1. Introduction

The concept of a 2- metric space was initially given by Gahler ([2],[3]) during 1960’s. Then about a decade after during 1970’s some basic fixed point results in such spaces have been established by Iseki ([4], [5]). There after some fixed point results are obtained in such spaces by Khan et al [6], Rhoades[7] and many others extending the fixed point results for contractive mappings from metric space to 2- metric space. In this subsection we give some preliminary definitions and results of aforesaid authors.

Definition 1.1 A 2-metric space is a space X in which, for each triple of points a, b, c there exists a real valued non negative function satisfying:

(1) For each pair of points a, b, a ≠ b of X, there exists a point c ε X such that ρ(a, b,c) ≠ 0.

(2) ρ(a, b, c) = 0 when at least two of the points are equal. (3) ρ(a, b, c) = ρ(a, c, b) = ρ(b, c, a) and (4) ρ(a, b, c) ≤ ρ(a, b, d) + ρ(a, d, c) + ρ(d, b, c)

Definition 1.2 A sequence {x_n} of X is called Cauchy sequence if

lim ρ(x_n , x_m ,a) = 0 for all a ε X.

Definition 1.3 A sequence {x_n} in X is convergent and x ε X is the limit of this

sequence if ρ(x_n, x ,a) = 0 for each a ε X.

Theorem 1.1 (Rhoades[7]) Let X be a complete 2-metric Space, f: X → X satisfying: there exists a h, 0 ≤ h < 1 such

ρ(f(x), f(y), a) ≤ h max {ρ(x, y, a), ρ(x, f(x), a), ρ(y, f(y), a), ρ (x,f(y),a), ρ (y,f(x),a)}that for each x, y, a ε X

Then f possesses a unique fixed point z and lim f ⁿ(x₀) = z for each x₀ ε X.

Theorem 1.2 (Rhoades[7]) Let f and g be mappings of a complete 2– metric space X into itself satisfying

ρ(f(x),g(y),a) ≤ h max {ρ(x,y,a), ρ(x,f(x),a), ρ(y,g(y),a), ρ(y,f(x),a), ρ(x,g(y),a)}

for all x,y ε X, h a fixed constant satisfying 0 ≤ h <1. Then f and g have a common fixed point z.

Theorem 1.3 (Rhoades[7]) Let X be a complete 2– metric space , {f_n}, n = 1,2,…. a sequence of mapping f_n : X → X, suppose there exists a sequence of non negative integers {m_n} and a number h, 0 ≤ h < 1 such that, for all x,y ∈ X and every pair i, j , i ≠ j and satisfying

ρ(f_i^mi(x),f_j^mj(y),a) ≤ h max {ρ(x,y,a), ρ(x,f_i^mi(x),a),ρ(y,f_j^mj(y),a), ρ(y,f_i^mi(x),a)}

Then the mappings {f_n} have a unique common fixed point.

2. Objective

The main objective of the paper is to establish few fixed point results in 2-metric space involving terms like [], [] ,

[] , [ρ(x,f_j^mj(y),a) + ρ(y,f_i^mi(x),a)]/2 etc in the inequality condition under max composition, which is more general than the inequality condition used by previous authors.

3. Method

The usual fixed point analysis methods for 2-metric space as used by earlier authors Gahler, Rhoades, Iseki has been used to prove our generalizations.

4. Main Results

In this paper few fixed point results are obtained for some generalized contractive mappings involving rational type terms in a 2- metric space. Taking a clue from Theorem 1.1, our first result goes as follows:

Theorem 4.1 Let X be a complete 2– metric space, f : X → X satisfying : there exists a h, 0 ≤ h < 1 such that for each x, y, a ε X,

ρ( f(x),f(y),a) ≤ h max {ρ(x,y,a), ρ(x, f(x),a), ρ(y,f(y),a), [],

[] } (i) Then f possesses a unique fixed point z.

Proof : Let x₀ ∈ X and define {x_n} by

x_n+1 = f(x_n), n = 0,1,2.. .

From (i) we have ,

ρ(x_n+1, x_m+1,a) = ρ(f(x_n), f(x_m),a) ≤ h max { ρ(x_n,x_m,a), ρ(x_n,x_n+1,a), ρ(x_m,x_m+1,a),

[], []}

≤ h ρ(x_n,x_m,a)

It cannot be that, ρ(x_n,x_m+1,a)/2 ≥ ρ(x_n, x_m,a) , ρ(x_n+1, x_m+1, a)

Again ,

ρ(x_n,x_m+1,a) ≤ ρ(x_n,x_m+1,x_n+1) + ρ(x_n,x_n+1,a) + ρ(x_n+1,x_m+1,a)

and we have , ρ(x_n,x_m+1,x_n+1) = 0.

