This paper devotes to the study of the classical Abel equation
dtdx​=g(t)x3+f(t)x2, where g(t) and f(t) are trigonometric
polynomials of degree m≥1. We are interested in the problem that whether
there is a uniform upper bound for the number of limit cycles of the equation
with respect to m, which is known as the famous Smale-Pugh problem. In this
work we generalize an idea from the recent paper (Yu, Chen and Liu,
arXiv:2304.13528, 2023) and give a new criterion to estimate the maximum
multiplicity of limit cycles of the above Abel equations. By virtue of this
criterion and the previous results given by {\'A}lvarez et al. and Bravo et
al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the
case when g(t) and f(t) are linear trigonometric, and obtain that the
maximum number of limit cycles, is three.Comment: 16 pages, 8 figure