Cheh-Chih Yeh

Lyapunov and Wirtinger Inequalities

Abstract In this paper, we prove the Lyapunov inequality for the second-order linear differential equation ( r ( t ) φ ( y ′ ( t ) ) ) ′ + p ( t ) φ ( y ( t ) ) = 0 , where (i) φ(s) = |s|αt-2s, α > 1 is a fixed real number (ii) r(t) and p(t) are integrable on [a, b] with r(t) > 0 on [a, b]. On the other hand, a generalized Wirtinger inequality is also given.

Young's inequality and related results on time scales

Abstract We establish the classical Young inequality on time scales as follows: a b ≤ ∫ 0 a g σ ( x ) Δ x + ∫ 0 b ( g − 1 ) σ ( y ) Δ y if g ∈ C r d ( [ 0 , c ] , R ) is strictly increasing with c > 0 and g ( 0 ) = 0 , a ∈ [ 0 , c ] , b ∈ [ 0 , g ( c ) ] . Using this inequality, we can extend Hőlder’s inequality and Minkowski’s inequality on time scales.

The existence of positive solutions for the Sturm-Liouville boundary value problems

Abstract For the Sturm-Liouville boundary value problem (p(t)u′(t))′+λf(t,u(t))=0, 0⩽t⩽1, α 1 u(0)−β 1 p(0)u′(0)=0, α 2 u(1)+β 2 p(1)u′(1)=0, where λ > 0, we shall use a fixed point theorem in a cone to obtain the existence of positive solutions for λ on a suitable interval.

Oscillation criteria for second-order neutral delay difference equations

In this paper, we establish some oscillation criteria of the second-order delay difference equation Δ[an−1Δ(xn−1+pn−1xn−1−σ)]+qnf(xn−τ)=0 , where σ and τ are nonnegative constants, {an}, and {pn, {qn} are nonnegative sequences and f ∈ C(ℜ;ℜ).

Oscillation theorems for second-order half-linear differential equations

Abstract Oscillation criteria for the second-order half-linear differential equation [r(t)|ξ′(t)| α−1 ξ′(t)]′ + p(t)|ξ(t)| α−1 ξ(t)=0, t ⩾ t 0 are established, where α > 0 is a constant and ∫ t ∞ p(s) exists for t ∈ [t0, ∞). We apply these results to the following equation: ∑ i=1 N D i (|Du(ξ)| n−2 D i u(ξ)) + c(|ξ|)|u(ξ)| n−2 u(ξ)= 0, ξ ∈ Ω α where D i = ∂ ∂x i , D = (D1,…, DN), Ωa = x ∈ R N : |x| ≥ a} is an exterior domain, and c ∈ C([a, ∞), R ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.

Anderson's inequality on time scales

Abstract We establish Anderson’s inequality on time scales as follows: ∫ 0 1 ( ∏ i = 1 n f i σ ( t ) ) Δ t ≥ ( ∫ 0 1 ( t + σ ( t ) ) n Δ t ) ( ∏ i = 1 n ∫ 0 1 f i ( t ) Δ t ) ≥ ( 2 n ∫ 0 1 t n Δ t ) ( ∏ i = 1 n ∫ 0 1 f i ( t ) Δ t ) if f i ( i = 1 , … , n ) satisfy some suitable conditions.

Existence of positive solutions for functional differential equations

Under suitable conditions on f(t, yt(θ)), the boundary value problem of second-order functional differential equation (FDE) with the form: (FDE) y″(t)+f(t, yt(θ))=0, for t ϵ[0,1], θϵ[−ρ,a]; (BC) αy(t) − βy′(t) = η(t), for t ϵ[−ρ,0], (BVP) γy(t)+δy′=e(t), for t ϵ[1,1+a], has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems which guarantee (BVP) has the multiple positive solutions.

A maximum principle for second order nonlinear differential inequalities and its applications

Abstract Let y(t) be a nontrivial solution of the second order differential inequality y(t){(r(t)y′(t))′ + ƒ(t,y(t))} ⩽ 0 We show that the zeros of y(t) are simple; y(t) and y′(t) have at most finite number of zeros on any compact interval [a, b] under suitable conditions on r and f. Using the main result, we establish some nonlinear maximum principles and a nonlinear Levins comparison theorem, which extend some results of Protter, Weinberger, and Levin.

Lasalle's inequality and uniqueness theorems

La Salles inequality has been extended to a general measure integral. Using this inequality, we prove that the initial value problem for the Volterra–Stieltjes integral equation a has at most one solution under some suitable assumptions on r, φ and f.

Oscillation of nonlinear functional differential equations of the second order

Abstract In this paper, we consider the oscillation and nonoscillation of solutions of the second order nonlinear functional differential equation [|y′(t)| α sgn y′ (t)]′ + q(t) ƒ (y(g(t))) = 0, t ≥ t 0 > 0 , where α > 0 is a constant, q(t) ∈ C([t0,∞): (0,∞)), f(y) ∈ C(R;R), g′(t) > 0 on [t0, ∞), and limt→∞ g(t) = ∞.