Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Dynamics of the difference equation

We study the invariant interval, the character of semicycles, the global stability, and the boundedness of the difference equation

where the initial conditions x_−k,…,x₋₁,x₀ are nonnegative, and p,q>0.     

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations

δ(a_nδx_n)=q_nx_n+1

and

δ(a_nδx_n)=q_nf(x_n+1),

where a_n > 0, q_n > 0, and f: R å R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Feedback affinization of nonlinear control systems

In this paper conditions for the nonlinear control system

to have a nonlinear feedback control

u=(x,v), vΩ′R^m′, m′m, 0Ω′

such that the nonlinear system takes a form of an affine system

are presented. All results require algebraic operations and differentiation of functions only.     

Stability and boundedness in difference systems with finite delay

In this paper, stability and boundedness theorems for delay difference systems with the condition

δV(n,x(n))≤-W₂(|x(n)|)+σ

are given, where Δ is the backward difference operator. Some known results are generalized.     

Estimating the distance to uncontrollability: A fast method and a slow one

For a linear control system (A,B), the distance to uncontrollability is characterized by

, where σ_n([A − λI,B]) is the smallest singular value of the augmented matrix [A − λI,B]. Two methods are developed to estimate the distance to uncontrollability, giving both a lower bound and an upper bound. One method is fast, requiring only one spectral decomposition of A and computations of three smallest singular values and being used for well-conditioned A. The other is slow, requiring computations of a large number of the smallest singular values, but it produces bounds as tight as possible and also a region containing global minimizers. Newton's method can be used to compute the global minimizers if one really wants.     

On systems of word equations with simple loop sets

Consider the infinite system S of word equations

For each , let T_k be the subsystem of S given by i{k,k+1,k+2}. We prove two properties of the above system.

(1) Let k≥1. If φ is a solution of T_k such that primitive roots of are of equal length, as well as primitive roots of , then φ is a solution of the whole S.

(2) If n=1 then, for any k≥2, a solution φ of T_k is also a solution of S.

On the stability of a weighted diamond of real polynomials

Recently, a subset of the robust stability literature concentrated on the so-called diamond of polynomials. In this paper, we study the weighted diamond Q_w defined as

, where the center q^*, the weights w₀, w₁, 3dot , w_n > 0 and the radius r > 0 are given. For this family, we show that an extreme point result holds if and only if a certain ‘interlacing condition’ on the weights is satisfied.     

Approximate fixed point sequence for a finite family of asymptotically pseudocontractive mappings

Let E be a real Banach space and K be a nonempty, closed, convex, and bounded subset of E. Let T_i:K→K, i=1,2,…,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε_n}, and asymptotically pseudocontractive mappings with sequences , where {ε_n} and , i=1,2,…,N, satisfy certain mild conditions. Let a sequence {x_n} be generated from x₁K by

for all integers n1, where T_n=T_n(modN), {u_n} be a sequence in K, and {λ_n}, {θ_n} and {μ_n} are three real sequences in [0,1] satisfying appropriate conditions; then x_n−T_lx_n→0 as n→∞ for each l{1,2,…,N}.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

Full-size image

, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

The query complexity of order-finding

We consider the problem where π is an unknown permutation on {0,1,…,2ⁿ−1}, y₀{0,1,…,2ⁿ−1}, and the goal is to determine the minimum r>0 such that π^r(y₀)=1. Information about π is available only via queries that yield π^x(y) from any x{0,1,…,2^m−1} and y{0,1,…,2ⁿ−1} (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.     

Oscillation theorems for certain neutral functional differential equations

In this paper, we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form

,here p and q are periodic functions.     

Factorization of disconjugate higher-order Sturm-Liouville difference operators

Using a recently proved equivalence between disconjugacy of the 2n^th-order difference equation

and solvability of the corresponding Riccati matrix difference equation, it is shown that the equation L(y) = 0 is disconjugate on a given interval if and only if the operator L admits the factorization of the form

L(y)_k+n=M^*(c_kM(y)_k)_k+n,

where M and its adjoint M* are certain n^th-order difference operators and c_k is a sequence of positive numbers.     

A unified presentation of some classes of meromorphically multivalent functions

The authors introduce and investigate various properties of a general class

U_k[p,a,β,A,B]

p, k ε N {1, 2, 3,…,}; 0 α < p; β 0

;

−1 A < B 1; 0 < B 1)

,which unifies and extends several (known or new) subclasses of meromorphically multivalent functions. The properties and characteristics of this general class, which are presented here, include growth and distortion theorems; they also involve Hadamard products (or convolution) of functions belonging to the class U_k[p,α,β,A,B].     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function

v(y,t)=(T−t)^{1/(2p+α−2)}u((T−t)^{(p−1)/(2p+α−2)}y,t),