Fixed points for occasionally weakly compatible maps

Fixed points for occasionally weakly compatible maps

Hakima Bouhadjera

Abstract

In this article, using occasionally weak compatibility due to Al-Thagafi and Shahzad [1], we generalize some common fixed point theorems of Greguˇs contraction type in a normed space.

2000 Mathematics Subject Classification: 47H10, 54H25.

Key words and phrases: Weakly compatible maps, Occasionally weakly compatible maps, Normed space, Common fixed point, Greguˇs contraction

type.

1 Introduction

Recently, Jungck [5] introduced the notion of weakly compatible maps as fol- lows:

Definition 1 Self-maps f and g of a metric space (X, d) are called weakly compatible if f t=gt for some t∈ X implies thatf gt=gf t.

More recently, Al-Thagafi and Shahzad [1] weakened the weak compatibil- ity by giving the so-called occasionally weak compatibility.

1Received 3 February, 2009

Accepted for publication (in revised form) 6 March, 2009

13

Definition 2 Self-mapsf andg of a set X are said to be occasionally weakly compatible if and only if there exists a point t ∈ X such that f t = gt and f gt=gf t.

In their paper [3], Djoudi and Nisse proved a common fixed point theorem of Greguˇs contraction type in a Banach space by using the weak compatibility.

Theorem 1 Let f, g, h and k be maps from a Banach space X into itself having the conditions

(1.1) f(X)⊂k(X) and g(X)⊂h(X), (1.2) the inequality

kf x−gyk^p ≤ ϕ(akhx−kyk^p+ (1−a) max{αkf x−hxk^p, βkgy−kyk^p,kf x−hxk^p2kf x−kyk^p2, kf x−kyk^p2kgy−hxk^p2,

1

2(kf x−hxk^p+kgy−kyk^p)});

for all x, y∈ X, where 0< a≤1, 0< α, β ≤1, p≥1 andϕ:R⁺→R⁺ such that ϕis upper semi-continuous, nondecreasing and ϕ(t)< t for anyt >0, (1.3) one of f(X) or g(X) is closed.

If the pairs {f, h} and {g, k} are weakly compatible, then f, g, h and k have a unique common fixed point in X.

In this work, we give some results which include the analogue of certain results in [2], [3], [4], [6], [7], [8] and references therein.

2 Main Results

Theorem 2 Let f, g, h, k be maps from a normed space (X,k.k) having inequality (1.2) for all x, y ∈ X, where 0 < a ≤ 1, α, β > 0, p ≥ 1 and ϕ :R⁺ → R⁺ such that ϕ(t) < t for any t > 0. If pairs of maps {f, h} and {g, k} are occasionally weakly compatible. Then f, g, h and k have a unique common fixed point.

Proof. Since pairs of maps {f, h} and {g, k} are occasionally weakly com- patible, then, there exist two elements u and v in X such that f u =hu and f hu=hf u;gv=kv andgkv =kgv.

First step: we prove that f u=gv. Suppose that kf u−gvk>0. Then, by using inequality (1.2) we get

kf u−gvk^p ≤ ϕ(akhu−kvk^p+ (1−a) max{αkf u−huk^p, βkgv−kvk^p,kf u−huk^p2kf u−kvk^p2, kf u−kvk^p2kgv−huk^p2,

1

2(kf u−huk^p+kgv−kvk^p)});

i.e.,

kf u−gvk^p ≤ ϕ(akf u−gvk^p+ (1−a)kf u−gvk^p)

= ϕ(kf u−gvk^p)

< kf u−gvk^p

which is a contradiction. Thus, we havehu=f u=gv=kv.

Second step: we claim thatf f u=f u=hf u. If not, then, kf²u−f uk>0 and the use of inequality (1.2) gives

kf²u−f uk^p = kf f u−gvk^p

≤ ϕ(akhf u−kvk^p+ (1−a) max{αkf f u−hf uk^p, βkgv−kvk^p,kf f u−hf uk^p2kf f u−kvk^p2, kf f u−kvk^p2kgv−hf uk^p2,

1

2(kf f u−hf uk^p+kgv−kvk^p)});

that is,

kf f u−f uk^p ≤ ϕ(kf f u−f uk^p)

< kf f u−f uk^p

this contradiction implies thatf f u=f u=hf u. Similarly, we can prove that ggv=gv =kgv. Putf u=hu=gv=kv=t, we conclude thattis a common fixed point of maps f,g,h andk.

Third step: Suppose that there is another common fixed point of mapsf,g, h and kcalled z, then, kt−zk>0. By inequality (1.2) we obtain

kt−zk^p = kf t−gzk^p

≤ ϕ(akht−kzk^p+ (1−a) max{αkf t−htk^p, βkgz−kzk^p,kf t−htk^p2kf t−kzk^p2, kf t−kzk^p2kgz−htk^p2,

1

2(kf t−htk^p+kgz−kzk^p)})

= ϕ(kt−zk^p)

< kt−zk^p.

The above contradiction demands that z=t.

