Abstract
In this work we are interested in the generalization of coincidence point and fixed point theorem for a 4-tuple of mappings satisfying a new type of implicit relation in complex valued $b-$metric spaces.
Highlights
Definition 1.4. [16] let f : C → C be a given mapping, we say that f is a non-decreasing mapping with respect if for every x, y ∈ C, x y implies f x f y
For all x, y ∈ X, where φ ∈ Fs, if one of F X, GX, f X or gX is a complete subspace of X, Cf,F = ∅, Cg,G = ∅ and f (Cf,F ) = F (Cf,F ) = g(Cg,G) = G(Cg,G) = {f x} = {gy} = {.}, for all x ∈ Cf,F, y ∈ Cg,G
For all x, y ∈ X, where φ ∈ Fs, if one of F X, GX, or X is a complete subspace of X, F and G have a unique common fixed point
Summary
Definition 1.4. [16] let f : C → C be a given mapping, we say that f is a non-decreasing mapping with respect if for every x, y ∈ C, x y implies f x f y. Let s ≥ 1 and Fs be the set of all functions φ (t1, t2, ..., t6) : C6+ −→ C satisfying the following conditions: (φ1) φ continuous on C6+, (φ2) ∃α, β ∈ R+ such that α + 2sβ < 1, ∀u, v, w ∈ C+ : φ (u, v, u, v, 0, w) 0 or φ (u, v, v, u, w, 0) 0 ⇒ |u| ≤ α|v| + β|w|, (φ3) ∃γ, μ ∈ R+ such that sγ + s2μ < 1, ∀u, v, w ∈ C+ : φ (u, 0, v, 0, 0, w) 0 ⇒ |u| ≤ γ|v| + μ|w|, (φ4) φ (u, 0, u, 0, 0, u) 0 or φ (u, u, 0, 0, u, u) 0 ⇒ u = 0.
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