Existence criteria for integral equations in banach spaces

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A new exploration on existence of Sobolev‐type Hilfer fractional neutral integro‐differential equations with infinite delay (2020, unknown): “15 [48, 49] Assume {un}n=1 is a sequence of Bochner integrable functions from I →X with ||un(t) || ≤ ε(t), for all t∈ I and every n≥ 1, where ε∈L(1)(I, R), then ρ(t)= β({un(t) : n≥ 1})∈ L(1)(I, R) and satisfies β ({∫ t 0 un(τ)dτ ∶ n ≥ 1 })≤2 ∫ t 0 ρ(τ)dτ.”
Existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order 1 < r < 2 (2020, background): “Lemma 2.10 [44] Assume that { N ( 𝜁 )} 𝜁 ∈ R be the strongly continuous cosine family in 𝒵 satisfying . for any bounded subset 𝒯 ⊆ 𝒟 ( 𝒬 ) , where Y is Banach space .”