Sentence examples for below notions from inspiring English sources

In this study, we introduce the below notions of the weaker Meir-Keeler function ϕ : [0, ∞) → [0, ∞) and stronger Meir-Keeler function ψ : [0, ∞) → [0, 1). Definition 5 We call ϕ : [0, ∞) → [0, ∞) a weaker Meir-Keeler function if the function ϕ satisfies the following condition ∀ η > 0 ∃ δ > 0 ∀ t ∈ 0, ∞ η ≤ t < δ + η ⇒ ∃ n 0 ∈ ℕ ϕ t n 0 < η.

We first define the below notion of φ-mapping.

We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler ( ψ x, φ ) -contraction.

By using the notions of generalized metrics and tυs-cone metrics, we introduced the below notion of tυs-generalized-cone metrics.

In 2012, Chen [17] introduced the below notion of cyclic orbital stronger Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

Then A ∩ B is nonempty and f has a unique fixed point in A ∩ B. Chen [17] also introduced the below notion of cyclic orbital weaker Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

We first introduce the below notion of stronger Meir-Keeler type mapping in a metric space.

Using the notions of the generalized cyclic contraction [1] and stronger Meir-Keeler type mapping, we introduce the below notion of generalized cyclic stronger Meir-Keeler contraction.

Using the notions of the generalized cyclic contraction and weaker Meir-Keeler type mapping, we introduce the below notion of generalized cyclic weaker Meir-Keeler contraction.

Using the notions of the cyclic orbital contraction (see, Definition 1) and stronger Meir-Keeler type mapping (see, Definition 2), we introduce the below notion of cyclic orbital stronger Meir-Keeler contraction.

Let E be a locally convex Hausdorff tυs with its zero vector θ, P a proper, closed, and convex pointed cone in E with intP ≠ ϕ and ≼ a partial ordering with respect to P. We introduce the below notion of the t υ s  -  H  - cone metric with respect to tυs-G-cone metric G. Definition 8 Let (X, G) be a tυs-G-cone metric space with a solid cone P and let be a collection of nonempty subsets of X.