Some new results on complete U n ∗ -metric space

Research Article

Some new results on complete U _n ^∗ -metric space

Akbar Dehghan Nezhad^a,∗, Najmeh khajuee^a

aDepartment of Mathematics, Yazd University, 89195–741, Yazd, Iran.

Communicated by S.M. Vaezpour

Abstract

In this paper, we give some new definitions ofU_n^∗-metric spaces and we prove a common fixed point theorem for two mappings under the condition of weakly compatible and establish common fixed point for sequence of generalized contraction mappings in completeU_n^∗-metric space. c2013 All rights reserved.

Keywords: U_n^∗-metric space, complete U_n^∗-metric space, sequence of contractive mapping.

2010 MSC: 47H10, 54H25.

1. Introduction and Preliminaries

Recently Sedghi et. al. [11] introduced the concept ofD^∗-metric spaces and proved some common fixed point theorems (see also [3]–[12]).

In the present work, we introduce a new notion of generalized D^∗-metric space called U^∗-metric space of dimension nand study some fixed point results for two self-mappings f and g on U_n^∗-metric spaces. Some fundamental properties of the proposed metric are studied.

Definition 1.1. [2] LetGbe an ordered group. An ordered group metric (or OG-metric ) on a nonempty set X is a symmetric nonnegative function dG from X×X into G such that dG(x, y) = 0 if and only if x =y and such that the triangle inequality is satisfied; the pair (X, d_G) is an ordered group metric space (or OG-metric space).

For n ≥ 2, let Xⁿ denotes the cartesian product X×. . .×X and R⁺ = [0,+∞). We begin with the following definition.

Definition 1.2. LetX be a non-empty set. LetU_n^∗ :Xⁿ −→G⁺ be a function that satisfies the following conditions:

∗Corresponding author

Email addresses: [email protected](Akbar Dehghan Nezhad),[email protected](Najmeh khajuee) Received 2012-12-14

(U1) U_n^∗(x₁, . . . , x_n) = 0 if x₁ =. . .=x_n,

(U2) U_n^∗(x₁, . . . , x_n)>0 for allx₁, ..., x_n withx_i6=x_j, for somei, j∈ {1, ..., n},

(U3) U_n^∗(x1, . . . , xn) =U_n^∗(xπ1, . . . , xπn), for every permutation (π₍₁₎, ..., π_(n)) of (1,2, ..., n), (U4) U_n^∗(x1, x2, . . . , xn)≤U_n^∗(x1, ..., xn−1, a) +U_n^∗(a, xn, ..., xn), for allx1, . . . , xn, a∈X.

The functionU_n^∗is called a universal ordered group metric of dimensionn, or more specifically anOU_n^∗-metric on X, and the pair (X, U_n^∗) is called anOU_n^∗-metric space.

For example we can place G⁺ =Z⁺ orR⁺. In the sequel, for simplicity we assume that G⁺=R⁺. Example 1.3. (a) Let (X, d) be a usual metric space, then (X, S_n) and (X, M_n) are U_n^∗-metric spaces, where

S_n(x₁, . . . , x_n) = 2 n(n−1)

X

1≤i<j≤n

d(x_i, x_j), M_n(x₁, . . . , x_n) = max{d(x_i, x_j) : 1≤i < j ≤n}.

(b) Let φ be a non-decreasing and concave function with φ(0) = 0. If (X, d) is a usual metric space, then (X, φ_n) defined by

φn(x1, ..., xn) =φ⁻¹

X

1≤i<j≤n

φ(d(xi, xj)

is aU_n^∗-metric.

(c) Let X=C([0, T]) be the set of all continuous functions defined on [0, T]. Defined I_n:Xⁿ−→R⁺ by In(x1, . . . , xn) = X

1≤i<j≤n

sup_t∈[0,T]|x_i(t)−xj(t)|.

Then (X, I_n) is a U_n^∗-metric space.

(d) Let X=Rⁿ defined Ln:Rⁿ−→R⁺ by L_n(x₁, . . . , x_n) = X

1≤i<j≤n

kx_i−x_jk^1r

For everyr ∈R⁺. Then (X, L_n) is a U_n^∗-metric space.

