... functio n r(t)=-||x (t)||2, t Î [0, 1], where x(t) is a soluti on of (3.6), for the rest part of the proof we proceed as in the proof of Theorem 3.1.Corollary 3.2 Let f be a scalar-valued function ... solutions of x = λTx, λ ∈[0, 1],(3:14)are bounded a priori, with the bound being independent of l. With this in mind, letx be a solution of (3.14). Note that x is also a solution of (3.6). ... authors of [20-24] consider this kind of “anti-periodic” conditions for differ-ential equations or impulsive differential equations. To the best of our knowledge it isthe first article to deal with...