Abstract

We explain the popularized concept of G-metric space and some current common properties of functions like compatibility and others. Particularly two self functions are specified and some most achieved fixed point theorems are generalized using some rational inequalities.

1. Introduction

Fixed point theory is a theory with enormous applications in multi branches of mathematics. Pioneering research work of this standard theory is published in 1922 by Stefan Banach ¹, and iterative method was used to fix the unique fixed points. This theory is very interesting to understand and is applicable in almost all fields of mathematics especially in ordinary differential equations. In 2004, Mustafa and Sims ⁶ introduced the concept of generalized metric space which is generalization of the ordinary metric space. It was sturdy generalization of metric spaces. The following concept of Ԍ-metric space and some basic definitions of Ԍ-metric space is given by some mathematicians (see, for example, ^{2, 3, 5, 6, 7, 8}).

Definition 1.1. Let be a set and : x x → [0, ∞) be a function satisfying the following axioms:

(Ԍ1) Ԍ(x, y, z) =0 if x = y = z,

(Ԍ2) 0 < Ԍ (x ,x, y) for all x, y ∈ X with x ≠ y

(Ԍ3) Ԍ (x, x, y) ≤ Ԍ (x, y, z), for all x, y, z ∈ with z ≠ y

(Ԍ4) Ԍ (x, y, z)= Ԍ (x, z, y)= Ԍ (y, z, x)

(symmetry in all three variables),

(Ԍ5) Ԍ (x, y, z)= Ԍ (x, a, a)+ Ԍ (a, y, z)

for all x, y, z, a ∈ , (rectangle inequality)

then the function Ԍ is called a generalized metric, or specifically a Ԍ -metric on and the pair (, Ԍ) is called a Ԍ -metric Space.

Definition 1.2. Let (, Ԍ) be a Ԍ-metric space and let {x_n} be a sequence of points in a point x in is said to be the limit of the sequence {x_n} if Ԍ(x, x_n, x_m) = 0, and one says that sequence {x_n} is Ԍ-convergent to x. Thus, if x_n x or x_n = x as n , in a Ԍ-metric space (, Ԍ), then for each ε> 0, there exists a positive integer N such that Ԍ(x, x_n, x_m) < ε for all m, n N.

Proposition 1.1. For a Ԍ-metric space (,Ԍ), the following are equivalent:

i. {x_n} is Ԍ-convergent to x,

ii. Ԍ(x_n, x_n, x) 0 as n ,

iii. Ԍ(x_n, x, x) 0 as n ,

iv. Ԍ(x_m, x_n, x) 0 as m, n .

Definition 1.3. Let ((, Ԍ), Ԍ) be a Ԍ-metric space. A sequence {x_n} is called Ԍ-Cauchy if, for each ε > 0, there exists a positive integer N such that

1. Ԍ(x_n, x_m, x_l) < ε, for all n, m, l N, i.e., if

2. Ԍ(x_n, x_m, x_l) 0 as n, m, l .

Definition 1.4. If (, Ԍ) and (^’, Ԍ^’) be two Ԍ-metric space and let f : (, Ԍ) (^’, Ԍ^’) be a function, then f is said to be Ԍ-continuous at a point x₀ if for given ε > 0, there exists > 0 such that for x, y and Ԍ(x₀, x, y) < implies Ԍ(f(x₀), f(x), f(y)) < ε. A function f is Ԍ-continuous at if and only if it is Ԍ-continuous at all x₀ or function f is said to be Ԍ-continuous at a point x₀ if and only if it is Ԍ-sequentially continuous at x₀, that is, whenever {x_n} is Ԍ-convergent to x₀, { f(x_n) } is Ԍ-convergent to f(x₀).

Definition 1.5. A Ԍ-metric space (, Ԍ) is said to be Ԍ-complete if every Ԍ-Cauchy sequence in (, Ԍ) is Ԍ-convergent in .

Proposition 1.2. A Ԍ-metric space (, Ԍ) is said to be Ԍ-complete if and only if (, d_Ԍ) is a complete metric space.

Proposition 1.3. Let (, Ԍ) be a Ԍ-metric space. Then, for any x, y, z and a in , it follows that:

(i) if Ԍ(x, y, z) = 0, then x = y = z,

(ii) Ԍ(x, y, z) Ԍ(x, x, y) + Ԍ(x, x, z),

(iii) Ԍ(x, y, y) 2Ԍ(y, x, x),

(iv) Ԍ(x, y, z)Ԍ(x, a, z) + Ԍ(a, y, z),

(v) Ԍ(x, y, z) (Ԍ(x, y, a) + Ԍ(x, a, z) + Ԍ(a, y, z)),

(vi) Ԍ(x, y, z) Ԍ(x, a, a) + Ԍ(y, a, a) + Ԍ(z, a, a).

Definition 1.6. Let f and g be two self mappings on a metric space (, Ԍ). The mappings f and g are said to be compatible if there exist a sequence {} such that =0, or =0, whenever {x_n} is a sequence in such that

Definition 1.7. Let f and g be two self mappings on a metric space (, Ԍ). The mappings f and g are said to satisfy the property (E.A), if there exist a sequence {x_n} in such that

for some x .

