COMMON FIXED POINT THEOREM FOR FOUR MAPPINGS DEFINED ON MENGER PM-SPACES

Vol. 43, No. 2, 2013, 39-49

COMMON FIXED POINT THEOREM FOR FOUR MAPPINGS DEFINED ON MENGER PM-SPACES

WITH NONLINEAR CONTRACTIVE TYPE CONDITION

Nataˇsa A. Babaˇcev¹

Abstract. This paper presents a common fixed point theorem for two pairs of self-mappings of which one is compatible and the other weakly compatible, defined on Menger PM-spaces, satisfying nonlinear general- ized contractive type condition involving Φ-functions.

AMS Mathematics Subject Classification(2010): 47H10, 54E70

Key words and phrases:Probabilistic metric spaces; Common fixed point;

Compatible mappings; Nonlinear contractive conditions; Φ-functions

1. Introduction

The notion of statistical metric spaces, as a generalization of metric spaces, was introduced by K. Menger [11] in 1942. Schweizer and Sklar [16] studied the properties of spaces introduced by K. Menger and gave some basic results on these spaces. They studied topology, convergence of sequences, continuity of mappings, defined the completeness of these spaces, etc. Following A. N.

Serstnev [20], H. Sherwood gave a notion of probabilistic metric spaces [18]ˇ and also proved a theorem of a characterization of nested, closed sequence of nonempty sets in complete probabilistic metric space.

Fixed point and common fixed point properties for mappings defined on probabilistic spaces were studied by many authors ([1], [17], [5], [19], [14], [8]).

Most of the properties which provide the existence of fixed point and common fixed point are of linear contractive type conditions. On the other hand, there are many generalizations ([15], [19]) of commutativity for the functions defined on spaces with non-deterministic distances (probabilistic metric spaces, fuzzy metric spaces, etc.) which have an important role in the statements providing the existence of a common fixed point.

The results in fixed point theory including nonlinear type contractive con- ditions were given by D.W. Boyd and J.S.W. Wong [2], S.N. Jeˇsi´c and N.A.

Babaˇcev [6], D. O’Regan and R. Sadaati [14] and recently by S.N. Jeˇsi´c et al.

Altering distance functions in Menger PM-spaces have been recently con- sidered by B.S. Choudhury and K. Das [3]. Some fixed point results involving altering distances in Menger PM-spaces were given by D. Mihet¸ in [12].

The purpose of this paper is to prove a common fixed point theorem for four mappings satisfying nonlinear contractive type condition involving altering distances in Menger PM-spaces.

1Department of Mathematics, Faculty of Electrical Engineering, Bulevar Kralja Aleksan- dra 73, P.O. Box 35-54, 11120 Beograd, Serbia, e-mail: [email protected]

2. Preliminaries

In the standard notation, let D+ be the set of all distribution functions F :R→[0,1],such thatF is a nondecreasing, left-continuous mapping, which satisfiesF(0) = 0 and sup_x∈RF(x) = 1. The spaceD+ is partially ordered by the usual point-wise ordering of functions, i.e.,F ≤Gif and only ifF(t)≤G(t) for all t ∈ R. The maximal element for D+ in this order is the distribution function given by

ε0(t) =

{ 0, t≤0, 1, t >0.

Definition 2.1. ([16]) A binary operationT : [0,1]×[0,1]→[0,1] is a contin- uoust-norm ifT satisfies the following conditions:

(a) T is commutative and associative;

(b)T is continuous;

(c) T(a,1) =afor alla∈[0,1];

(d)T(a, b)≤T(c, d) whenever a≤c andb≤d,anda, b, c, d∈[0,1].

Examples oft-norm areT(a, b) = min{a, b} andT(a, b) =ab.

Thet-norms are defined recursevly byT¹=T and

Tⁿ(x1, . . . , xn+1) =T(Tⁿ⁻¹(x1, . . . , xn), xn+1).

forn≥2 andxi∈[0,1] for all i∈ {1, . . . , n+ 1}.

Definition 2.2. A Menger probabilistic metric space (briefly, Menger PM- space) is a triple (X,F, T) where X is a nonempty set, T is a continuous t-norm, and F is a mapping from X×X into D⁺ such that, ifFx,y denotes the value ofF at the pair (x, y),the following conditions hold:

(PM1)Fx,y(t) =ε0(t) if and only ifx=y;

(PM2)Fx,y(t) =Fy,x(t);

(PM3)Fx,z(t+s)≥T(Fx,y(t), Fy,z(s)) for allx, y, z∈X and s, t≥0.