Corollary 1 Let f, g, h, k be as in Theorem 2. Suppose that these maps satisfy instead of inequality (1.2) the next one

kf x−gyk^p ≤ ϕ(akhx−kyk^p+ (1−a) max{αkf x−hxk^p, βkgy−kyk^p,kf x−hxk¹²kf x−kyk¹², kf x−kyk¹²kgy−hxk¹²,

1

2(kf x−hxk+kgy−kyk)}^p);

for all x, y ∈ X, where ϕ, a, α, β and p are as in Theorem 2, then, the four maps have a unique common fixed point.

Corollary 2 If we replace inequality (1.2) in Theorem 2 with the following one

kf x−gyk^p≤ϕ(akhx−kyk^p+ (1−a)kf x−kyk^p2kgy−hxk^p2);

for all x, y ∈ X, where ϕ, a and p are as in Theorem 2, then, f, g, h and k have a unique common fixed point.

We finish our work by giving the next result.

Theorem 3 Let {fi}, i = 1,2, . . ., h and k be self-maps of a normed space (X,k.k) such that

(i) pairs of maps{f1, h}and{fn, k},n >1are occasionally weakly compatible,

(ii) the inequality

kf1x−fnyk^p ≤ϕ(akhx−kyk^p+ (1−a) max{αkf1x−kyk^p, βkfny−hxk^p}) holds for allx, y∈ X, whereα, β, ϕ, pare as in Theorem 2,0< a <1provided that a+ (1−a) max{α, β}<1, then, all fi, h and k have a unique common fixed point.

Proof. Since pairs {f1, h} and {fn, k}, n = 2,3, . . . are occasionally weakly compatible, then, as in proof of Theorem 2, there are two elementsu andv in X such thatf1u=huand f1hu=hf1u;fnv=kv andfnkv=kfnv.

First, we prove that f1u = fnv. Indeed, let f1u 6= fnv, then, inequality (ii) gives

kf1u−fnvk^p ≤ ϕ(akhu−kvk^p+ (1−a) max{αkf1u−kvk^p, βkfnv−huk^p})

= ϕ([a+ (1−a) max{α, β}]kf1u−fnvk^p)

< [a+ (1−a) max{α, β}]kf1u−fnvk^p

< kf1u−fnvk^p

which is a contradiction. Hence, we have f1u=fnv=hu=kv.

Now, if f_n²v6=fnv, then, by condition (ii) we have kfnv−f_n²vk^p = kf1u−fnfnvk^p

≤ ϕ(akhu−kfnvk^p+(1−a) max{αkf1u−kfnvk^p, βkfnfnv−huk^p})

= ϕ([a+ (1−a) max{α, β}]kfnv−f_n²vk^p)

< [a+ (1−a) max{α, β}]kfnv−f_n²vk^p

< kfnv−f_n²vk^p

a contradiction. Thus, fnfnv = fnv =kfnv. Similarly, f1f1u =f1u =hf1u.

Put hu = f1u = fnv = kv = t, then, t is a common fixed point of maps {fi}i≥1,h andk.

The uniqueness of the common fixed point follows immediately from inequality (ii).

Remark 1 In this paper, we proved a unique common fixed point of several maps in normed spaces by using the weaker condition of compatibility called occasionally weak compatibility due to Al-Thagafi and Shahzad without calling inclusions between images of maps. Hence, our result is more general than results in [3] and references therein and say that the weak compatibility is the least condition of maps to have common fixed points is not true.

References

[1] M.A. Al-Thagafi, N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sin. (Engl. Ser.) 24, 2008, no. 5, 867-876.

[2] M.L. Diviccaro, B. Fisher, S. Sessa, A common fixed point theorem of Greguˇs type, Publ. Math. Debrecen 34, 1987, no. 1-2, 83-89.

[3] A. Djoudi, L. Nisse, Greguˇs type fixed points for weakly compatible map- pings, Bull. Belg. Math. Soc. Simon Stevin 10, 2003, no. 3, 369-378.

[4] B. Fisher, S. Sessa, On a fixed point theorem of Greguˇs, Int. J. Math.

Math. Sci. 9, 1986, no. 1, 23-28.

[5] G. Jungck,Common fixed points for noncontinuous nonself maps on non- metric spaces, Far East J. Math. Sci. 4, 1996, no. 2, 199-215.

[6] P.P. Murthy, Y.J. Cho, B. Fisher, Common fixed points of Greguˇs type mappings, Glas. Mat. Ser. III 30(50), 1995, no. 2, 335-341.

[7] H.K. Pathak, Y.J. Cho, S.M. Kang, B. Madharia,Compatible mappings of type (C) and common fixed point theorems of Greguˇs type, Demonstratio Math. 31, 1998, no. 3, 499-518.

[8] H.K. Pathak, M.S. Khan, Compatible mappings of type(B) and common fixed point theorems of Greguˇs type, Czechoslovak Math. J. 45(120), 1995, no. 4, 685-698.

H. Bouhadjera

Laboratoire de Math´ematiques Appliqu´ees

Universit´e Badji Mokhtar B. P. 12, 23000, Annaba Alg´erie e-mail: b [email protected]