# A common fixed point theorem using implicit relation and property (E.A) in metric spaces

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### "A common fixed point theorem using ..." refers background in this paper

...1) compatible [3] if limn→∞d(ASxn, SAxn) = 0, (1....

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...The concept of weakly commuting mappings of Sessa [7] is sharpened by Jungck [3] and further generalized by Jungck and Rhoades [4]....

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...Introduction The concept of weakly commuting mappings of Sessa [7] is sharpened by Jungck [3] and further generalized by Jungck and Rhoades [4]....

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642 citations

### "A common fixed point theorem using ..." refers background in this paper

...Mappings A and S are said to be weakly commuting[7] if d(SAx, ASx) ≤ d(Ax, Sx), for all x ∈ X, (1....

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...The concept of weakly commuting mappings of Sessa [7] is sharpened by Jungck [3] and further generalized by Jungck and Rhoades [4]....

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...Introduction The concept of weakly commuting mappings of Sessa [7] is sharpened by Jungck [3] and further generalized by Jungck and Rhoades [4]....

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525 citations

### "A common fixed point theorem using ..." refers background or methods or result in this paper

...Therefore our main Theorem 4.1 extends Theorem B of Aamri and Moutawakil, since they considered ψ to be nondecreasing and such that 0 < ψ(t) < t,t > 0, which clearly implies ψ(0) = 0....

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...([1]) Let A, B, S and T be self-mappings of a metric space (X, d) such that (a) d(Ax,By) ≤ φ(max {d(Sx, Ty), d(By, Sx), d(By, Ty)}, ∀(x, y) ∈ X(2), where φ : R+ → R+ is a non-decreasing function on R+ such that 0 < φ(t) < t, for each t ∈ (0,∞)....

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...Compare it with condition (ii) in Theorem 1 [1]....

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...Our theorem improves and generalizes the main Theorems of Popa [6] and Aamri and Moutawakil [1]....

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...A) of [1] and implicit relation of [6] to unify under property(E....

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132 citations

### "A common fixed point theorem using ..." refers background in this paper

...For a related problem, see Theorem 1 and Corollary 3 in [2]....

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63 citations

### "A common fixed point theorem using ..." refers background in this paper

...F (d(Ax, By), d(x, y), d(Ax, x), d(By, y), d(By, x), d(Ax, y)) ≤ 0, for all x, y ∈ X, where F must satisfy all the conditions of implicit relation (see [8])....

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...≤ ψ(G(max{d(x, y), d(Ax, x), d(By, y), 1 2 (d(By, x) + d(Ax, y))})), for all x, y ∈ X, which is similar to condition in Theorem 1 [8]....

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...As a particular case, we can consider G(u) = ∫ u 0 φ(s) ds, where φ ≥ 0 is Lebesgue-integrable and such that ∫ 2 0 φ(t)dt > 0, for every 2 > 0, and ψ(t) = kt, with 0 ≤ k < 1 (see [8])....

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