and Applied Analysis 3 Definition 1.10 see 5 . Let X,G be a G-metric space and f, g : X → X be given mappings. We say that f is a generalized weakly G-contraction mapping of type B with respect to g if for all x, y, z ∈ X, the following inequality holds: ψ ( G ( fx, fy, fz )) ≤ ψ ( G ( gx, gx, fy ) G ( gy, gy, fz ) G ( gz, gz, fx ) 3 ) − φGgx, gx, fy, Ggy, gy, fz, Ggz, gz, fx, 1.2