Abstract
In this paper, we introduce first the concept of a Pompeiu-Hausdorff b-metric-like space. We also establish some best proximity points and stability results for controlled proximal contractive set valued mappings in the class of b-metric-like spaces and partial b-metric spaces. Moreover, we provide some examples and many nice consequences from our obtained results.
Highlights
Introduction and preliminariesMarkins [ ] and Nadler [ ] initiated the study of fixed point theorems for set valued operators
We are interested first to initiate the concept of a Pompeiu-Hausdorff b-metric-like and to prove some best proximity points and stability results
Hussain et al [ ] established some fixed point theorems in the setting of b-metric-like spaces
Summary
Suppose that (i) A = ∅; (ii) for each x ∈ A , we have Tx ⊆ B ; (iii) the pair (A, B) satisfies the weak P-property; (iv) there exists x ∈ A such that √T is a proximal contraction on Bσ (x , r) and δσ (Tx , {x }) + σ (A, B) ≤ ( – α)r. Suppose that (ii) for each x ∈ A , we have Tx ∈ B ; there σ (x , exists x Tx ) + σ (∈AA, B )s≤uc hs t –hsa (t T–i√s aαsp)rro;ximal contraction Proof It suffices to take s = and T as a single-valued mapping in Theorem. Let (X, σ ) be a complete b-metric-like space, r > , and T : X → Cb(X) be a multi-valued mapping
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