Definition 1.8. Let f and g be two self mappings on a metric space (, Ԍ). The mappings f and g are said to satisfy the common limit range () property if there exist a sequence {x_n } in such that =gx for some x .

Definition 1.9. Two maps are said to be weakly compatible if they commute at coincidence points.

2. Main Result

This is the result for unique fixed point for pair of weak compatible maps.

Theorem 2.1. Let (, Ԍ) be a complete Ԍ -metric space and f, g : be the self mapping on (,Ԍ) satisfying the following conditions:

(1.1) f() Ԍ()

(1.2) f or g is continuous,

(1.3)

for all x, y, z , where 0 ≤ < Including the fact that f and g are compatible maps then f and g have a unique common fixed point in

Proof. Let x₀ be a random selection, then by (1.1), there may exist one point or one may choose some point x₁ such that fx₀=gx_1. In general one can choose x_n+1 such that

y_n = fx_n = gx_n+1, n=0, 1, 2, 3.

From (1.3), we have

Using the preposition Ԍ(x, y, y) 2Ԍ(y, x, x), we have

Case 1.

If the max is taken as,

Then using (1.3), we Ԍet

Repetitively we have

Hence for every natural n and m where n < m, we have

For limiting values of n and m as infinity, we have = 0. Therefore {} is a Ԍ-Cauchy sequence.

Case 2.

If

Then (1.3) becomes,

Which is a contradiction because 0 ≤ <

Hence the sequence is again a G-Cauchy sequence.

Case 3.

If the max is taken as,

Then using (1.3), we Ԍet

Where , it is obvious that p < 1. Also

0 ≤ <

It is clear from case 1 that it is again a G-Cauchy sequence in G-metric space.

For all the cases we have G-Cauchy sequence, also it is mentioned that the given G-metric space is complete.

Therefore the Cauchy sequence is convergent and the point of convergence belongs to the given metric space.

i.e.

We have

Because of the continuity of one of the two maps f or g, let us assume that the map g is continuous and hence we have,

Additionally, f and g are compatible, hence

gives

Now, from (2.1), we have

taking n approaches to infinity,

Case 1

It is a contradiction since

Case 2

which is a contradiction since

Hence =, next we are to show that g μ =f μ = μ, to prove this substitute x =, y = z = μ in (1.3)

If n approaches to infinity,

Ԍ(,f,f)=, a contradiction as

.

Hence =. Thus is a fixed point of f and g. by the compatibility of f and g, it is obvious that fg= gf=.

Uniqueness

It is to be assume that that are two distinct fixed points

of f and g then from (1.3). we have

hence since <

Theorem 2.2. Let (, Ԍ) be a complete Ԍ-metric space and f, g : be weakly compatible self mapping on (,Ԍ) satisfying (1.1) and (1.3) also

(1.4) one of the subspace f(X) or g(X) is complete then f and g have unique common fixed point.

proof

it is clear from theorem (2.1) that {}is G-Cauchy sequence, without loss of generality it can be assumed that g(X) is complete. Then the subsequence is going to have a limit in g(X). let it be .let then

g=. Since {} is G-Cauchy sequence, containing a convergent subsequence therefore the sequence is also

convergent. It is to prove that =.

Let x=, y=and z=

From (1.3), we have

Taking n approaches to infinity, we have

= as < ,

i.e., now. It is to be shown that = . Suppose that now put on setting x= y = z =in (1.3), we have

Using the prepositions (1.3) and using the fact that f =g, we have

Clearly we have,

It proves that , i.e., is a common fixed point. Uniqueness is confirmed by the uniqueness of theorem 2.1.

Theorem 2.3. Let (, Ԍ) be a complete Ԍ-metric space and f, g : be the self mapping on (,Ԍ) satisfying the conditions (1.3) and the following conditions

(1.6) f and g satisfy property (E.A)

(1.5) g() is complete subspace of

Then f and Ԍ have a unique common fixed point in provided f and Ԍ are compatible maps.

Proof. Since e property (E.A) is satisfied by f and g, therefore, there exists one sequence in . Such that

for some . By the closeness of g() in . = for some

implies =.

We declare that

Using the equation (1.3)

which is a contradiction because of range of hence

Hence is a fixed point of f and g.

Theorem 2.4. Let (, Ԍ) be a complete Ԍ-metric space and f, g: be the self mapping on (, Ԍ) satisfying the following conditions (1.2) and (1.3) and the condition

(1.4) any one of the subspaces f () or g() is complete.

Proof. Since the self maps f and g satisfies the common limit range property of type g i.e. (CLR_g), therefore there exists a sequence {} in in such a way

for some .

We declare that f=g= using the equation (1.3) and taking limiting value of n as infinity, we have

This equation is a contradiction because of the values of , hence

Thus is a common fixed point of f and g.

References