(e) Let X=RdefinedK_n:Rⁿ−→Rby Kn(x1, ..., xn) =

0 ifx1=· · ·=xn

Mox{x₁,· · · , xn} otherwise Then (X, K_n) is a U_n^∗-metric space.

Remark 1.4. In aU_n^∗-metric space, we prove thatU^∗(x, ..., x, y) =U^∗(x, y, ...y). For (i) U^∗(x, ..., x, y)≤U^∗(x, ..., x) +U^∗(x, y, ..., y) =U^∗(x, y, ..., y) and similary

(ii) U^∗(y, ...y, x)≤U^∗(y, ..., y) +U^∗(y, x, ..., x) =U^∗(y, x, ..., x).

Hence by (i),(ii) we getU^∗(x, ..., x, y) =U^∗(x, y, ...y).

Proposition 1.5. Let (X, U) and(Y, V) be twoU_n^∗-metric spaces. Then (Z, W) is also a U_n^∗-metric space, where Z = X×Y and W(z₁, ..., z_n) = max{U(x₁, ..., x_n), V(y₁, ..., y_n)} for z_i = (x_i, y_i) ∈ Z with x_i ∈ X, yi∈Y, i= 1, ..., n.

Proof. Obviously (U1-U3) conditions are satisfied. To prove the (U4) inequality. Let z₁, ..., z_n ∈ Z, with c= (a, b), zi= (xi, yi), i= 1, ..., n,

W(z1, ..., zn) =max{U(x1, ..., xn), V(y1, ..., yn)}) ≤ max{U(x1, ..., xn−1, a) +U(a, xn, ..., xn), V(y1, ..., yn−1, b) +V(b, yn, ..., yn)}

≤max{U(x₁, ..., xn−1, a), V(y₁, ..., yn−1, b)}

+max{U(a, x_n, ..., x_n), V(b, y_n, ..., y_n)}

=W(z₁, ..., zn−1, c) +W(c, z_n, ..., z_n).

Hence (Z, W) is a U_n^∗-metric space.

Definition 1.6. A U_n^∗-metric space X is said to be bounded if there exists a constant M > 0 such that U_n^∗(x₁, ..., x_n)≤M for allx₁, ..., x_n∈X. AU_n^∗-metric spaceX is said to be unbounded if it is not bounded.

Proposition 1.7. Let (X, U_n^∗) be a U_n^∗-metric space and let M >0 be a fixed positive real number. Then (X, V) is a bounded U_n^∗-metric space with bound M, where the functionV is given by

V(x₁, ..., x_n) = M U^∗(x₁, ..., x_n) (k+U^∗(x1, ..., xn)) for allx1, ..., xn∈X and with k >0.

Proof. Obviously (U1-U3) conditions are satisfied. We only prove the (U4) inequality. Let x1, ..., xn∈X, V(x1, ..., xn) = M U^∗(x₁, ..., x_n)

(k+U^∗(x1, ..., xn)) = M− M k

(k+U^∗(x1, ..., xn))

≤ M− M k

(k+U^∗(x1, ..., xn−1, a) +U^∗(a, xn, ..., xn))

= M(U^∗(x1, ..., xn−1, a) +U^∗(a, xn, ..., xn)) (k+U^∗(x₁, ..., xn−1, a) +U^∗(a, x_n, ..., x_n))

= M(U^∗(x₁, ..., xn−1, a))

(k+U^∗(x1, ..., xn−1, a) +U^∗(a, xn, ..., xn)) + M(U^∗(a, x_n, ..., x_n))

(k+U^∗(x₁, ..., xn−1, a) +U^∗(a, x_n, ..., x_n))

≤ M(U^∗(x1, ..., xn−1, a))

(k+U^∗(x₁, ..., xn−1, a) + M(U^∗(a, xn, ..., xn)) (k+U^∗(a, x_n, ..., x_n))

= V(x₁, ..., xn−1, a) +V(a, x_n, ..., x_n).

Hence (X, V) is a U_n^∗-metric space.

