and iff ()is satisfied.

Proposition . Let(X,p)be a d-metric space and let(xn)n∈Nbe a sequence of points of X such that

p(xn+,xn+)≤ϕ

p(xn+,xn)

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holds for aϕ∈P.Then we have

lim

n→∞p(xn+,xn) =nlim→∞p(xn,xn) = . ()

Proof Let us adoptan=p(xn+,xn). Clearly,an+≤ϕ(an),n∈Nholds, andlimn→∞an= , asϕ∈P. Now,p(xn,xn)≤p(xn+,xn) = ancompletes (). In the followingf(X) is replaced byfX,f(x) is replaced byfx, etc. The precise order in max{p(a,b),p(c,a),p(d,b)}informs on the variables of the mappingh, and they are not shown in the proofs.

Lemma . Let(X,p)be a d-metric space and let f,g,i,j be self mappings in X satisfying the following conditions:

fX⊂jX, gX⊂iX and at least one of these sets is-complete, () p(fx,gy)≤ϕmaxp(ix,jy),p(fx,ix),p(gy,jy),h(ix,jy,fx,gy), ()

for a h:X→[,∞)such that(a), (b)hold,and aϕ∈P(see Corollary.).Then there exist sequences(xn)n∈N, (yn)n∈Nin X such that

xk+=gyk for xk=jyk and

xk=fyk– for xk–=iyk–, k∈N,

()

and(xn)n∈Nconverges to a point x such that p(x,x) = ,and

x= lim

n→∞xn=klim→∞fyk–=klim→∞iyk–

= lim

k→∞gyk=klim→∞jyk.

()

Proof From (), (), and (a) (forb=c) it follows that

for eachx∈Xthere exists ay∈Xsuch thatfx=jyand p(gy,fx) =p(fx,gy)≤ϕmaxp(ix,fx),p(gy,fx), and (fora=d)

for eachx∈Xthere exists ay∈Xsuch thatgx=iyand p(fy,gx)≤ϕmaxp(gx,jx),p(fy,gx).

Ife.g. p(fx,ix)≤p(gy,fx) holds, thenp(gy,fx)≤ϕ(p(gy,fx)) means thatp(gy,fx) = , asϕ∈ . Consequently, () and () yield

for eachx∈Xthere exists ay∈Xsuch that fx=jyandp(gy,fx)≤ϕp(fx,ix)

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and

for eachx∈Xthere exists ay∈Xsuch that gx=iyandp(fy,gx)≤ϕp(gx,jx).

()

For anx∈Xlet us adoptx=gx=iy,x=fy, wherep(x,x)≤ϕ(p(x,jx)) (see ()). Now, we deﬁnex=gyforysuch thatx=fy=jyandp(x,x)≤ϕ(p(x,x)) (see ()). In turn x=fy fory such thatx=gy=iy andp(x,x)≤ϕ(p(x,x)) (see ()). By induction we deﬁne two sequences (xn)n∈N, (yn)n∈Nsatisfying () and

p(xn+,xn+)≤ϕ

p(xn+,xn)

, n∈N.

In view of Proposition . we havelimn→∞p(xn+,xn) = .

Suppose that there exists an inﬁnite setK⊂Nsuch that for eachk∈Kthere exists an n∈Nfor which  <α<p(xk,xn++k) holds. Letn=n(k) >  be the smallest numbers satisfying this inequality fork∈K. We have (see (c))

α–p(xk,xk–) –p(xn+k,xn++k)

<p(xk,xn++k) –p(xk,xk–) –p(xn+k,xn++k)

≤p(xk–,xn+k)

≤p(xk–,xk) +p(xk,xn–+k) +p(xn–+k,xn+k)

≤p(xk–,xk) +α+p(xn–+k,xn+k)

forn=n(k), and therefore (see ()),

lim

k∈Kp(xk–,xn+k) =α.

Now, (), (), (), and the above equality yield

p(xk,xn++k) =p(fyk–,gyn+k)

≤ϕmaxp(iyk–,jyn+k),p(fyk–,iyk–),p(gyn+k,jyn+k),h(·)

=ϕmaxp(xk–,xn+k),p(xk,xk–),p(xn++k,xn+k),h(·)

=ϕmaxp(xk–,xn+k),h(·)

for largek. In turn, (b) yields

lim sup

k∈K

h(·)≤lim

k∈Kp(xk–,xn+k) =α,

and hence we obtainmax{p(xk–,xn+k),h(·)}<α+for largek. Fromϕ(β) <αforβ≤α, and () we get

α<p(xk,xn++k)≤ϕ

maxp(xk–,xn+k),h(·)

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for largek, a contradiction. Therefore

lim

k,n→∞p(xk,xn++k) = 

holds, and () with the triangle inequality (c) yieldlimm,n→∞p(xn,xm) = . Consequently (see ()), there exists anx∈Xsuch thatlimn→∞p(x,xn) =p(x,x) = , and () holds. Let us recall (see []) that a pair (f,i) of mappingsf,i:X→Xis calledweakly compatible if

for eachx∈X,fx=ixyieldsﬁx=ifx.

