Shiueh-Ling Yu

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.

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 periodic solutions of higher-order differential equations

We established some sufficient conditions for the existence of solutions of high-order periodic boundary value problem for n >= 2 (E)u^(^n^)(t)+D(t,u(t),...,u^(^n^-^2^)(t))u^(^n^-^1^)(t)+g(t,u(t),...,u^(^n^-^2^)(t))=h(t),fort@?[0,T],(BC){u^(^i^)(0)=0,i=0,1,2,...,n-3,(BVP)u^(^n^-^2^)(0)=u^(^n^-^2^)(T),u^(^n^-^1^)(0)=u^(^n^-^1^)(T), where h @e L^1(0,T), D @e C([0,T]x R^n^-^1,R), and g:[0,T] x R^n^-^1 -> R be a Caratheodory function, T-periodic in the first variable.

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.

Positive solutions of integro-differential inequalities

Abstract In this paper, we shall show that under suitable conditions on f and K , the inequalities −λ n + ∫ 0 ∞ e λs K(s)ds > 0, for all λ>0,(n = 1,2,4) imply that the integro-differential inequalities (−1) n+1 y (n) (t) + ∫ 0 t f(t − s,y(s)) ds ≤ 0, on (0,∞), (n = 1,2,4) have no positive solutions, respectively. Moreover, we shall demonstrate that f cannot be a superlinear function.