Letx₁, ..., x_n∈X, Then we have, V(x₁, ..., x_n) = M U^∗(x1, ..., xn)

(k+U^∗(x₁, ..., x_n)) ≤ M U^∗(x1, ..., xn) (U^∗(x₁, ..., x_n)) =M This show that (X, V) is bounded withU_n^∗-bound M.

Definition 1.8. Let (X, U_n^∗) be a U_n^∗-metric space, then forx0∈X, r >0, the U_n^∗-ball with centerx0 and radiusr is

B_U^∗(x₀, r) ={y∈X :U_n^∗(x₀, y, ..., y)< r}.

Definition 1.9. Let (X, U_n^∗) be a U_n^∗-metric space andY ⊂X.

(1) If for everyy∈Y there exist r >0 such thatBU^∗(y, r)⊂Y, then subset Y is called open subset ofX.

(2) SubsetY ofX is said to beU^∗-bounded if there existsr >0 such thatU^∗(x, y, ..., y)< rfor allx, y∈Y. (3) A sequence {x_k} inX converges tox if and only if

U^∗(xk, ..., xk, x) =U^∗(x, ..., x, xk)→0 as k→ ∞.

That is for eachε >0 there exists N ∈Nsuch that

∀k≥N =⇒U^∗(x, ..., x, x_k)< ε (?).

This is equivalent with, for eachε >0 there exists N ∈Nsuch that

∀l₁, ..., ln−1 ≥N =⇒U^∗(x, x_l1, ..., x_ln−1)< ε (??).

(4) Let (X, U_n^∗) be aU_n^∗-metric space, then a sequence{x_k} ⊆X is said to beU_n^∗-Cauchy if for everyε >0, there exists N ∈ Nsuch that U_n^∗(xk, xm, ..., xl) < ε for all k, m, ..., l≥N. The U_n^∗-metric space (X, U_n^∗) is said to bo complete if every Cauchy sequence is convergent.

Remark 1.10. (i) Let τ be the set of all Y ⊂ X with y ∈ Y if and only if there exists r > 0 such that B_U^∗(y, r)⊂Y. Thenτ is a topology onX induced by the U_n^∗-metric.

(ii) If have (?) of Definition 1.9, then for each ε >0 there exists, N1 ∈Nsuch that for every l1≥N1 =⇒U^∗(x, ..., x, x_l1)< ε

n−1, N₂ ∈Nsuch that for every l₂≥N₂ =⇒U^∗(x, ..., x, x_l2)< ε

n−1,

and similary there exist Nn−1 ∈N such that for everyln−1≥Nn−1 =⇒U^∗(x, ..., x, x_ln−1)< ε n−1. LetN0 =max{N₁, ..., Nn−1} andK0 =min{l₁, ..., ln−1}. ForK0 > N0 we have

U^∗(x, xl1, ..., xln−1) ≤ U^∗(x, xl1, ..., xln−2, x) +U^∗(x, xln−1, ..., xln−1)

≤ U^∗(x, x, x_l1, ..., x_ln−3, x) +U^∗(x, x_ln−2, ..., x_ln−2) + U^∗(x, x_ln−1, ..., x_ln−1)

≤ ...

≤

n−1

X

i=1

U^∗(x, xli, ..., xli)

< (n−1)ε n−1 =ε.

Conversely, setl1 =· · ·=ln−1=k in (??) we haveU^∗(x, ..., x, x_k)< ε.

Proposition 1.11. In aU_n^∗-metric space, (X, U_n^∗), the following are equivalent.

(i) The sequence {x_k} is U_n^∗-Cauchy.

(ii) For everyε >0, there exists N ∈N such that U_n^∗(x_k, ..., x_k, x_l) < ε, for all k, l≥N. Lemma 1.12. Let (X, U^∗) be a U_n^∗-metric space.

(1) If r >0, then the ballB_U^∗(x, r) with centerx∈X and radius r is the open ball.

(2) If sequence{x_k} in X converges to x, then x is unique.

(3) If sequence{x_k} in X converges to x, then sequence {x_k} is a Cauchy sequence.

