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Table 2 Calculations for maps in Example 2.11

From: Remarks on the fixed point problem of 2-metric spaces

x

a

θ ( r ) ⋅ d ( x , T x , a ) ≤ d ( x , y , a )

y

d ( T x , T y , a ) ≤ r ⋅ d ( x , y , a )

1

2

θ(r)⋅d(1,T 1,2)=0 ≤ d(1,y,2)

1

d(T 1,T 1,2)=0 ≤ r⋅d(1,1,2)

2

d(T 1,T 2,2)=0 ≤ r⋅d(1,2,2)

3

d(T 1,T 3,2)=0 ≤ r⋅d(1,3,2)

1

3

θ(r)⋅d(1,T 1,3)=θ(r)⋅d(1,2,3)≤d(1,y,3)

2

d(T 1,T 2,3)=0 ≤ r⋅d(1,2,3)

2

3

θ(r)⋅d(2,T 2,3)=0 ≤ d(2,y,3)

1

d(T 2,T 1,3)=0 ≤ r⋅d(2,1,3)

2

d(T 2,T 2,3)=0 ≤ r⋅d(2,2,3)

3

d(T 2,T 3,3)=0 ≤ r⋅d(2,3,3)

2

1

θ(r)⋅d(2,T 2,1)=θ(r)⋅d(2,3,1)≤d(2,y,1)

3

d(T 2,T 3,1)=0 ≤ r⋅d(2,3,1)

3

1

θ(r)⋅d(3,T 3,1)=0 ≤ d(1,y,3)

1

d(T 3,T 1,1)=0 ≤ r⋅d(3,1,1)

2

d(T 3,T 2,1)=0 ≤ r⋅d(3,2,1)

3

d(T 3,T 3,1)=0 ≤ r⋅d(3,3,1)

3

2

θ(r)⋅d(3,T 3,2)=θ(r)⋅d(3,1,2)≤d(2,y,3)

1

d(T 3,T 1,2)=0 ≤ r⋅d(3,1,2)