Existence of solutions for infinite systems of differential equations by densifiability techniques

Abstract

A novel technique to state the existence of solutions for certain infinite systems of differential equations is proposed. Our main tool will be the so called degree of nondensifiability, which seems to work under more general conditions than the measures of noncompactness. In fact, in our main result, the required conditions proposed for the existence of solutions of such system are more general than others required in most of the results based on such measures.

A fixed point result for mappings on the $\ell_{\infty}$-sum of a closed and convex set based on the degree of nondensifiability

“…It is worth saying that, despite the DND shares some properties (see [10,13]) similar to those of the so-called measures of noncompactness (MNC), see, for instance [1,4], the DND is not a MNC. Moreover, in [10,11,14] we have proved several fixed point results based on the DND which works under conditions that similar fixed point results based on the MNCs do not work.…”

“…X continuous, with C 2 B.X / closed and convex, such that f .B/ 2 B.X/ for each non-empty B C . For a given r 2 OE0; 1/, we will say that f is an r-DND-contraction if .f .B// Ä r .B/; for all non-empty, closed and convex B C .The following fixed point result was proved in[11, Corollary 3.3].Theorem 2.6. Let f W C !…”

A quantitative version of the Arzelà-Ascoli theorem based on the degree of nondensifiability and applications

“…for each x ∈ C. Then, T is continuous (in fact, affine) and T (C) ⊂ C. One can check (see [3, Example 2, p. 169]) that T is 1 2 -κ-contractive and 1-χcontractive, because of χ(T (C)) = χ(C) = 1/2, whereas in [11] we show that T is 1 2 -φ d -contractive. To end this section, note that the above examples show that φ d and the MNCs χ and κ are essentially different.…”

“…As it is shown in [10,11,12], we can use φ d as an alternative to the MNCs in certain fixed point problems, because in some cases it seems to work out under more general conditions than the MNCs. So, with a suitable combination of a Sadovskiȋ fixed point theorem type for φ d (see Theorem 2.11) and Theorem 3.2, we will prove in Section 4 a result regarding the existence of solutions of certain Volterra integral equations.…”

On Implicit Neutral Tempered ψ-Caputo Fractional Differential Equations with Delay via Densifiability Techniques

“…The development of this concept owes much to Cherruault [17] and Mora [39]. Very recently, García [20] established a novel fixed-point result based on the DND, showcasing its applicability in broader contexts compared to the Darbo fixed-point theorem (DFPT) and its recognized extensions. For a deeper understanding of the utility of DND in investigating the existence of solutions to specific differential or integral equations within certain Banach spaces, we suggest exploring the works [19,20,21,22].…”