a criterion for the monotonicity of the ratio of two abelian integrals in piecewise-smooth differential systems

In this paper, we present a new criterion function for investigating the monotonicity of the ratio of two abelian integrals in piecewise-smooth differential systems, and then, apply it to deal with some examples. more precisely, we consider the abelian integrals of the form ik(h) = ꭍ fk(x)ydx, k = 0, 1, with γh = γl h + γr h , where γl h = {(x, y) ∈ r 2 | 1 2 y 2 + ψ2(x) = h, x < 0} and γr h = {(x, y) ∈ r² | 1/2 y² + ψ1(x) = h, x > 0}. we prove that the monotonicity of the presented criterion function implies the monotonicity of the ratio i1(h)/i0(h) and provide a few examples to explain the application of this criterion.