New criteria for upper bounds on the number of limit cycles in Liénard systems

We study upper bounds for isolated periodic orbits of generalized Liénard systems on a single potential well. Common-curvature and common-intercept conditions yield uniqueness and hyperbolicity, while a convex–concave condition on the potential-profile slopes gives an upper bound of two. The statements allow finite unequal potential endpoints; appropriate versions also admit degenerate restoring forces and reduced regularity. We prove a uniqueness extension across slope interfaces, a quantitative joint weak-damping criterion, and a compensation criterion controlled by two derivatives. A separate response comparison at neutral cycles, combined with a connected parameter-family argument, gives a coefficient criterion for a two-cycle bound. Applications include classical quartic systems, negative-parameter and positive-parameter Selkov systems, a uniqueness conjecture for arbitrary-degree polynomial systems, and explicit smooth and nonsmooth examples. Existence is proved separately whenever an exact count is asserted. Restrictions to a potential well, a parameter region, or a compact width interval are retained throughout.