Abstract:
We consider a higher-order three-point boundary value problem on time scales.A new existence result is first obtained by using a fixed point theorem due to Krasnoselskii andZabreiko. Later, under certain growth conditions imposed on the nonlinearity, several sufficientconditions for the existence of a nonnegative and nontrivial solution are obtained by usingLeray-Schauder nonlinear alternative. Our conditions imposed on nonlinearity are all very easyto verify; as an application, some examples to demonstrate our results are given.

Abstract:
We study the existence of positive solutions for a class of -point boundary value problems on time scales. Our approach is based on the monotone iterative technique and the cone expansion and compression fixed point theorem of norm type. Without the assumption of the existence of lower and upper solutions, we do not only obtain the existence of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

Abstract:
The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations. 1. Introduction In recent years, motivated by some ecological problems, much attention has been attached to the existence of sign-changing solutions for nonlinear partial differential equations (see [1–4] and the references therein). We note that the proofs of main results in [1–4] depend upon critical point theory. However, some concrete nonlinear problems have no variational structures [5]. To overcome this difficulty, in [6], Zhang studied the existence of sign-changing solution for nonlinear operator equations by using the cone theory and combining uniformly positive condition. Xu [7] studied multiple sign-changing solutions to the following -point boundary value problems: where , , . We list some assumptions as follows.(A1)Suppose that the sequence of positive solutions to the equation ？is ;(A2) , is a continuous function, , and for all ;(A3)let and . There exist positive integers and such that (A4)there exists such that for all with . Theorem 1 (see [7]). Suppose that conditions are satisfied. Then the problem (1) has at least two sign-changing solutions. Moreover, the problem (1) also has at least two positive solutions and two negative solutions. Based on [7], many authors studied the sign-changing solutions of differential and difference equations. For example, Yang [8] considered the existence of multiple sign-changing solutions for the problem (1). Compared with Theorem 1, Yang employed the following assumption which is different from .(A′4)There exists such that Pang et al. [9] investigated multiple sign-changing solutions of fourth-order differential equation boundary value problems. Moreover, Wei and Pang [10] established the existence theorem of multiple sign-changing solutions for fourth-order boundary value problems. Y. Li and F. Li [11] studied two sign-changing solutions of a class of second-order integral boundary value problems by computing the eigenvalues and the algebraic multiplicities of the corresponding linear problems. He et al. [12] discussed the existence of sign-changing solutions for a class of discrete boundary value problems, and a concrete example was also given. Very recently, Yang [13] investigated the following discrete fourth Neumann

Abstract:
By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales , ; , ; , ; , , where , , , , , , , and is rd-continuous.

Abstract:
By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales uΔn(t)+g(t)f(u(t),uΔ(t),…,uΔn 2(t))=0, 00, β≥0, γ>0, δ≥0, ξ, η∈(0,T), ξ<η, and g:(0,T)→[0,+∞) is rd-continuous.

Abstract:
In this paper, we study the existence of positive solutions to nonlinear m-point boundary-value problems for a p-Laplacian dynamic equation on time scales. We use fixed point theorems in cones and obtain criteria that generalize and improve known results.

Abstract:
By means of fixed point index, we establish sufficient conditions for the existence of positive solutions to p-Laplacian difference equations. In particular, the nonlinear term is allowed to change sign.

Abstract:
Several existence theorems of positive solutions are established for nonlinear -point boundary value problem for the following dynamic equations on time scales ((Δ))？

Abstract:
This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique.

Abstract:
This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique.