where k ∈ [ 0, 1 ). Then T has a unique fixed point. Remark 1.1 We notice that condition (2) implies condition (3). The converse is true only if k ∈ [ 0, 1 2 ). For details see [4].
where k ∈ [ 0, 1 ). Then T has a unique fixed point. Remark 18 The condition (1) implies the condition (2). The converse is true only if k ∈ [ 0, 1 2 ). For details, see [36].
where k ∈ [ 0, 1 ). Then T has a unique fixed point. Remark 1.1 We notice that the condition (2) implies the condition (3). The converse is true only if k ∈ [ 0, 1 2 ). For details, see [5].
for all x, y ∈ X, where φ : [ 0, + ∞ ) → [ 0, + ∞ ) is continuous with φ − 1 ( { 0 } ) = { 0 }. Then T has a unique fixed point. Remark 5 An additional nondecreasing function ψ could be added in (3.6) to become ψ ( d ( T x, T y ) ) ≤ ψ ( d ( x, y ) ) − φ ( d ( x, y ) ). However, it was shown in [40] that it is redundant, hence we will not use it here.
Then f has a unique fixed point in X. Remark 3.8 If ψ ( t ) = k t, where k ∈ ( 0, 1 ) in Corollary 3.7, we get the Banach contraction principle.
Then f has a unique fixed point in X. Remark 2.6 Corollary 2.5 generalizes and extends the corresponding results in Mustafa and Sims [[8], Theorem 2.1].
Then Φ has a unique fixed point in X. Remark 2.3 If C is a nonempty closed (convex) subset of a complete metric space ( X, d ), then the conclusion of Theorem MK is still true.
∀ x, y, z ∈ X, where k ∈ [ 0, 1 2 ), m ∈ N, then f has a unique fixed point in X. Remark 2.13 Corollary 2.12 generalizes and extends the corresponding results in Mustafa and Sims [[8], Corollary 2.3].
Then there exist x ∈ X such that x = F ( x, x ) ; that is, F admits a unique fixed point in X. Remark 3.4 Comparing Theorem 3.2 with Theorem 3.1 in [8], we can see that Theorem 3.2 is a genuine generalization of Theorem 3.2.
By Theorem 3.1, T has a unique fixed point in X. Remark 3.3 Since in Example 3.1 the underlying solid cone P in the Banach algebra is not normal, we can conclude that any of the theorems in [15] cannot cope with Example 3.1, which shows that the main results without the assumption of normality in this paper are meaningful.
The strict monotonicity of and Remark 4.7 yields a unique.