(4) The function of U_n^∗ is continuous onXⁿ.

Proof. proof 1)

Let w ∈ BU^∗(x, r) so that U^∗(x, w, ..., w) < r. If set U^∗(x, w, ..., w) = δ and r⁰ = r−δ then we prove that B_U^∗(w, r⁰) ⊆ B_U^∗(x, r). Let y ∈ B_U^∗(w, r⁰), by (U4) we have U^∗(x, y, ..., y) = U^∗(y, ..., y, x) ≤ U^∗(y, ..., y, w) +U^∗(w, x, ..., x)< r⁰+δ =r.

proof 2)

Letx_k −→y andy 6=x. Since {x_k} converges tox and y, for each ε >0 there exists, N1 ∈Nsuch that for every k≥N1 =⇒U^∗(x, ..., x, xk)< ε

and 2

N₂ ∈Nsuch that for every k≥N₂ =⇒U^∗(y, ..., y, x_k)< ε 2. If set N0=mox{N₁, N2}, then for everyk≥N0 by (U4) we have

U^∗(x, ..., x, y)≤U^∗(x, ..., x, x_k) +U^∗(x_k, y, ...., y)< ε 2+ ε

2 =ε.

thenU^∗(x, ..., x, y) = 0 is a contradiction. So x=y.

proof 3)

Since x_k−→x for each ε >0 there exists,

N1 ∈Nsuch that for every k≥N1 =⇒U^∗(xk, ..., xk, x)< ε and 2

N₂ ∈Nsuch that for every l≥N₁ =⇒U^∗(x, x_l..., x_l)< ε 2.

If set N₀=mox{N₁, N₂}, then for everyk, l≥N₀ by (U₄) we have

U^∗(xk, ..., xk, xl)≤U^∗(xk, ..., xk, x) +U^∗(x, xl, ..., xl)< ε 2 +ε

2 =ε.

Hence sequence{x_k} is a Cauchy sequence.

proof 4)

Let the sequence

(x1)k, ...,(xn)k inXⁿ converges to a point (z1, ..., zn) i.e.

k→∞lim(xi)_k=zi i= 1, ..., n for each ε >0 there exists,

N1 ∈Nsuch that for every k > N1 =⇒U^∗ z1, ..., z1,(x1)_k

< ε n N2 ∈Nsuch that for every k > N2 =⇒U^∗ z2, ..., z2,(x2)_k

< ε .. n

.

N_n∈N such that for everyk > N_n=⇒U^∗ z_n, ..., z_n,(x_n)_k

< ε n. If set N₀=mox{N₁, ..., N_n}, then for everyk≥N₀ we have

U^∗ (x1)k, ...,(xn)k

≤ U^∗ (x1)k, ...,(xn−1)k, zn

+U^∗ zn,(xn)k, ...,(xn)k

≤ U^∗ (x1)k, ...,(xn−2)k, zn, zn−1

+U^∗ zn−1,(xn−1)k, ...,(xn−1)k

+ U^∗ z_n,(x_n)_k, ...,(x_n)_k

≤ ...

≤ U^∗(z₁, ..., z_n) +

n

X

i=1

U^∗ z_i,(x_i)_k, ...,(x_i)_k

≤ U^∗(z1, ..., zn) +nε

n =U^∗(z1, ..., zn) +ε.

Hence we have

U^∗ (x₁)_k, ...,(x_n)_k

−U^∗(z₁, ..., z_n)< ε

U^∗(z₁, ..., z_n) ≤ U^∗ z₁, ..., zn−1,(x_n)_k

+U^∗ (x_n)_k, z_n, ..., z_n

≤ U^∗ z₁, ..., zn−2,(x_n)_k,(xn−1)_k

+U^∗ (xn−1)_k, zn−1, ..., zn−1 + U^∗ (xn)_k, zn, ..., zn

≤ ...

≤ U^∗ (x1)k, ...,(xn)k

+

n

X

i=1

U^∗ (xi)k, zi, ..., zi

≤ U^∗ (x₁)_k, ...,(x_n)_k +nε

n =U^∗ (x₁)_k, ...,(x_n)_k +ε.

