We investigate the problem with perturbed periodic boundary values $$\left\}\begin{array}{c}{y}^{\text{'}\text{'}\text{'}}\left(x\right)+a_2\left(x\right)y^{\text{'}\text{'}}\left(x\right)+a_1\left(x\right)y^{\text{'}}\left(x\right)+a_0\left(x\right)y\left(x\right)=f\left(x\right),y^{\left(i\right)}\left(T\right)=c{y}^{\left(i\right)}\left(0\right),i=0,1,2;0<c<1\hfill \end{array}\right.$$ with $a_2,a_1,a_0\in C[0,T]$ for some arbitrary positive real number $T$, by transforming the problem into an integral equation with the aid of a piecewise polynomial and utilizing the Fredholm alternative theorem to obtain a condition on the uniform norms of the coefficients $a_2$, $a_1$ and $a_0$ which guarantees unique solvability of the problem. Besides having theoretical value, this problem has also important applications since decay is a phenomenon that all...

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.