Fixed point theory (locally (α,β,ψ) dominated contractive condition)

     (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.