That is,

U^∗(z1, ..., zn)−U^∗ (x1)_k, ...,(xn)_k

< ε.

Therefore we have|U^∗ (x1)k, ...,(xn)k

−U^∗(z1, ..., zn)|< ε, that is

k→∞lim U^∗ (x₁)_k, ...,(x_n)_k

=U^∗(z₁, ..., z_n).

Definition 1.13. ([6]) Let f and g be mappings from a U_n^∗-metric space (X, U_n^∗) into itself. Then the mappings are said to be weak compatible if they commute at their coincidence point, that is f x = gx implies thatf gx=gf x.

Definition 1.14. Let (X, U_n^∗) be aU_n^∗-metric space, forA1, ..., An⊆X, define

∆_U^∗(A₁, ..., A_n) =sup{U^∗(a₁, ..., a_n)|a_i ∈A_i, i= 1, ..., n}.

Remark 1.15. It follows immediately from the definition that (i) IfA_i consists of a single point a_i we write

∆^∗_U(A1, ..., Ai−1, Ai, Ai+1, ..., An) = ∆^∗_U(A1, ..., Ai−1, ai, Ai+1, ..., An).

IfA₁, ..., A_n also consists of a single point a₁, ..., a_n respectively, we write

∆^∗_U(A1, ..., An) = ∆^∗_U(a1, ..., an).

Also we have

∆U^∗(A1, ..., An) = 0⇐⇒A1 =· · ·=An={a},

∆U^∗(A1, ..., An) = ∆U^∗(Aπ1, ..., Aπn), for for every permutation (π₍₁₎, ..., π_(n)) of (1,2, ..., n).

In particular for∅6=A1=· · ·=An⊆X,

∆_U^∗(A₁) =sup{U^∗(b₁, ..., b_n)|b₁, ..., b_n∈A₁}.

(ii) If A⊆B, then ∆_U^∗(A)≤∆_U^∗(B).

(iii) For a sequence A_k = {x_k, x_k+1, x_k+2,· · · } in U_n^∗-metric space (X, U_n^∗), let a_k = ∆_U^∗(A_k) for k ∈ N. Then

(a) : SinceA_k+1⊆A_k hence ∆_U^∗(A_k+1)≤∆_U^∗(A_k), for everyk≥1.

(b) : U^∗(x_l1, ..., x_ln)≤∆_U^∗(A_k) =a_k for everyl₁, ..., l_n≥k, (c) : 0≤∆U^∗(Ak) =ak.

Therefore,{a_k}is decreasing and bounded for allk∈N, and so there exists an 0≤asuch that limk→∞a_k= a.

Lemma 1.16. Let (X, U_n^∗) be an U_n^∗-metric space. If limk→∞a_k = 0, then sequence {x_k} is a Cauchy sequence.

Proof. Since limk→∞a_k= 0, we have that for everyε >0, there exists aN₀ ∈Nsuch that for everyk > N₀,

|a_k−0|< ε. That isak= ∆U^∗(Ak)< ε. Then for l1, ..., ln≥k > N0 by (b) of Remark 1.15 we have U^∗(x_l1, ..., x_ln)≤sup{U^∗(xi, ..., xj)|xi, ..., xj ∈A_k}=a_k< ε.

Therefore, {x_k}is a Cauchy sequence in X.

2. Main results

Theorem 2.1. Let X be a U_n^∗-complete metric space

I) If f and g be self-mappings of a complete U_n^∗-metric space (X, U_n^∗) satisfying:

i) g(X)⊆f(X), andf(X) is closed subset of X, ii) the pair (f, g) is weakly compatible,

iii) U^∗(gz₁, ..., gz_n)≤ψ(U^∗(f z₁, ..., f z_n)),for every z₁, ..., z_n∈X, where ψ:R⁺−→R⁺ is a nondecreasing continuous function with ψ(t)< t for everyt >0.

Thenf and g have a unique common fixed point in X.