Now, we are ready to prove our theorem.

Theorem . Let(X,p)be a d-metric space such that(b)holds.Assume that f,g,i,j are self mappings in X satisfying

(f,i),(g,j) are weakly compatible, ()

(),and()for aϕ∈P(see Corollary.)and a mapping h:X→[,∞)for which the system of conditions(a)-(e)holds.Then f,g,i,j have a unique common ﬁxed point,say x,and p(x,x) = .

Proof Let (xn)n∈N, (yn)n∈N, and x be as in Lemma .. Assumee.g.that x∈jX (for -completejX orfX). Then there exists avsuch thatx=jv. Let us prove thatgv=x. We have

p(gv,x) =p(x,gv)≤p(x,xk) +p(xk,gv).

Supposep(gv,x) > . Then we obtain

p(xk,gv) =p(fyk–,gv)

≤ϕmaxp(iyk–,jv),p(fyk–,iyk–),p(gv,jv),h(·)

=ϕmaxp(xk–,x),p(xk,xk–),p(gv,x),h(·)

=ϕp(gv,x),

for largek(see (), (c). Hence,

p(gv,x)≤ lim

k→∞p(x,xk) +ϕ

p(gv,x)

yieldsp(gv,x) =  (ϕ∈), andgv=x. Thus we havex=gv=jv.

FromgX⊂iXit follows that there exists awsuch thatx=iw. Let us show thatfw=x. We have

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Supposep(fw,x) > . Then we have

p(fw,gyk)≤ϕ

maxp(iw,jyk),p(fw,iw),p(gyk,jyk),h(·)

=ϕmaxp(x,xk),p(fw,x),p(xk+,xk)

=ϕp(fw,x)

for largek(see ()), (d)). Hence, we obtain

p(fw,x)≤ϕp(fw,x)+ lim

k→∞p(xk+,x) =ϕ

p(fw,x),

andp(fw,x) = ,i.e. x=fw=iw. Now, () yields

jx=jgv=gjv=gx and ix=ifw=ﬁw=fx.

Let us show thatfx=gx. We have (see (), (e), (b))

p(fx,gx)≤ϕmaxp(ix,jx),p(fx,ix),p(gx,jx),h(·)

=ϕmaxp(fx,gx),p(fx,fx),p(gx,gx)=ϕp(fx,gx),

and thereforep(fx,gx) = ,i.e. fx=gx(let us note thatp(fx,fx) =  not necessarily holds if pis ad-metric).

Now, it is clear thatfx=gx=ix=jxholds.

Let us prove thatgx=x. We have (see (), (e), (b))

p(x,gx) =p(fw,gx)≤ϕmaxp(iw,jx),p(fw,iw),p(gx,jx),h(·) =ϕmaxp(x,gx),p(x,x),p(gx,gx)=ϕp(x,gx),

and consequently,p(x,gx) = ,i.e. gx=x. We have proved thatx=fx=gx=ix=jx.

Ifyis a common ﬁxed point of our mappings, then (see (), (e), (b))

p(x,y) =p(fx,gy)≤ϕmaxp(ix,jy),p(fx,ix),p(gy,jy),h(·)=ϕp(x,y)

yieldsp(x,y) = ,i.e. x=y.

The authors of [] consider (in metric spaces) formulas M, M, andM instead of ‘max. . .’ from our condition (). It would be rather exhausting to citeMorM, never-theless all three formulas are applied in conditions that are particular cases of () forh satisfying (a)-(e).

The next theorem is a consequence of Corollary ., and Theorem . (see also Exam-ple .).

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The above theorem is a far extension of [], Theorem . in its part concerning a unique common ﬁxed point; our mappingϕneed not be continuous (see also Example .).

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The work has been supported by the Polish Ministry of Science and Higher Education.

Received: 6 August 2016 Accepted: 10 November 2016 References

1. Liu, Z, Zhang, X, Ume, JS, Kang, SM: Common ﬁxed point theorems for four mappings satisfyingψ-weakly contractive conditions. Fixed Point Theory Appl.,201520 (2015)

2. Pasicki, L: Fixed point theorems for contracting mappings in partial metric spaces. Fixed Point Theory Appl.2014, 185 (2014)

3. Pasicki, L: Dislocated metric and ﬁxed point theorems. Fixed Point Theory Appl.201582 (2015) 4. Pasicki, L: The Boyd-Wong idea extended. Fixed Point Theory Appl.2016, 63 (2016)

5. Hitzler, P, Seda, AK: Dislocated topologies. J. Electr. Eng.51(12), 3-7 (2000) 6. Matthews, SG: Partial metric topology. Ann. N.Y. Acad. Sci.728, 183-197 (1994)

7. Romaguera, S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl.2010, Article ID 493298 (2010)

8. Jungck, G, Rhoades, BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math.29, 227-238 (1998)