Thus,

ρ(x_n+1,x_m+1,a) ≤ h ρ(x_n, x_m, a)

Similarly, ρ(x_n,x_m,a) ≤ h ρ(x_n-1, x_m-1,a)

≤ h² ρ(x_n-2, x_m-2,a)

……………

≤ hⁿ ρ(x₀, x_k,a)

Therefore for integers n, m where n > m ≥ 0,

ρ(x_n, x_m, a) ≤ hⁿ ρ(x₀, x_k,a) (ii)where k is a suitable integer satisfying 0 < k ≤ m. using property 4 of definition 1

and (ii),we get

ρ(x₀, x_k,a) ≤ ρ(x₀, x_k,x₁) + ρ(x₀, x₁,a) + ρ(x₁, x_k, a)

≤ ρ(x_0, x_k, x₁) + ρ(x₀, x₁, a) + h ρ(x₀, x_k, a)

≤ ………………

≤ρ(x₀, x₁, a)

Therefore, ρ(x_n, x_m, a) ≤ρ(x₀, x₁, a)

So it can be easily shown that

ρ(x₀, x_k, x₁) = 0 , (see[7]) where k^/ is a suitable integer satisfying 0≤ k^/≤ k. Therefore {x_n} is Cauchy sequence, hence convergent. Let us consider the limit z, using (i), for any a ε X.

ρ( x_n+1, f(z), a) = ρ ( f(z),a, x_n+1) = ρ( f(z), x_n+1, a)

≤ h max{ ρ( z, x_n, a), ρ(z, f(z), a), ρ(x_n, x_n+1, a), [],

[] }

Taking the limit of both sides as n → ∞, we have,

ρ(z, f(z),a) ≤ h ρ(z, f(z),a)

which implies z = f(z).

Suppose z, w are fixed points of f. Then from (i), each a ε X, we have

ρ(z, w, a) ≤ h ρ(z, w, a).

Since 0 ≤ h < 1, and using ρ(a, b, c) ≠ 0.So we get, z = w. Therefore f has a unique fixed point z.

Theorem 4.2 Let X be a complete 2- metric space, {f_n} a sequence of mappings of X into X with fixed points z_n, and f a mappings of X into X satisfying

ρ(f(x),f(y),a) ≤ h max {ρ(x,y,a), ρ(x,f(x),a), ρ(y,f(y),a), ρ(x,f(y),a), ρ(y,f(x),a)} (iii)

with fixed point z, such that f_n → f uniformly on { z_n : n = 1,2, ….}. Then z_n → z.

Proof: Let ε > 0. From the uniform convergence of {f_n} on {z_n : n = 1,2,..} there exists an integer N such that for all n ≥ N,

ρ( f(z_n), f_n(z_n), a) < , for all z_n , where M =

Now, ρ(z_n, z, a) = ρ( f_n(z_n), f(z),a)

≤ ρ(f_n(z_n),f(z),f(z_n) +ρ(f_n(z_n), f(z_n),a) + ρ(f(z_n),f(z),a) (iv)

Again from (iii),

ρ(f(z_n),f(z),a) ≤ h max{ρ(z_n,z,a),ρ(z_n,f(z_n),a),ρ(z,f(z),a),ρ(z_n,f(z),a)ρ(z,f(z_n),a)}

≤ h max{ρ(z_n, z, a), ρ(z_n, f(z_n),a)}

so that

ρ(f_n(z_n), f(z), f(z_n)) = ρ(f(z), f(z_n),z_n)

≤ h max {ρ(z_n, z, z_n), ρ(z_n, f(z_n),z_n)} = 0.

Now (iv) becomes

ρ(z_n, z, a) ≤ ρ(f_n(z_n),f(z_n),a) + h max{ρ( z_n, z, a), ρ(z_n, f(z_n),a)}

which implies

ρ( z_n, z, a) ≤ < ε.

Thus z_n → z.

Our next result generalizes theorem 1.2 of Rhoades[7].

Theorem 4.3 Let f and g be mappings of a complete 2–metric space X into itself satisfying

ρ(f(x),g(y),a) ≤ h max {ρ(x,y,a), ρ(x,f(x),a), ρ(y,g(y),a), ρ(y,f(x),a), ρ(x,g(y),a),

[]} (v)

for all x,y ε X, h a fixed constant satisfying 0 ≤ h <1 . Then f and g have a common fixed point z and (fg)ⁿ(x₀) → z and (gf)ⁿ(x₀) → z for each x₀ ε X.