II) If f_k:X −→X be a sequence maps such that

U^∗(f_iz₁, f_jz₂, ..., f_lzn−1, z_n)≤βU^∗(z₁, ..., z_n) for alli6=j and z₁, ..., z_n∈X with0≤β < 1

2. Then {f_k} have a unique common fixed point.

Proof. proof I)

Let x₀ be an arbitrary point in X. By (i), we can choose a point x₁ in X such that y₀ = gx₀ =f x₁ and y₁ =gx₁ =f x₂. In general, there exists a sequence {y_k} such that, y_k =gx_k =f x_k+1, for k= 0,1,2,· · ·. We prove that sequence{y_k}is a Cauchy sequence. LetAk={y_k, yk+1, yk+2,· · · }andak= ∆U^∗(Ak), k∈N.

Then we know limk→∞a_k=afor somea≥0.

Takingz_i =x_li+l in (iii) for l≥1 andl₁, ..., l_n≥0 U^∗(yl1+l, ..., yln+l) = U^∗(gxl1+l, ..., gxln+l)

≤ ψ(U^∗(f x_l1+l, ..., f x_ln+l))

= ψ(U^∗(y_l1+l−1, ..., y_ln+l−1))

Since U^∗(yl1+l−1, ..., yln+l−1)≤al−1, for everyl1, ..., ln≥0 andψ is increasing int, we get U^∗(y_l1+l, ..., y_ln+l)≤ψ(U^∗(y_l1+l−1, ..., y_ln+l−1)).

Therefore

sup

l1,...,ln≥0

{U^∗(y_l1+l, ..., y_ln+l)≤ψ(al−1).

Hence, we have a_l ≤ψ(al−1). Letting l → ∞, we get a≤ ψ(a). If a6= 0, then a ≤ψ(a) < a, which is a contradiction. Thus a= 0 and hence limk→∞ak = 0. Thus Lemma 1.16 {y_k} is a Cauchy sequence in X.

By the completeness of X, there exists a v∈X such that

k→∞lim y_k= lim

k→∞gx_k = lim

k→∞=f x_k+1 =v.

Letf(X) is closed, there existw∈Xsuch thatf w=v, Now we show that gw=v For this it is enough set x_k, ..., x_k, w replacingz1, ..., zn respectively, in inequality (iii) we get

U^∗(gx_k, ..., gx_k, gw)≤ψ(U^∗(f x_k, ..., f x_k, f w))

Takingk→ ∞, we get

U^∗(v, ..., v, gw)≤ψ(U^∗(0)) = 0, it implies gw=v.

Since the pair (f, g) are weakly compatible, hence we get, gf w=f gw. Thusf v =gv. Now we prove that gv=v. If we substitute z₁, ..., z_n in (iii) byx_k, ..., x_k and v respectively, we get

U^∗(gx_k, ..., gx_k, gu)≤ψ(U^∗(f x_k, ..., f x_k, f v)) Takingk→ ∞, we get

U^∗(v, ..., v, gv)≤ψ(U^∗(v, ..., v, gv)).

Ifgv 6=v, then U^∗(v, ..., v, gv)< U^∗(v, ..., v, gv), is contradiction. Therefore, f v=gv=v.

For the uniqueness, let v and v⁰ be fixed points off, g. Takingz1 =...=zn−1 =v and zn=v⁰ in (iii), we have

U^∗(v, ..., v, v⁰) = U^∗(gv, ..., gv, gv⁰)

≤ ψ(U^∗(f v, ..., f v, f v⁰))

= ψ(U^∗(v, ..., v, v⁰))

< U^∗(v, ..., v, v⁰), which is a contradiction. Thus we havev=v⁰. proof II)

Let x0 ∈ X be any fixed arbitrary element define a sequence {x_k} in X as. x_k+1 = f_k+1x_k for all k = 0,1,2,· · · .