Remark 2.3. [17] Every metric space is a PM-space. Let (X, d) be a metric space and T(a, b) = min{a, b} is a continuoust-norm. DefineF_x,y(t) =ε₀(t− d(x, y)) for allx, y∈X andt >0.The triple (X,F, T) is a PM-space induced by the metricd.

Definition 2.4. Let (X,F, T) be a Menger PM-space.

(1) A sequence {xn}n in X is said to be convergent to x in X if, for every ε >0 andλ > 0 there exists a positive integerN such thatFx_n,x(ε)>1−λ whenevern≥N.

(2) A sequence {xn}n in X is called Cauchy sequence if, for everyε >0 and λ >0 there a exists positive integerN such that Fx_n,x_m(ε)>1−λwhenever n, m≥N.

(3) A Menger PM-space is said to be complete if every Cauchy sequence inX is convergent to a point inX.

The (ε, λ)-topology ([16]) in the Menger PM-space (X,F, T) is introduced by the family of neighbourhoods Nxof a pointx∈X given by

Nx={Nx(ε, λ) : ε >0, λ∈(0,1)} where

Nx(ε, λ) ={y∈X: Fx,y(ε)>1−λ}.

The (ε, λ)-topology is a Hausdorff topology. In this topology the function f is continuous in x0 ∈X if and only if for every sequencexn →x0 it holds that f(xn)→f(x0).

The following lemma is proved by B. Schweizer and A. Sklar.

Lemma 2.5. [16]Let (X,F, T)be a Menger PM-space. Then the functionF is lower semi-continuous for every fixed t >0, i.e. for every fixed t >0 and every two convergent sequences {xn},{yn} ⊆X such that xn → x, yn →y it follows that

lim inf

n→∞ Fx_n,y_n(t) =Fx,y(t).

Definition 2.6. Let (X,F, T) be a Menger PM-space andA⊆X.Closure of the setA is the smallest closed set containingA,denoted byA.

Obviously, having in mind the Hausdorff topology and the definition of converging sequences we have that the next remark holds.

Remark 2.7. x∈Aif and only if there exists a sequence{xn} in Asuch that xn→x.

Definition 2.8. [4] Let (X,F, T) be a Menger PM-space and A ⊆ X. The probabilistic diameter of set Ais given by

δ_A(t) = inf

x,y∈Asup

ε< t

F_x,y(ε).

The diameter of the setA is defined by δA= sup

t>0

inf

x,y∈Asup

ε< t

Fx,y(ε).

If there exists λ∈(0,1) such that δ_A = 1−λ, the setAwill be called proba- bilistic semi-bounded. Ifδ_A= 1, the setAwill be called probabilistic bounded.

Lemma 2.9. Let (X,F, T) be a Menger PM-space. A set A ⊆ X is proba- bilistic bounded if and only if for each λ ∈(0,1) there exists t >0 such that F_x,y(t)>1−λfor allx, y∈A.

Proof. The proof follows from the definitions of supAand infAof non-empty sets.

It is not difficult to see that every metrically bounded set is also probabilistic bounded if it is considered in an induced PM-space.

H. Sherwood has proved the following theorem.

Theorem 2.10. [18] Let(X,F, T)be a Menger PM-space and {Fn} a nested sequence of nonempty, closed subsets of X such that δFn → ε0 as n → ∞. Then, there is exactly one point x₀∈F_n,for every n∈N.

It is easy to show that the following lemma is satisfied.

Lemma 2.11. Let(X,F, T)be a Menger PM-space. A collection{Fn}n∈Nhas probabilistic diameter zero i.e. for each r∈(0,1) and each t > 0 there exists n0∈N such thatFx,y(t)>1−r for all x, y∈Fn₀ if and only ifδF_n →ε0 as n→ ∞.

Lemma 2.12. Let(X,F, T)be a Menger PM-space with the continuoust-norm T which satisfies T(a, a)≥a for everya∈[0,1]. Then, for everyx, y, z ∈X and allt >0 holds

(1) Fx,y(2t)≥min{Fx,z(t), Fy,z(t)}.