Proof: Let x₀ ε X and we define {x_n} by

x_2n+1 = f(x_2n) and x_2n+2 = g(x_2n+1)

From (vii), we get

ρ(x_2n+1, x_2n+2, a) = ρ(f(x_2n), g(x_2n+1), a)

≤ h max {ρ(x_2n,x_2n+1,a), ρ(x_2n,x_2n+1,a), ρ(x_2n+1,x_2n+2,a),

ρ(x_2n+1,x_2n+1,a), ρ(x_2n,x_2n+2,a), [] }

≤ h max { ρ(x_2n,x_2n+1,a), ρ(x_2n, x_2n+2,a)}

Again ,

ρ(x_2n,x_2n+2,a) ≤ ρ(x_2n,x_2n+2,x_2n+1) + ρ(x_2n,x_2n+1,a) + ρ(x_2n+1,x_2n+2,a)

and we have ,

ρ(x_2n,x_2n+2,x_2n+1) = 0.

Thus, ρ(x_2n+1,x_2n+2,a) ≤ h ρ(x_2n, x_2n+1, a)

Similarly, ρ(x_2n,x_2n+1,a) ≤ h ρ (x_2n-1, x_2n,a)

For arbitrary n, we have

ρ(x_n, x_n+1, a) ≤ hⁿ ρ(x₀, x₁, a) (vi)

For any m > n and using property 4 of definition 1 and (vi)

ρ(x_m, x_n,a) ≤

= hⁿ(1-h)^-1.[ρ(x₀, x₁, x_m) + ρ(x₀, x₁, a)]

we can easily shown that, ρ(x₀, x₁, x_m) = 0. ([7] )

So that {x_n} is a Cauchy sequence and hence convergent.Let us consider the limit z.

Now,

ρ(f(z), z, a) ≤ ρ(f(z), z, x_2n+2) + ρ(f(z), x_2n+2, a) + ρ(x_2n+2, z, a) (vii)

From (vii), we get

ρ(f(z), x_2n+2, a) = ρ(f(z), g(x_2n+1), a)

≤ h max{ ρ(z, x_2n+1, a), ρ(z, f(z), a), ρ(x_2n+1, x_2n+2, a), ρ(x_2n+1,f(z),a),

ρ(z,x_2n+2,a), []} (viii)

Substituting (vii) into (viii) and taking limit as n → ∞, we have,

ρ(f(z), z, a) ≤ h ρ(z, f(z), a)

as 0 ≤ h < 1 , we get z = f(z).

Therefore, z is a fixed point of f. Similarly, we can show that z is also a fixed point of g. For uniqueness, suppose z and w are common fixed points of f and g.

Now from (v)

ρ(z, w, a) = ρ(f(z), g(w), a)

≤ h max {ρ(z, w, a), ρ(z,f(z),a), ρ(w,g(w),a),ρ(z,g(w),a),ρ(w,f(z),a),

[}

≤ h max { ρ(z,w,a), 0 ,0 ,ρ(z,w,a), ρ(w,z,a),ρ(z,w,a)}

or, ρ(z,w,a) ≤ h ρ(z,w,a),

which implies z = w. Thus z is a unique common fixed point of f and g .

An extension of theorem 1.3 of Rhoades[7] goes as follows:

Theorem 4.4 Let X be a complete 2–metric space , {f_n}, n = 1,2,…. a sequence of mapping f_n: X → X, suppose there exists a sequence of non negative integers {m_n} and a number h, 0 ≤ h < 1 such that, for all x,y ε X and every pair i ,j , i ≠ j and satisfying

ρ(f_i^mi(x),f_j^mj(y),a) ≤ h max {ρ(x,y,a), ρ(x,f_i^mi(x),a),ρ(y,f_j^mj(y),a), ρ(y,f_i^mi(x),a),

ρ(x,f_j^mj(y), [ρ(x,f_j^mj(y),a) + ρ(y,f_i^mi(x),a)] /2 } (ix)

Then the mappings {f_n} have a unique common fixed point.

Proof : Taking g_i = f_i^mi, i = 1,2,3,…. in (ix), we have

ρ(g_i(x), g_j(y),a) ≤ h max {ρ(x,y,a), ρ(x,g_i(x),a), ρ(y,g_j(y),a), ρ(y,g_i(x),a),

ρ(x,g_j(y),a), [} (x)

Let us consider x₀ ε X and we define

x_n = g_n(x_n-1), n = 1,2,…

Now from (x), we get

ρ(x_n, x_n+1,a) = ρ( g_n(x_n-1), g_n+1(x_n),a)

≤ h max { ρ(x_n-1,x_n,a), ρ(x_n-1,x_n,a), ρ(x_n,x_n+1,a),ρ(x_n,x_n,a), ρ(x_n-1,x_n+1,a),

[]}

≤ h max { ρ(x_n-1,x_n,a), ρ(x_n-1,x_n,a), ρ(x_n,x_n+1,a),0, ρ(x_n-1,x_n+1,a),

}

as in the proof of Theorem 2.3, we led to the conclusion that

ρ(x_2n,x_2n+1,a) ≤ h ρ(x_2n-1,x_2n,a)

and in general

ρ(x_n,x_n+1,a) ≤ hⁿ ρ(x₀,x₁,a).

