Probabilistic ( Quasi ) metric Versions for a Stability Result of Baker Dorel

and Applied Analysis 3 Definition 2.3 see 16 . Let X,F, T be a probabilistic metric space. A mapping f : X → X is said to be a Sehgal contraction or B-contraction if the following relation holds: Ff p f q kt ≥ Fpq t , ( p, q ∈ X, t > 0. 2.1 Theorem 2.4 see 17 . Let X,F, T be a complete probabilistic metric space with T of Hadžić-type and f : X → X be a B-contraction. Then f has a fixed point if and only if there is p ∈ X such that Fpf p ∈ D . If Fpf p ∈ D , then p∗ : limn→∞f p is the unique fixed point of f in the set Y {q ∈ X : Fpq ∈ D }. The following lemma completes Theorem 2.4 with an estimation relation, in the case T TM. Lemma 2.5 see 18 . Let X,F, TM be a complete probabilistic metric space and f : X → X be a k − B contraction. Suppose that Fpf p ∈ D and let p∗ limn→∞f p . Then Fpp∗ t 0 ≥ Fpf p 1 − k t , ∀t > 0. 2.2 This lemma can be extended to the case of probabilistic metric spaces under a continuous t-norm of H-type. Lemma 2.6. Let X,F, T be a complete probabilistic metric space, with T a continuous t-norm of H-type and bn n be a strictly increasing sequence of idempotents of T . Suppose f : X → X is a B-contraction with Lipschitz constant k ∈ 0, 1 . If there exists p ∈ X such that Fpf p ∈ D , then p∗ limn→∞ f p is the unique fixed point of f in the set { q ∈ X : Fpq ∈ D } . 2.3 Moreover, if t > 0 is so that Fpf p 1 − k t ≥ bn, then Fpp∗ t 0 ≥ bn. Proof. We have to prove only the last part of the theorem. We show by induction on m that Fpf p 1 − k s ≥ bn implies Fpfm p s ≥ bn, for all m ≥ 1. The case m 1 is obvious. Now, suppose that Fpfm p s ≥ bn. Then Fpfm 1 p s ≥ T ( Fpf p 1 − k s , Ff p fm 1 p ks ) ≥ TFpf p 1 − k s , Fpfm p s )