Let $T$ be a $\gamma$-contraction on a Banach space $Y$ and let $S$ be an almost $\gamma$-contraction, i.e. sum of an $\left(\epsilon ,\gamma \right)$-contraction with a continuous, bounded function which is less than $\epsilon$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u=Tu.$ If moreover there exists $v$ in $Y$ with $v=Sv$, then we will give estimates for $\parallel u-v\parallel .$ Finally, we establish some inequalities related to the Cauchy problem.

The properties of solutions of the nonlinear differential equation $x^{\text{'}}=A\left(s\right)x+f(s,x)$ in a Banach space and of the special case of the homogeneous linear differential equation $x^{\text{'}}=A\left(s\right)x$ are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.

The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M + are considered over a regoin (a, b) on the real line, −∞ ≤ a < b ≤ ∞, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L2|w (a, b) which are regularly solvable with respect to the minimal operators T0 (M ) and T0...

The objective of this note is the announcement of two results of Ambrosetti-Prodi type concerning the existence of periodic (respectively bounded) solutions of the first order differential equation x' = f (t,x).

Totally bounded differential systems in ${\mathbb{R}}^2$ are defined as having all trajectories bounded. By Dulac’s finiteness theorem it is proved that totally bounded polynomial systems exhibit an unbounded «annulus» of cycles. The portrait of the remaining trajectories is examined in the case the system has, in ${\mathbb{R}}^2$, a unique singular point. Work is in progress concerning the study of totally bounded polynomial systems with two singular points.

The existence of bounded solutions for equations $x^{\text{'}}=A\left(t\right)x+f(t,x)$ in Banach spaces is proved. We assume that the linear part is trichotomic and the perturbation $f$ satisfies some conditions expressed in terms of measures of noncompactness.