(Locally (α,β,ψ) dominated contractive condition) Let α,β:X×X→[0,+∞), r>0, x₀∈X, ψ∈Ψ, (X,d) be an (α,β)-complete metric space and S,T:X→X are locally (α,β)-continuous and (α,β)-dominated triangular mappings on B(x₀,r). We say that the pair (S,T) satisfies the locally (α,β,ψ) dominated contractive condition on B(x₀,r), if
for all x,y∈B(x₀,r) with α(x,y)≥β(x,y) or α(y,x)≥β(y,x), and
∑_{i=0}^{j}ψ^{i}(d(x₀,Sx₀))≤r for all j∈ℕ∪{0}. #1.2
2. Result for locally (α,β,ψ) dominated contractive condition Theorem 2.1 Let α,β:X×X→[0,+∞), r>0, x₀∈X, ψ∈Ψ, (X,d) be an (α,β)-complete metric space and S,T:X→X. If the following conditions hold: 1) S and T are (α,β)-continuous, 2) The pair (S,T) satisfies the locally (α,β,ψ) dominated contractive condition on B(x₀,r), 3) If x and y belongs to set of common fixed points of S and T, then α(x,y)≥β(x,y). Then S and T have a unique common fixed point.
According to 1.1 condition I want to solve this theorem in families of mappings.
(Locally (α,β,ψ) dominated contractive condition)
Let α,β:X×X→[0,+∞), r>0, x₀∈X, ψ∈Ψ, (X,d) be an (α,β)-complete metric space and S,T:X→X are locally (α,β)-continuous and (α,β)-dominated triangular mappings on B(x₀,r). We say that the pair (S,T) satisfies the locally (α,β,ψ) dominated contractive condition on B(x₀,r), if
d(Sx,Ty)≤ψ(max{d(x,y),d(x,Sx),d(y,Ty),((d(x,Ty)+d(y,Sx))/2)}), #1.1
for all x,y∈B(x₀,r) with α(x,y)≥β(x,y) or α(y,x)≥β(y,x), and
∑_{i=0}^{j}ψ^{i}(d(x₀,Sx₀))≤r for all j∈ℕ∪{0}. #1.2
2. Result for locally (α,β,ψ) dominated contractive condition
Theorem 2.1 Let α,β:X×X→[0,+∞), r>0, x₀∈X, ψ∈Ψ, (X,d) be an (α,β)-complete metric space and S,T:X→X. If the following conditions hold:
1) S and T are (α,β)-continuous,
2) The pair (S,T) satisfies the locally (α,β,ψ) dominated contractive condition on B(x₀,r),
3) If x and y belongs to set of common fixed points of S and T, then α(x,y)≥β(x,y).
Then S and T have a unique common fixed point.
According to 1.1 condition I want to solve this theorem in families of mappings.