Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays

<abstract><p>We consider the existence and stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. In order to overcome the incompleteness of the space composed of Weyl almost periodic functions, we first obtain the existence of a bounded continuous solution of the system under consideration by using the fixed point theorem, and then prove that the bounded solution is Weyl almost periodic by using a variant of Gronwall inequality. Then we study the global exponential stability of the Weyl almost periodic solution by using the inequality technique. Even when the system we consider degenerates into a real-valued one, our results are new. A numerical example is given to illustrate the feasibility of our results.</p></abstract>

Global existence and asymptotic behavior of solutions to fractional ( p , q )-Laplacian equations

The aim of this paper is to consider the following fractional parabolic problem u t + ( − Δ ) p α u + ( − Δ ) q β u = f ( x , u ) ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p α is the fractional p-Laplacian with 0 < α < 1 < p < ∞, ( − Δ ) q β is the fractional q-Laplacian with 0 < β < α < 1 < q < p < ∞, r > 1 and λ > 0. The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

Continuous Dependence on the Heat Source of 2D Large-Scale Primitive Equations in Oceanic Dynamics

In this paper, we consider the initial-boundary value problem for the two-dimensional primitive equations of the large-scale oceanic dynamics. These models are often used to predict weather and climate change. Using the differential inequality technique, rigorous a priori bounds of solutions and the continuous dependence on the heat source are established. We show the application of symmetry in mathematical inequalities in practice.

Finite-time stabilization and H∞ control of Port-controlled Hamiltonian systems with disturbances and saturation

The finite-time stabilization and finite-time H∞ control problems of Port-controlled Hamiltonian (PCH) systems with disturbances and input saturation (IS) are studied in this paper. First, by designing an appropriate output feedback, a strictly dissipative PCH system is obtained and finite-time stabilization result for nominal system is given. Second, with the help of the Hamilton function method and truncation inequality technique, a novel output feedback controller is developed to make the PCH system finite-time stable when IS occurs. Further, a finite-time H∞ controller is designed to attenuate disturbances for PCH systems with IS, and sufficient conditions are presented. Finally, a numerical example and a circuit example are given to reveal the feasibility of the obtained theoretical results.

Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks

In this paper, the fixed-time synchronization analysis is addressed for a class of discontinuous neutral-type neural networks. The focus is mainly on the design of useful control laws such that the constructed error system converges to zero in a fixed time. The major difficulty is to cope with the discontinuous neuron activations, D operators, time-varying discrete, and distributed delays simultaneously. To accomplish the target, a new and effective framework is firstly established. By means of functional differential inclusions theory, inequality technique and Lyapunov–Krasovskii functional, novel discontinuous feedback controllers are designed and some new verifiable algebraic criteria are derived to design the control gains. In contrast to the existed results on the neutral-type neural networks, the theoretical results of this paper are more general and rigorous. Finally, numerical examples and simulations are presented to illustrate the correctness of the main results.

Dynamical analysis of a stochastic three-species predator–prey system with distributed delays

A stochastic two-prey one-predator system with distributed delays is proposed in this paper. Firstly, applying the linear chain technique, we transform the predator–prey system with distributed delays to an equivalent system with no delays. Then, by use of the comparison method and the inequality technique, we investigate the stability in mean and extinction of species. Further, by constructing some suitable functionals, using M-matrix theory and three important lemmas, we establish sufficient conditions assuring the existence of distribution and the attractivity of solutions. Finally, some numerical simulations are given to illustrate the main results.

Oscillation Criteria for Third-Order Nonlinear Neutral Dynamic Equations with Mixed Deviating Arguments on Time Scales

Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the inequality technique, new oscillation criteria are established, which improve and generalize related results in the literature. Several examples are given to illustrate the main results.

New methods in spectral theory of N-body Schrödinger operators

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

Blow-up results of the positive solution for a class of degenerate parabolic equations

This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.

On the Oscillation of Conformable Impulsive Vector Partial Differential Equations

In this paper, we study the oscillations of a class of conformable impulsive vector partial functional differential equations. For this class, our approach is to reduce the multi-dimensional oscillation problems to that of one dimensional impulsive delay differential inequalities by applying inner product reducing dimension method and an impulsive differential inequality technique. We provide an example to illustrate the effectiveness of our main results.