One of the most important theorems in Ordinary Di↵erential Equations is Picard’s Existence and Uniqueness Theorem for first-order ordinary di↵erential equations. Why is Picard’s Theorem so important? One reason is it can be generalized to establish existence and uniqueness results for higher-order ordinary di↵erential equations and for systems of di↵erential equations. Another is that it is a g...

Abstract General nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem existence uniqueness solution to second-type value given. Finally, for numerical solution, two different approaches illustrated some examples included...

In this paper, we study the global stability and the existence of almost periodic solution of high-order Hopfield neural networks with distributed delays of neutral type. Some sufficient conditions are obtained for the existence, uniqueness and global exponential stability of almost periodic solution by employing fixed point theorem and differential inequality techniques. An example is given to...

Considering the ω-distance function defined by Kostić in proximity space, we prove Matkowski and Boyd–Wong fixed-point theorems space using ω-distance, provide some examples to explain novelty of our work. Moreover, characterize Edelstein-type theorem compact space. Finally, investigate an existence uniqueness result for solution a kind second-order boundary value problem via obtained Matkowski...

This paper is devoted to studying the existence of positive solutions for a class of higher-order neutral dynamic equations on time scales. First, all various types of the eventually positive solutions are given by means of their asymptotic behaviors. Then, by means of Kranoselskii’s fixed point theorem, the existence criteria are provided for these type solutions. Some examples in the last sec...

We study the gradient flow of the Riemannian functional F(g) := ∫ M |Rm| . This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that, as with the Ricci flow, the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L2 derivative estimates of Bernstein-Bando-...