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Uniform asymptotic stability implies exponential stability for two-dimensional half-linear difference systems

The purpose of this talk to expand the results on the stability of differential equations to the theory of difference equations. First, we consider a two-dimensional differential system: x′ = a11(t)x + a12(t)ϕp (y), y′ = a21(t)ϕp(x) + a22(t)y, (HS) where coefficients a11(t), a12(t), a21(t) and a22(t) are continuous; p and p∗ are positive numbers satisfying (p − 1)(p∗ − 1) = 1; the real-valued function ϕq(z) is defined by ϕq(z) = { |z|q−2z if z ̸= 0, 0 if z = 0, z ∈ R with q = p or q = p∗. Note that the function ϕp is the inverse function of ϕp. System (HS) has the zero solution (x(t), y(t)) ≡ (0, 0). In the case that a11(t) ≡ 0 and a12(t) ≡ 1, system (HS) becomes the differential equation (ϕp(x′)) ′ − a22(t)ϕp(x′) − a21(t)ϕp(x) = 0. (H) If x(t) is a solution of (H), then cx(t) is also a solution of (H) for any c ∈ R; that is, the solution space of (H) is homogeneous. However, even if x1(t) and x2(t) are two solutions of (H), the function x1(t) + x2(t) is not always a solution of (H); that is, the solution space of (H) is not additive. For this reason, equation (H) is often called ”halflinear differential equation”(see [1, 2, 5, 6]). Moreover, system (HS) is called ”half-linear differential system”(see [2, 5, 6]).

2012 / 04 --- 日本冈山理工大学， 应用数学系讲师， 2010/ 04 - 2012 / 03 日本都城工业高等专科学校，数学系讲师， 2010 / 04 日本岛根大学 博士毕业， 在国际学术刊物上发表论文十余篇。