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<p/> <p>The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales <inline-formula> <graphic file="1687-2770-2011-591219-i1.gif"/></inline-formula>, subject to the multi-point boundary condition <inline-formula> <graphic file="1687-2770-2011-591219-i2.gif"/></inline-formula><inline-formula> <graphic file="1687-2770-2011-591219-i3.gif"/></inline-formula>, where <inline-formula> <graphic file="1687-2770-2011-591219-i4.gif"/></inline-formula> is an increasing homeomorphism and satisfies the relation <inline-formula> <graphic file="1687-2770-2011-591219-i5.gif"/></inline-formula> for <inline-formula> <graphic file="1687-2770-2011-591219-i6.gif"/></inline-formula>, which generalizes the usually <it>p</it>-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearity <it>f</it> contains the first-order derivative explicitly and the operator <inline-formula> <graphic file="1687-2770-2011-591219-i7.gif"/></inline-formula> is not necessarily odd.</p>