Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:

t_∂u−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Periodic solutions of a nonlinear second-order differential equation with deviating argument

With the help of the coincidence degree continuation theorem, the existence of periodic solutions of a nonlinear second-order differential equation with deviating argument

x^″(t)+f₁(x(t))x^′(t)+f₂(x(t))(x^′(t)⁾²+g(x(t−τ(t)))=0,     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

A new analytical method for the linearization of dynamic equation on measure chains

In this paper, by introducing the concept of topological equivalence on measure chain, we investigate the relationship between the linear system xΔ=A(t)x and the nonlinear system xΔ=A(t)x+f(t,x). Some sufficient conditions are obtained to guarantee the existence of a equivalent function H(t,x) sending the (c,d)-quasibounded solutions of nonlinear system xΔ=A(t)x+f(t,x) onto those of linear system xΔ=A(t)x. Our results generalize the Palmer's linearization theorem in [K.J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl. 41 (1973) 753-758] to dynamic equation measure chains. In the present paper, we give a new analytical method to study the topological equivalence problem on measure chains. As we will see, due to the completely different method to investigate the topological equivalence problem, we have a considerably different result from that in the pioneering work of Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191]. Moreover, we prove that equivalent function H(t,x) is also ω-periodic when the systems are ω-periodic. Hilger [S. Hilger, Generalized theorem of Hartman-Grobman on measure chains, J. Aust. Math. Soc. Ser. A 60 (2) (1996) 157-191] never considered this important property of the equivalent function H(t,x).     

Max-algebraic attraction cones of nonnegative irreducible matrices

It is known that the max-algebraic powers A^r of a nonnegative irreducible matrix are ultimately periodic. This leads to the concept of attraction cone Attr(A, t), by which we mean the solution set of a two-sided system λ^t(A)A^r⊗x=A^r+t⊗x, where r is any integer after the periodicity transient T(A) and λ(A) is the maximum cycle geometric mean of A. A question which this paper answers, is how to describe Attr(A,t) by a concise system of equations without knowing T(A). This study requires knowledge of certain structures and symmetries of periodic max-algebraic powers, which are also described. We also consider extremals of attraction cones in a special case, and address the complexity of computing the coefficients of the system which describes attraction cone.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:

(t_∂−A)u+x_∂q(u)=f(u)+g(u)Ft,x     

Oscillation of neutral differential equation with positive and negative coefficients

In this paper, we provide oscillation properties of every solution of the neutral differential equation with positive and negative coefficients

[x(t)−R(t)x(t−r)^]′+P(t)x(t−τ)−Q(t)x(t−σ)=0,     

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

We consider the boundary value problems: (?p(x^′′(t)))+q(t)f(t,x(t),x(t−1),x^′(t))=0, ?p(s)=|s|^p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems.     

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.     

Pointwise Estimate for Linear Combinations of Bernstein-Kantorovich Operators

For linear combinations of Bernstein-Kantorovich operators K_n, r(f, x), we give an equivalent theorem with ω^2r_?^λ(f, t). The theorem unites the corresponding results of classical and Ditzian-Totik moduli of smoothness.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Numerical range and quasi-sectorial contractions

A method developed in Arlinski? (1987) [1] is applied to study the numerical range of quasi-sectorial contractions and to prove three main results. Our first theorem gives characterization of the maximal sectorial generator A in terms of the corresponding contraction semigroup {exp(−tA)}_t?0. The second result establishes for these quasi-sectorial contractions a quite accurate localization of their numerical range. We give for this class of semigroups a new proof of the Euler operator-norm approximation: exp(−tA)=limn→∞(I+tA/n)⁻ⁿ, t?0, with the optimal estimate: O(1/n), of the convergence rate, which takes into account the value of the sectorial generator angle (the third result).     

Boundedness in Nonlinear Asymmetric Oscillations

In this paper, the boundedness of all solutions of the nonlinear equation (?_p(x′))′+(p-1)[α?_p(x⁺)−β?_p(x⁻)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C⁷ is 2π_p-periodic, f,g are bounded C⁶ functions, ?_p(u)=∣u∣^p−2u, p?2,α,β are positive constants, x⁺=max{x,0},x⁻=max{−x,0}.     

On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.