On abstract degenerate neutral differential equations

We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.     

Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.     

Global Solutions for Abstract Differential Equations with Non-Instantaneous Impulses

In this note we study the existence of global solutions for a class of impulsive abstract differential equations with non-instantaneous impulses. Specifically, we establish the existence of mild solutions on \({[0, \infty)}\) and the existence of \({\mathcal{S}}\)-asymptotically \({\omega}\)-periodic mild solutions. Our results are based on the Hausdorff measure of non-compactness. Some applications involving partial differential equations are considered.     

Abstract quasilinear equations of second order with Wentzell boundary conditions

We introduce a general framework to treat abstract quasilinear equations of second order with Wentzell boundary conditions. As an example we study a wave equation for a second order quasilinear differential operator on with Wentzell boundary conditions.     

A new coercivity condition applied to semipositone integral equations with nonpositive,unbounded nonlinearities and applications to nonlocal BVPs

We consider the perturbed Hammerstein integral equation

$$\begin{aligned} y(t)=\gamma _1(t)H_1(\varphi _1(y))+\gamma _2(t)H_2(\varphi _2(y))+\lambda \int _0^1G(t,s)f(s,y(s)) \mathrm{d}s \end{aligned}$$

and demonstrate the existence of at least one positive solution in the case where this problem is semipostione—i.e., f is allowed to be negative on its domain. As applications of this abstract result, we demonstrate existence of at least one positive solution to a variety of boundary value problems in the ordinary differential equations setting as well as the setting of elliptic partial differential equations on annuli. Finally, two novel aspects of this study are that, first, our results allow for f to be strictly negative on its entire domain, and second, it can actually be the case that \(\lim _{y\rightarrow +\infty }f(t,y)=-\infty \), uniformly for \(t\in [0,1]\). In addition, these results can hold even if \(H_1\) and \(H_2\) are piecewise linear on their domains.

     

Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.     

Degenerate first-order inverse problems in Banach spaces

We are concerned with an inverse problem for a degenerate linear evolution equation of first-order. Both hyperbolic and parabolic cases will be considered. We indicate sufficient conditions for the existence and uniqueness of a solution. All the results can be applied to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

The Bolzano mean-value theorem and partial differential equations

We study the existence of solutions to abstract equations of the form $0=Au+F(u)$, $u\in K?E$, where A is an abstract differential operator acting in a Banach space E, K is a closed convex set of constraints being invariant with respect to resolvents of A and a perturbation F satisfies a certain tangency condition. Such problems are closely related to the so-called Poincaré–Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results should be regarded as infinite-dimensional variants of Bolzano and Miranda–Poincaré theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction–diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.     

<Emphasis FontCategory="NonProportional">pyomo.dae</Emphasis>: a modeling and automatic discretization framework for optimization with differential and algebraic equations

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differential equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.     

An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime

In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters. Employing an abstract formulation of differential equations on function spaces, our framework includes Schrödinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients. We illustrate the general mechanism on the basis of the first-order Lie splitting and the second-order Strang splitting method. Further, we specify the local error representation for a fourth-order splitting scheme by Yoshida. From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrödinger equations in the semi-classical regime with critical parameter 0<ε?1, provided that the time stepsize h is sufficiently smaller than \(\sqrt[p]{\varepsilon}\), where p denotes the order of the splitting method.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009,where the abstract theory of fold completeness in Banach spaces has been presented.Using obtained there abstract results,we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.Moreover,equations and boundary conditions may contain abstract operators as well.So,we deal,generally,with integro-differential equations,functional-differential equations,nonlocal boundary conditions,multipoint boundary conditions,integro-differential boundary conditions.We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach Lq-framework,in contrast to previously known results in the Hilbert L 2-framework.Some concrete mechanical problems are also presented.     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

Multiple positive solutions for time scale boundary value problems on infinite intervals

The study of dynamic equations on time scales is an area of mathematics. It has been created in order to unify the study of differential and difference equations. In this paper, we consider the time-scale boundary value problems

Property (A) of third-order advanced differential equations

In the paper we offer criteria for property (A) of the third-order nonlinear functional differential equation with advanced argument $(a(t)(x'(t))^\gamma )'' + p(t)f(x(\sigma (t))) = 0,$ , where $\mathop \smallint \limits^\infty a^{ - 1/\gamma } (s)ds = \infty $ . We establish new comparison theorems for deducing property (A) of advanced differential equations from that of ordinary differential equations without deviating argument. The presented comparison principle fill the gap in the oscillation theory.     

On the Possibility of Stabilization of Evolution Systems of Partial Differential Equations on ℝ n × [0, + ∞) Using One-Dimensional Feedback Controls

We establish conditions for the stabilizability of evolution systems of partial differential equations on by one-dimensional feedback controls. To prove these conditions, we use the Fourier-transform method. We obtain estimates for semialgebraic functions on semialgebraic sets by using the Tarski–Seidenberg theorem and its corollaries. We also give examples of stabilizable and nonstabilizable systems.     

Disjoint Hypercyclic and Disjoint Topologically Mixing Properties of Degenerate Fractional Differential Equations

The main purpose of this paper is to analyze the classes of disjoint hypercyclic and disjoint topologically mixing abstract degenerate (multi-term) fractional differential equations in Banach and Fréchet function spaces. We focus special attention on the analysis of abstract degenerate differential equations of first and second order, when we also consider disjoint chaos as a linear topological dynamical property. We provide several illustrative examples and applications of our abstract results.     

Global Wellposedness for a Certain Class of Large Initial Data for the 3D Navier–Stokes Equations

In this article, we consider a special class of initial data to the 3D Navier–Stokes equations on the torus, in which there is a certain degree of orthogonality in the components of the initial data. We showed that, under such conditions, the Navier–Stokes equations are globally wellposed. We also showed that there exists large initial data, in the sense of the critical norm ${B^{-1}_{\infty,\infty}}$ that satisfies the conditions that we considered.     

Multipoint problems for degenerate abstract differential equations

Multipoint boundary value problems for degenerate differential-operator equations of arbitrary order are studied. Several conditions for the separability in Banach-valued L _p -spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular, the sectoriality of this operator is established. As applications, the boundary value problems for degenerate quasielliptic partial differential equations and infinite systems of differential equations on cylindrical domain are studied.     

Regular Degenerate Separable Differential Operators and Applications

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in L _p -spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.