We analyze the strength of Helly’s selection theorem (HST), which is the most important compactness theorem on the space of functions of bounded variation (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W , compatible with the—so called—weak∗ topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß prin...

Abstract In this article we combine the study of solutions PDEs with asymptotic properties via compactification domain. We define new spaces functions on which equations, prove a version Ascoli–Arzelà Theorem, develop fixed point index results necessary to existence and multiplicity in these also illustrate applicability theory an example.

This article studies the existence theory of an innovation type generalized proportional fractional differential equations with assistance technique Kuratowski measure on noncompactness combined fixed-point theorem Mönch. Also, we use Lebesgue’s dominated convergence and Arzelá–Ascoli fixed point uniqueness results. An application is also presented by employing two illustrative iks, which enric...

Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether ...

‎In this paper‎, ‎first‎, ‎we investigate the construction of compact sets of $ C^k$ and $ C_0^k$‎ ‎by proving ``$C^k‎, ‎C_0^k-version$‎" ‎of Arzel`{a}-Ascoli theorem‎, ‎and then introduce new measures of noncompactness on these spaces‎. ‎Finally‎, ‎as an application‎, ‎we study the existence of entire solutions for a class of the functional integral-differential equations by using Darbo's fixe...

In this paper, we investigate infinite interval problems for the fractional evolution equations with Hilfer derivative. By using generalized Ascoli–Arzelà theorem and some new techniques, prove existence of mild solutions when semigroup is compact as well noncompact. addition, an example provided to illustrate results.