When we consider the bifurcation problem x + (x2 - v2)x + (x3 - x) = 0, according to the Melnikov's method, to any finite region 'which is given in advance and 0 < | e|≤1, there are two and only two closed orbits in the neighborhood of curves H(x, y) = y2/2 - x2/2 + x4/4 = h1and H (x, y) = h2. This result is incorrect to infinite point neighborhood. In this paper, first of all. we give a theorem about existence of at most two limit cycles of Lienard's equation, then, for all > 0, we discuss the upper bound problem about the number of Limit Cycles.
