Wei-Cheng Lian

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.

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type (|u′|m−2u′)′ + f(t,u,u′)=0, m ⩾ 2 are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.

Levin's comparison theorems for nonlinear second order differential equations

Abstract The nonlinear Levins comparison theorems for nonlinear second order differential equations have been established by using a modified Levins technique.

Existence of Positive Solutions for Generalized p-Laplacian BVPs

Using Kransnoskiis fixed point theorem, the authors obtain the existence of multiple solutions of the following boundary value problem

Hermite-Hadamard's Inequality on Time Scales

Recently, new developments of the theory and applications of dynamic derivatives on time scales were made. The study provides an unification and an extension of traditional differential and difference equations and, in the same time, it is a unification of the discrete theory with the continuous theory, from the scientific point of view. Moreover, it is a crucial tool in many computational and numerical applications. Based on the wellknown Δ delta and ∇ nabla dynamic derivatives, a combined dynamic derivative, socalled α diamond-α dynamic derivative, was introduced as a linear combination of Δ and ∇ dynamic derivatives on time scales. The diamond-α dynamic derivative reduces to the Δ derivative for α 1 and to the ∇ derivative for α 0. On the other hand, it represents a “weighted dynamic derivative” on any uniformly discrete time scale when α 1/2. See 1–5 for the basic rules of calculus associated with the diamond-α dynamic derivatives. The classical Hermite-Hadamard inequality gives us an estimate, from below and from above, of the mean value of a convex function. The aim of this paper is to establish a full analogue of this inequality if we compute the mean value with the help of the delta, nabla, and diamond-α integral. The left-hand side of the Hermite-Hadamard inequality is a special case of the Jensen inequality. Recently, it has been proven a variant of diamond-α Jensen’s inequality see 6 .

Uniqueness of positive solutions for generalized Laplacian boundary value problems

Abstract In this paper, we establish the uniqueness of positive solutions of generalized Laplacian boundary value problems where α i , β i ≥ 0 and α i 2 + β i 2 ≠ 0 ( i = 1, 2).