Sentence examples for modular space where from inspiring English sources

Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.

Corollary 3.3 Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.

Theorem 3.2 Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.

However, since is a modular space where for all, Theorem 4.8 implies the existence of a fixed point when we define mappings on a -a.e.

Let ω ∗ be another fixed point of T such that ρ ( c < ∞. Then we get ρ ( c = ρ ( c ( T ω − T ω ∗ ) ) ≤ ψ ( ρ ( c , which implies ρ ( c = 0, i.e., ω = ω ∗. We complete the proof of our theorem. □. The following corollary is an immediate consequence of Theorem 3.6. Corollary 3.7 [6]. Let X ρ be a ρ-complete modular space, where ρ satisfies the Fâtou property.

Theorem 3.18 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be a ( G ˜, ϕ, ψ ) ρ -contraction.

Theorem 3.11 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be a Hardy-Rogers type ( G ˜ ) ρ -contraction.

Theorem 3.6 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be a ( G ˜, A ) ρ -contraction.

Theorem 3.19 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be a ( G ˜, ϕ, ψ ) ρ -contraction, which is orbitally G ρ -continuous.

Theorem 3.12 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be a Hardy-Rogers type ( G ˜ ) ρ -contraction, which is orbitally G ρ -continuous.

Corollary 3.8 Let X ρ be a ρ-complete modular space endowed with a graph G, where ρ satisfies the Δ 2 -condition and let T : X ρ → X ρ be edge-preserving, the set X T nonempty and graph G be weakly connected and a C ρ -graph.