Letd_k=U^∗(x_k, x_k+1, ..., x_k+1) for allk= 0,1,2,· · · . Now

d_k+1 = U^∗(x_k+1, x_k+2, ..., x_k+2)

= U^∗(f_k+1x_k, f_k+2x_k+1, ..., f_k+2x_k+1, x_k+2)

≤ βU^∗(xk, xk+1, ..., xk+1, xk+2)

≤ βU^∗(x_k, x_k+1, ..., x_k+1, x_k+1) +βU^∗(x_k+1, x_k+2, ..., x_k+2)

= βd_k+βd_k+1. Hence

dk+1 ≤ β 1−βdk, d_k ≤ β

1−βdk−1 for alln= 1,2,· · ·. Letα= β

1−β, we have d_k ≤α dk−1 ≤α^kd₀→0 as k→ ∞. Therefore

limk→βd_k= 0. Thus

limk→βU^∗(xk, xk+1, ..., xk+1) = 0.

Now we shall prove that{x_k} is aU_n^∗-Cauchy sequence inX.

Letl > k > N₀ for someN₀ ∈N. Now

U^∗(x_k, ..., x_k, x_l) ≤ U^∗(x_k, ..., x_k, x_k+1) +U^∗(x_k+1, ..., x_k+1, x_l)

≤

l−1

X

t=∞

U^∗(x_t, ..., x_t, x_t+1)→0 ask, l→ ∞

Hence limk,l→∞U^∗(x_k, ..., x_k, x_l) = 0.

Thus{x_k} is U_n^∗-Cauchy sequence inX.

Since X isU_n^∗-complete x_k →xin X. We prove that x is a fixed point off_k for all k suppose there exist a k⁰ such that f_k⁰x6=x. Then

U^∗(f_k⁰, x, ..., x) = lim

k→∞U^∗(f_k⁰x, x_k+1, ..., x_k+1, x)

= lim

k→∞U^∗(f_k⁰x, f_k+1x_k, ..., f_k+1x_k, x)

≤ β lim

k→∞U^∗(x, xk+1, ..., xk+1, x) = 0.

ThereforeU^∗(f_k⁰, x, ..., x) = 0, Therefore f_kx=xfor all k. Thus xis common fixed point of {f_k} for allk.

For the uniqueness, supposex6=y such thatf_ky=y for allk. Then U^∗(x, y, ..., y) = U^∗(f_ix, f_jy, ..., f_jy, y)

≤ β U^∗(x, y, ..., y) This implies (1−β)U^∗(x, y, ..., y)≤0.

Since x6=y we have U^∗(x, y..., y)>0 her (1−β)<0.

This implies β >1 which contraction toβ < 1 2. Thus{f_k} have a unique common fixed point.

Corollary 2.2. Let f be self-mapping of a complete U_n^∗-metric space (X, U_n^∗) satisfying:

U^∗(z₁, ..., z_n)≤ψ(U^∗(f^mz₁, ..., f^mz_n)),

for every z₁, ..., z_n ∈ X, f is surjective and m ∈ N, where ψ : R⁺ −→ R⁺ is a nondecreasing continuous function with ψ(t)< t for every t >0.

Thenf have a unique fixed point in X.

Proof. If we define g =I identity map in Theorem 2.1. There exists a unique v ∈ X such that f^mv =v.

Thus

f^m(f v) =f(f^mv) =f v.

Since vis unique, we havef v=v.

Corollary 2.3. Let g be self-mapping of a complete U_n^∗-metric space (X, U_n^∗) satisfying:

U^∗(g^mz1, ..., g^mzn)≤ψ(U^∗(z1, ..., zn)),

for every z1, ..., zn ∈ X and m ∈ N, where ψ : R⁺ −→ R⁺ is a nondecreasing continuous function with ψ(t)< t for every t >0.

Theng have a unique fixed point inX.

Proof. If we define f = I identity map in Theorem 2.1. There exists a unique v ∈X such that g^mv =v.

Thus

g^m(gv) =g(g^mv) =gv.

Since vis unique, we havegv =v.

Corollary 2.4. Let f and g be self-mappings of a complete U_n^∗-metric space (X, U_n^∗) satisfying:

(i) g^r(X)⊆f^s(X), andf^s(X) is closed subset of X,

(ii) the pair(f^s, g^r) is weakly compatible and f^sg=gf^s, g^rf =f g^r,

(iii) U^∗(g^rz1, ..., g^rzn)≤ψ(U^∗(f^sz1, ..., f^szn)),for every z1, ..., zn∈X andr, s∈N where ψ:R⁺−→R⁺ is

a nondecreasing continuous function with ψ(t)< t for everyt >0.

