Abstract
Fractals have gained great attention from researchers due to their wide applications in engineering and applied sciences. Especially, in several topics of applied sciences, the iterated function systems theory has important roles. As is well known, examples of fractals are derived from the fixed point theory for suitable operators in spaces with complete or compact structures. In this article, a new generalization of Hausdorff distance on , is a class of all nonempty compact subsets of the metric space ( , ). Completeness and compactness of are analogously obtained from its counterparts of ( , ). Furthermore, a fractal is presented under a finite set of generalized -contraction mappings. Also, other special cases are presented.
Highlights
If any Cauchy sequence < pn > ⊂ Ω converge to p ∈ Ω Ω is called complete ∅-metric space.”
We find that supq∈B d∅ (p, B) = supq∈Bd∅ (20, B) = d∅ (20, 50) = 302 = 900
We prove the conditions (i-iii) in Definition 1.1 are satisfied
Summary
Let Ω, ∅ and d∅ in Proposition 2.3 and {An} be a sequence in H (Ω) and A = {p ∈ Ω : ∃{pn} converges to p and pn ∈ An, ∀n}. To prove that A ∈ H (Ω), it only remains to prove that A is totally bounded To get those results, the following proposition is required. Let Ω, ∅ and d∅ in Proposition 2.3 and {Cn} be a sequence of totally bounded sets in Ω and C ⊆ Ω. By Proposition 2.5, A is totally bounded, A is complete since it is closed subset of a complete space,. Let ǫ > 0, to show that {An} converges to A ∈ H (Ω), we must prove ∃r > 0 ∋ d∅ (An, A) < ǫ, ∀ n ≥ r, A ⊆ An + ǫ and An ⊆ A + ǫ by Proposition 2.2.
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