Proof. For every t-norm T which satisfies T(a, a) ≥ a, for every a, b ∈ [0,1]

it holds T(a, b) ≥ T(min{a, b},min{a, b}) ≥ min{a, b}. From the previous, property (PM3) and the fact that T is nondecreasing we have that for every x, y, z∈X and allt >0 holdsF_x,y(2t)≥min{F_x,z(t), F_y,z(t)}.

Khan et al. in [10] introduced the concept of altering distance functions that alter the distance between two points in metric spaces. Recently, B.S.

Choudhury and K. Das extended this concept to the probabilistic fixed point theory in [3], and proved a fixed point theorem for the t-norm T = min. D.

Mihet¸ proved some fixed point results that generalize the results given in [3], considering continuoust-norm.

Definition 2.13. [3] A functionϕ: [0,∞)→[0,∞) is said to be a Φ-function if the following conditions hold:

(i)ϕ(t) = 0 if and only if t= 0 ;

(ii)ϕis strictly increasing andϕ(t)→ ∞as t→ ∞; (iii) ϕis left-continuous in (0,∞);

(iv)ϕis continuous at 0.

The class of allϕ-functions will be denoted by Φ.

Lemma 2.14. Let (X,F, T)be a Menger PM-space. Let ϕ: [0,∞)→[0,∞) be aΦ-function. Then, the following statement holds.

If for x, y∈X,0< c <1, we have thatFx,y(t)≥Fx,y(ϕ(s/c))for allt >0 and somes >0 such that t > ϕ(s)>0,then x=y.

Proof. Sinceϕ(0) = 0 andϕis continuous in 0, there existss >0 such thatt >

ϕ(s)>0.From the fact thatϕis strictly increasing, and sincec∈(0,1), by in- duction we get thatFx,y(t)≥Fx,y(ϕ(s))≥Fx,y(ϕ(s/c))≥ · · · ≥Fx,y(ϕ(s/cⁿ)).

Taking lim inf asn→ ∞we getFx,y(t)≥1, i.e. x=y.

In fixed point theory, a very important role is played by the generalizations of commutativity. The concept of compatible mappings was introduced by

G. Jungck ([9]) and S.N. Mishra ([13]). There are many generalizations of compatibility in different senses. Recently, B. Singh et al. introduced the concept of weak compatibility in [19].

Definition 2.15. [13] Let (X,F, T) be a Menger PM-space and S and R self-mappings on X.We say that the mappingsS and Rare compatible if

(2) lim inf

n→∞ FSRx_n,RSx_n(t) = 1 for every t >0, holds whenever (x_n)_n∈Nis a sequence inX such that lim

n→∞Sx_n= lim

n→∞Rx_n= z∈X holds.

Definition 2.16. [19] Let (X,F, T) be a Menger PM-space andSandRself- mappings on X.We say that the mappingsS and Rare weakly compatible if for some z∈X holds thatSz=Rz thenSRz=RSz.

It is easy to see that the class of compatible mappings is broader than the class of commuting mappings. Indeed, every pair of commuting mappings is also compatible, while the converse is not true ([19]). Also, every pair of compatible mappings is weakly compatible, as the following remark shows.

Remark 2.17. LetS and R be compatible mappings on a Menger PM-space (X,F, T). Then, the following holds:

If for somez∈X we have Sz=Rz thenSRz =RSz.

Proof. This follows directly from Definition 2.15 takingx_n =zfor everyn∈N for some point z∈X.

Examples of compatible and weak compatible mappings can be found in [9], [13] and [19].

3. Main results

Lemma 3.1. Let(X,F, T) be a Menger PM-space with continuoust-norm T which satisfies T(a, a) ≥a for every a ∈ [0,1] and S and R compatible self- mappings on X and let Sxn and Rxn converge to some point z ∈ X for a sequence {xn}n∈N inX.If S is continuous, then lim

n→∞RSxn=Sz.

Proof. Let λ∈(0,1) and t >0 be arbitrary. Since S and R are compatible, it follows that F_RSxn,SRxn(t)>1−λ forn∈N large enough. Also,Sx_n and Rx_n converge toz, soF_Rxn,z(t)>1−λandF_Sxn,z(t)>1−λ.From Lemma 2.12 and the continuity ofS it follows that

F_RSxn,Sz(2t)≥min{F_RSxn,SRxn(t), F_SRxn,Sz(t)} ≥min{1−λ,1−λ}= 1−λ holds. Since λ ∈ (0,1) is arbitrary, we get that lim inf

n→∞ F_RSxn,Sz(t) = 1, i.e.

nlim→∞RSxn=Sz.