Thenf and g have a unique common fixed point in X.

Proof. By Theorem 2.1 there exists a fixed point v ∈X such that f^sv =g^rv =v. On the other hand, we have

gv=g(g^rv) =g^r(gv) andgv=g(f^sv) =f^s(gv).

Since vis unique, we havegv =v. Similarly, we havef v=v.

Corollary 2.5. Let f,g andh be self-mappings of a complete U_n^∗-metric space (X, U_n^∗) satisfying:

(i) g(X)⊆f h(X), andf h(X) is closed subset of X,

(ii) the pair(f h, g) is weakly compatible and f h=hf, gh=hg,

(iii) U^∗(gz₁, ..., gz_n) ≤ ψ(U^∗(f hz₁, ..., f hz_n)), for every z₁, ..., z_n ∈ X, where ψ : R⁺ −→ R⁺ is a nonde- creasing continuous function with ψ(t)< t for everyt >0.

Thenf, g and h have a unique common fixed point in X.

Proof. By Theorem 2.1 there exists a fixed pointv∈X such thatf hv =gv=v.

Now, we prove thathv =v. Ifhv6=vin (iii), then we have U^∗(hv, v, ..., v) = U^∗(hgv, gv, ..., gv)

= U^∗(ghv, gv, .., gv)

≤ ψ(U^∗(f hhv, f hv, ..., f hv))

= ψ(U^∗(hv, v, ..., v))

< U^∗(hv, v, ..., v),

which is a contradiction. Thus we havehv=v. Therefore,

f v=f hv=v=hv=gv.

Acknowledgements:

The authors thank the editor and the referees for their useful comments and suggestions.

References

[1] A. Bagheri Vakilabad and S. Mansour Vaezpour, Generalized contractions and common fixed point theorems concerningτ- distance, J. Nonlinear Sci. Appl.3(3) (2010), 78-86.

[2] L. W. Cohen and C. Goffman,The topology of ordered Abelian groups, Trans. Amer. Math. Soc.67(1949), 310-319.

1.1

[3] A. Dehghan Nezhad and Z. Aral, The topology of GB-metric spaces, ISRN. Mathematical Analysis, Hindawi, (2011). 1

[4] A. Dehghan Nezhad and H. Mazaheri,New results in G-best approximation in G-metric spaces,Ukrainian Math.

J.,62(4), (2010), 648-654.

[5] B.C. Dhage,A common fixed point principle inD-metric spaces, Bulletin of the Calcutta Mathematical Society.

91(6) (1999), 475-480.

[6] G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl.

Math.29(3) (1998), 227-238. 1.13

[7] N. V. Luong and N. X. Thuan , Common fixed point theorems in compact D^∗-metric spaces, Internationnal Mathematical Forum.6(13) (2011),605-612.

[8] H.K. Nashine,Coupled common fixed point results in orderedG-metric spaces, J. Nonlinear Sci. Appl.1(2012), 1-13.

[9] V. Popa and A.M. Patriciu,A general fixed point theorem for pairs of weakly compatible mappings inG-metric spaces, J. Nonlinear Sci. Appl.5(2012), 151-160.

[10] S. Sedghi, M.S. Khan and N. Shobe,Fixed point theorems for six weakly compatible mappings inD^∗-metric spaces, J. Appl. Math. Informatics.27(2) (2009), 351-363.

[11] S. Sedghi, S. Nabi and Z. Haiyun,A common fixed point theorems for inD^∗-metric spaces,Hindawi Publishing Corporation. Fixed point Theory and Applications, Article ID 27906, (2007), p. 13, doi: 10.1155. 1

[12] S. Shaban, S. Nabi, Z. Haiyun and S. Shahram,Common fixed point theorems for two mappings inD^∗-metric spaces, Journal of prime research in mathematics.4(2008), 132–142. 1