Fu-Hsiang Wong

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.

Existence of positive solutions for non-positive higher-order BVPs

Abstract We shall provide conditions on non-positive function f ( t , u 1 ,…, u n −1 ) so that the boundary value problem has at least one positive solution. Then, we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions.

Existence of positive solutions for higher order difference equations

Abstract We shall provide conditions on the function ƒ(i,u 1 ,u 2 ,…,u n−1 ) , so that the boundary value problem has at least one positive solution. Then, we shall apply this result to establish several existence results which guarantee the multiple positive solutions.

Positive solutions for nonlinear singular boundary value problems

Under suitable conditions on f(t,u), it is shown that the two-point boundary value problem u″(t)+λƒ(t,u(t))=0, in(0,1), u(0)=u(1)=0 , has at least one positive solution for λ in a compatible interval.

Existence of positive solutions for higher-order generalized p-Laplacian BVPs

Abstract In this paper, we provide suitable conditions on the function f(t, u1, …, un−1) such that the higher-order generalized p-Laplacian boundary value problem has at least one positive solution. Moreover, we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions.

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.

Existence of solutions to (k, n-k-2) boundary value problems

We provide very general existence criteria to solutions for a class of higher order boundary value problems. Our results supplement as well as improve several recent results established in the literature.

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.