Let f,R,S,T X→X be four mappings such that f(X)⊆R(X)∪S X)∪T X) and dominating map f is a weak annihilator of R, S, and T. Suppose that for every three comparable elements x,y,z∈X, ψ G ( fx, fy, fz ) ≤ ψ M 1 ( x, y, z ) − φ M 1 ( x, y, z ), Open image in new window (56).
Let f,g,R,S X→X be four mappings such that f(X)⊆R(X) and g(X)⊆S X) and dominating maps f and g are weak annihilators of R and S, respectively.
Let f,g,h,T X→X be four mappings such that f(X)∪g(X)∪h(X)⊆T X) and dominating maps f, g, and h are weak annihilators of T. Suppose that for every three comparable elements x,y,z∈X, ψ G ( fx, gy, hz ) ≤ ψ M 2 ( x, y, z ) − φ M 2 ( x, y, z ), Open image in new window (57).
Let be a metric type space, and let be four mappings such that and, and suppose that at least one of these four subsets of is complete.
Let (f,g,R,S Xrightarrow X) be four mappings such that (f(X subseteq R(X)) and (g(X subseteq S X)).
Corollary 2.4 Let ( X, ⪯, G ) be a partially ordered G-complete G-metric space, and let f, g, h, R : X → X be four mappings such that f ( X ) ∪ g ( X ) ∪ h ( X ) ⊆ R ( X ).
Let (T:Xrightarrow X) and (G Xrightarrow X) be two mappings such that T is a G-isotone mapping and (T X subseteq G(X)).
Let f,T X→X be two mappings such that f(X)⊆T X), dominating map f is a weak annihilator of T. Suppose that for every three comparable elements x,y,z∈X, ψ G ( fx, fy, fz ) ≤ ψ M 4 ( x, y, z ) − φ M 4 ( x, y, z ), Open image in new window (59).
Assume f : X → X and γ : X × X → [ 0, ∞ ) be two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist ψ ∈ Ψ, α ∈ Φ α, and β ∈ Φ β such that ψ ( t ) − α ( s ) + β ( s ) > 0 for all t > 0 and s = t or s = 0. (2.12).
Assume f : X → X and γ : X × X → [ 0, ∞ ) be two mappings such that f is a non-decreasing and γ-admissible mapping. Assume that there exist ψ ∈ Ψ, α ∈ Φ α, and β ∈ Φ β such that ψ ( t ) − α ( s ) + β ( s ) > 0 for all t > 0 and s = t or s = 0 (2.1).