Theorem 3.2. Let(X,F, T)be a complete Menger PM-space with continuous t-norm T which satisfies T(a, a)≥afor every a∈[0,1], let c∈(0,1) be fixed and let A, B, S and R be self-mappings on X and there exists x₀ ∈ X such thatO(A, x₀) ={Aⁿx₀, n∈N∪ {0}}andO(B, x₀) ={Bⁿx₀, n∈N∪ {0}}are probabilistic bounded sets. Let the following conditions hold:

(a)A(X)⊆R(X), B(X)⊆S(X),

(b) One of the mappingsA andS is continuous,

(c) The pair{A, S} is compatible and{B, R} is weakly compatible, (d) There exists aΦ-functionϕsuch that

(3) FAx,By(ϕ(t))≥FSx,Ry(ϕ(t/c)),

for every t >0 and all x, y∈X. ThenA, B, S and R have a unique common fixed point.

Proof. From(a) it follows that, forx0 there existsx1∈X such thatA(x0) = R(x1) and for such a pointx1 there exists x2∈ X such thatB(x1) =S(x2).

By induction we can construct the following sequence{zn}n∈N

(4)

{ z2n−1=Rx2n−1=Ax2n−2

z_2n=Sx_2n =Bx_2n−1 .

Let us consider a nested sequence of non-empty, closed sets defined by Fn={zn, zn+1, . . .}, n∈N.

We now prove that the family{Fn}n∈Nhas the probabilistic diameter zero.

Let λ∈(0,1) andt > 0 be arbitrary. From Fk ⊆ O(A, x0)∪ O(B, x0), it follows thatFk is a probabilistic bounded set for arbitraryk∈N.

Let x, y ∈ Fk be arbitrary. There are sequences {z_n(i)}, {z_n(j)} in Fk

(n(i), n(j)≥n, i, j∈N) such that lim

i→∞z_n(i)=xand lim

j→∞z_n(j)=y.

Case I. Let us assume that n(i)∈ 2N−1 andn(j) ∈ 2N or vice-versa, for large enoughi, j∈Ni.e. z_n(i)=Ax_n(i)−1 andz_n(j)=Bx_n(j)−1.

Since ϕ(0) = 0 and ϕ is continuous in 0, there exists r > 0 such that t > ϕ(r)>0. From (3) and the fact thatF is nondecreasing, it follows that

F_zn(i),zn(j)(t) =F_{Axn(i)−1,Bxn(j)−1}(t)≥F_{Axn(i)−1,Bxn(j)−1}(ϕ(r))

≥F_{Sxn(i)−1,Rxn(j)−1}(ϕ(r/c)) =F_{Axn(j)−2,Bxn(i)−2}(ϕ(r/c))

=Fz_n(i)−1,z_n(j)−1(ϕ(r/c)).

By induction, form∈N, we get that

Fz_n(i),z_n(j)(t)≥Fz_n(i)−m,z_n(j)−m(ϕ(r/c^m)).

Since{z_n(i)−m},{z_n(j)−m}are sequences inFk we have that Fz_n(i)−m,z_n(j)−m(ϕ(r/c^m))≥δF_k(ϕ(r/c^m))

holds. SinceFk is probabilistic bounded andϕis a Φ-function, lettingm→ ∞ we get

Fz_n(i),z_n(j)(t)≥δF_k(ϕ(r/c^m))→ϵ0. It follows that

(5) Fz_n(i),z_n(j)(t)>1−λ, forn(i)∈2N−1, n(j)∈2N, or vice-versa.

Case II.Let us assume that bothn(i) and n(j) are from the set 2N−1 and letn(l)≥kbe an arbitrary positive integer andn(l)∈2N.

Analoguously as in Case I, by replacingtwith ₂^t, we show that F_{Axn(j)−1,Bxn(l)−1}(t/2)>1−λ and F_{Axn(i)−1,Bxn(l)−1}(t/2)>1−λ.

From Lemma 2.12 and the previous, we conclude that Fz_n(i),z_n(j)(t) =FAx_n(i)−1,Ax_n(j)−1(t)

≥min{

FAx_n(i)−1,Bx_n(l)−1(t/2), FAx_n(j)−1,Bx_n(l)−1(t/2)}

≥min{1−λ,1−λ}= 1−λ holds, i.e.

