中文摘要：

考虑了非线性项是变号的m-点奇异P—Laplacian动力方程（ψp（u^△（t）））↓△＋q（t）f（t，u（t））=0，t∈（0，T）T，u（0）=0，ψp（u^△（T））=∑i=2 ^m-2 ψi（u^△（ξi）），其中ψp（s）=｜s｜p^-2s，P〉1，ψi：R→R是连续的、不增的，0〈ξ1〈∈2〈…〈ξm-2〈p（T）．利用Schauder不动点定理和上下解方法，证明了上述边值问题正解的一些存在性法则．这些结果在相应的微分方程（T=R）、差分方程（T=Z）以及通常的测度链上都是新的．特别是，如果非线性项容许变号，那么Sun和Li的结果[Appl．Math．Comput．，2006，182：478—491]仅仅是我们所得结果在相应微分方程（T=R）的一种特殊情形．作为应用，给出了一个例子验证了主要结果．

英文摘要：

This paper is concerned with the following m-point singular p-Laplacian dy- namic equation （ψp（u^△（t）））↓△＋q（t）f（t,u（t））=0,t∈（0,T）T,u（0）=0,ψp（u^△（T））=∑i=2 ^m-2 ψi（u^△（ξi）） , where ψp（s）=｜s｜p^-2s with P〉1,ψi：R→R is continuous and nondecreasing, 0〈ξ1〈∈2〈…〈ξm-2〈p（T）. The nonlinearity term is allowed to change sign. By using the Schauder fixed point theorem together with upper and lower solutions method, some existence criteria are established for positive solutions of the boundary value problem. These results are new even for the corresponding differential （T = R and difference equation （T = Z）, as well as general time scales setting. In particular, if the nonlinear term is allowed to change sign, then the problem of Sun and Li [Appl. Math. Comput., 2006, 182：478 491] is only a special case of our problem for the corresponding differential equation （T = R）. As an application, an example is given to illustrate these results.