Therefore, {x_n} is Cauchy sequence and converges to a limit z. Now we get from (x),

ρ(g_n(x),g_m+1(x_m),a) ≤ h max { ρ(z,x_m,a), ρ(z,g_n(z),a), ρ(x_m,x_m+1,a),

ρ(x_m,g_n(z),a), ρ(z,x_m+1,a), }

Taking limit as m → ∞, we obtain,

ρ(g_n(z),z,a) ≤ h ρ(g_n(z),z,a) = h ρ(z,g_n(z),a)

which implies that g_n(z) = z, as 0 ≤ h < 1.

For each n, we have f_n(z) = f_n(g_n(z)) = f_n(f_n^mnz), which shows that f_n(z) is a fixed point

of g_n. Uniqueness of this theorem follows easily from (ix) and by uniqueness, we

have f_n(z) = z .Hence z is a unique common fixed point of f_n.

Theorem 4.5 Let T and S be two self mappings of a complete metric space (X,d) satisfying

ad(Tx,Sy,u) + bd(x,Tx,u) + cd(y,Sy,u) ≤ q max{d(x,y,u), d(x,Tx,u), d(y,Sy,u)} (xi)

for x, y ε X and u ε X; a + c > q and a > q . Then T and S have a unique common fixed point.

Proof: Let x₀ ε X and we define {x_n} by x_2n+1 = Tx_2n and x_2n+2 = Sx_2n+1.

Now putting x = x_2n and y = x_2n+1 in (xi) we have

ad(Tx_2n, Sx_2n+1, u) + bd(x_2n, Tx_2n, u) + cd(x_2n+1, Sx_2n+1, u)

≤ q max{d(x_2n, x_2n+1, u), d(x_2n, Tx_2n, u), d(x_2n+1,Sx_2n+1,u)}

or, ad(x_2n+1, x_2n+2, u) + bd(x_2n, x_2n+1, u) + cd(x_2n+1, x_2n+2, u)

≤ q max{d(x_2n, x_2n+1, u), d(x_2n, x_2n+1, u), d(x_2n+1, x_2n+2,u)}

or, (a+c) d(x_2n+1, x_2n+2, u) + bd(x_2n, x_2n+1, u) ≤ qd(x_2n, x_2n+1, u)

or, d(x_2n+1, x_2n+2, u) ≤d(x_2n, x_2n+1, u) = h d(x_2n, x_2n+1, u), where h =

Similarly, d(x_2n, x_2n+1, u) ≤ h d(x_2n-1, x_2n, u)

Therefore for any arbitrary n

d(x_n, x_n+1, u) ≤ hⁿ d(x₀, x₁, u) (xii)

From (xii) using the property (4) of definition 1, we get, for any m > n

d(x_m, x_n,u) ≤

≤ hⁿ (1-h)^-1[d(x₀, x₁, x_m) + d(x₀, x₁, u)]

We can easily shown that d(x₀, x₁, x_m) = 0 ( Rhoades[7] )

So that {x_n} is a Cauchy sequence, hence convergent and {Tx_2n},{Sx_2n+1} also converge to z. From (xi),

ad(Tx_2n, Sz, u) + bd(x_2n, Tx_2n, u) + cd(z, Sz, u)

≤ q max{d(x_2n, z, u), d(x_2n, Tx_2n, u), d(z, Sz, u)}

In the limiting case, we get

ad(z, Sz, u) + cd(z, Sz, u) ≤ qd(z, Sz, u)

or, (a + c -q) d(z, Sz, u) ≤ 0.

Therefore, Sz = z since a+c > q.

Thus z is a fixed point of S. Similarly we can show that z is also a fixed point of T. Hence z is a common fixed point of T and S.

The uniqueness of the common fixed point can be easily shown by using (xi). This completes the proof of the theorem.

Omitting the term bd(x,Tx,u) and cd(y,Sy,u) from the left hand side of theorem 4.5 we get the following result as a corollary of the above Theorem.

Corollary 4.1 Let the self mappings T and S of a complete metric space (X, d) satisfy

d(Tx, Sy, u) ≤ q max{d(x, y, u), d(x, Tx, u), d(y, Sy, u)} for x, y, u ε X. Then T and S have a unique common fixed point.

5. Discussion and Conclusions

Our results obtained in 2-metric space with inquality condition involving rational terms have generalized the earlier resuls of Gahler, Rhoades, Iseki etc in terms of inquality condition as well as in terms of a pair of mappings and a family of mappings.