(6) Fz_n(i),z_n(j)(t)≥1−λ, forn(i), n(j)∈2N−1.

Similarly we can prove that (6) holds forn(i), n(j)∈2N. Finally, from (5) and (6) we conclude that

Fz_n(i),z_n(j)(t)≥1−λ

holds for everyi, j∈N.Taking liminf wheni, j→ ∞and applying Lemma 2.5 we get thatF_x,y(t)>1−λfor every x, y∈F_k. From Lemma 2.11 it follows that the collection{F_n}n∈Nhas probabilistic diameter zero.

Applying Theorem 2.10 we conclude that this collection has a non-empty intersection that consists of exactly one point z.Since the collection{F_n}n∈N

has probabilistic diameter zero and z ∈ F_n for every n ∈ N, then for every λ∈(0,1) and for allt >0 there existsn0∈Nsuch that for everyn≥n0 we have that Fz_n,z(t)> 1−λ. From this it follows that for every λ∈ (0,1) we have that lim inf

n→∞ Fz_n,z(t)>1−λ.Takingλ→0 we get that lim inf

n→∞ F_zn,z(t) = 1 i.e. lim

n→∞zn = z. From the definition of the sequences {Ax2n−2}, {Sx2n}, {Bx2n−1}and {R2n−1} it follows that every one of these sequences converges to z.

We shall prove thatzis a common fixed point of the mappingsA, B, Sand R.Let us first assume thatS is continuous. Then we have that lim

n→∞SSx2n=

Sz.From the compatibility of the pair{A, S} and from Lemma 3.1 it follows that lim

n→∞ASx2n=Sz.

From the properties of the functionϕit follows that there existsr >0 such thatt > ϕ(r)>0. Using the condition (3) we get that the following inequality holds:

FASx_2n,Bx_2n−1(t)≥FASx_2n,Bx_2n−1(ϕ(r))≥FSSx_2n,Rx_2n−1(ϕ(r/c)).

Taking lim inf asn→ ∞we get that

FSz,z(t)≥FSz,z(ϕ(r/c)).

From Lemma 2.14 it follows that Sz = z. Using condition (3) again, we get that

FAz,Bx_2n−1(t)≥FAz,Bx_2n−1(ϕ(r))≥FSz,Rx_2n−1(ϕ(r/c)) and taking lim inf asn→ ∞we get that

FAz,z(t)≥FSz,z(ϕ(r/c)) =Fz,z(ϕ(r/c)) = 1.

This means thatAz=z.SinceA(X)⊆R(X),there exists a pointu∈X such thatz=Az=Ruand we have that

F_z,Bu(t)≥F_z,Bu(ϕ(r)) =F_Az,Bu(ϕ(r))≥F_Sz,Ru(ϕ(r/c)) =F_z,z(ϕ(r/c)) = 1, which means thatBu=z. From the weak compatibility of the pair{B, R}it follows thatRz =RBu=BRu=Bz.Also, from (3) it follows that

F_Ax2n,Bz(t)≥F_Ax2n,Bz(ϕ(r))≥F_Sx2n,Rz(ϕ(r/c)).

Taking lim inf whenn→ ∞and from Lemma 2.14, we get thatBz=z.Thus, z is a common fixed point of the mappingsA, B, S andR.

Now, let us assume that A is a continuous mapping. Then we have that

nlim→∞AAx2n=Az.From the compatibility of the pair{A, S} and Lemma 3.1 it follows that lim

n→∞SAx_2n=Az.Using the condition (3) we get that F_{AAx2n,Bx2n−1}(t)≥F_{AAx2n,Bx2n−1}(ϕ(r))≥F_{SAx2n,Rx2n−1}(ϕ(r/c)).

Taking lim inf asn→ ∞we get that

F_Az,z(t)≥F_Az,z(ϕ(r/c)).

From Lemma 2.14 it follows that Az=z.SinceA(X)⊆R(X), there exists a pointv∈X such thatz=Az=Rv. From (3) we have that

F_AAx2n,Bv(t)≥F_AAx2n,Bv(ϕ(r))≥F_SAx2n,Rv(ϕ(r/c)).

