Abstract
In this paper we introduce the notion of proximal ρ-normal structure of pair of ρ-admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.
Highlights
Fixed point theory is powerful tools in different fields such as differential equations, dynamical systems, optimal control, and many other scientific branches; it treats equations of type Tx = x where T : X → X is a map of a nonempty set to itself.Let A, B ⊂ X and T a cyclic mapping on A ∪ B; that is, T : A ∪ B → A ∪ B and T(A) ⊆ B, T(B) ⊆ A; in this case, T does not necessarily possess a fixed point if, for instance, A ∩ B = 0
In this paper we introduce the notion of proximal ρ-normal structure of pair of ρ-admissible sets in modular spaces
Recall that a map T : A ∪ B → A ∪ B is called relatively nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x ∈ A and y ∈
Summary
On the other hand, Eldred et al in [7], after generalizing the geometric concept of normal structure for a pair of subsets (A, B) in Banach space introduced earlier by Brodski and Milman (see [8]), proved the existence of best proximity points for relatively nonexpansive mappings in Banach space. The best proximity points results was investigated by many authors and found extension and generalization for different class of mappings and spaces; for a recent account of the theory we refer the reader to [11,12,13,14,15,16,17,18]. We give existence results of a best proximity point in the setting of proximal ρ-admissible subsets in modular space.
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