Taking lim inf asn→ ∞we get that

FAz,Bv(t)≥FAz,Rv(ϕ(r/c)) =Fz,z(ϕ(r/c)) = 1,

which means thatz=Bv.Since the pair{B, R}is weakly compatible we have that Rz=RBv=BRv=Bz.Also, using condition (3) we have

F_Ax2n,Bz(t)≥F_Ax2n,Bz(ϕ(r))≥F_Sx2n,Rz(ϕ(r/c)).

Taking lim inf asn→ ∞we get that

Fz,Bz(t)≥Fz,Bz(ϕ(r))≥Fz,Rz(ϕ(r/c)) =Fz,Bz(ϕ(r/c)).

This means thatz=Bz=Rz.SinceB(X)⊆S(X),there exists a pointw∈X such thatz=Bz=Sw.From (3) it follows that

F_Aw,z(t)≥F_Aw,z(ϕ(r)) =F_Aw,Bzϕ(r))≥F_Sw,Rz(ϕ(r/c)) =F_z,z(ϕ(r/c)) = 1, i.e. Aw = z. Since the pair {A, S} is compatible and z = Aw = Sw, from Remark 2.17 we have that Az = ASw = SAw = Sz. Thus, z is a common fixed point for the mappingsA, B, S andR.

Let us now show that z is a unique common fixed point. Let us assume that there exists another common fixed pointy. From (3) it follows that

Fz,y(t)≥Fz,y(ϕ(r)) =FAz,By(ϕ(r))≥FSz,Ry(ϕ(r/c)) =Fz,y(ϕ(r/c)).

Finally, from Lemma 2.14 it follows that z=y.

Example 3.3. Let (X,F, T) be a complete Menger PM-space induced by a metric d(x, y) = |x−y| on X = [0,+∞) ⊂ R given in Remark 2.3. Let ϕ(t) =t, t >0, c= ¹₂ and

Ax= x

1 +x, Sx= 2x,

Bx= { _x

1+x, x∈[0,1]

0, x >1 , Rx=

{ 2x, x∈[0,1]

0, x >1

Note thatϕis a Φ-function. We shall prove that all the conditions of Theorem 3.2 are satisfied. First notice that A(X) = [0,1)⊂[0,2] =R(X) andB(X) = [0,¹₂) ⊂[0,+∞) = S(X). The sets A(X) and B(X) are metrically bounded, i.e. probabilistic bounded as subsets of the Menger PM-space. BecauseASx=

2x

1+2x andSAx=_1+x^2x we conclude thatAandS are not commuting.

We now prove that they are compatible mappings. Note that FASx,SAx(t) =ε0

(

t− 2x² (1 + 2x)(1 +x)

)

and FAx,Sx(t) =ε0

(

t−2x²+x 1 +x

)

Since (1+x)(1+2x)^2x2 ≤^x+2x_1+x² holds for allx≥0 we get FASx,SAx(t)≥FSx,Ax(t)

for all x, t ≥ 0. For a sequence {xn} in [0,+∞) such that lim

n→∞Axn =

nlim→∞Sxn = z, from the previous inequality it follows that lim inf

n→∞ FASx_n,SAx_n(t) = 1.

Now we prove that the mappings BandRare weakly compatible. IfBz= Rz then z= 0 orz >1.In the case when z= 0 we getRB(0) =BR(0) = 0.

On the other hand, ifz >1 then RB(z) =R(0) = 0 and BR(z) =B(0) = 0, i.e. the conditionRBz=BRz from Definition 2.16 is satisfied.

We now prove that the condition (3) is satisfied, too. Note that for all x, y∈X we have that _(1+x)(1+y)¹ ≤1.

a) For x, y∈[0,1] we get FAx,By(t) =ε0

(

t− |x−y| (1 +x)(1 +y)

)

≥ε0(2t−2|x−y|) =FSx,Ry(2t).

b) Forx >1 andy >1 we get FAx,By(t) =ε0

( t− x

1 +x )

≥ε0(2t−2x) =FSx,Ry(2t).

c) Ifx∈[0,1] andy >1 the proof is reduced to b). Ifx >1 andy∈[0,1]

the proof is reduced to a).

From the above we conclude that condition (3) is satisfied. We get that all the mappings have a unique common fixed point. It is easy to see that this point isx= 0.

Acknowledgement

This research was supported by Ministry of Education, Science and Tech- nological Development of the Republic of Serbia, Project grant number 174032.

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Received by the editors October 10, 2011