(untitled) (JOURNAL OF ADVANCES IN MATHEMATICS, 2018)
Consider the equation u’(t) = A (t, u (t)), u(0)= U0 ; u’ := du/dt (1). Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u (t)|| < µ (t) -1 for every t belongs to R+ = [0,infinity). Here µ(t) > 0, µ belongs to C1 (R+), is a suitable function and the norm ||u || is the norm in a Banach space X with the property ||u (t) ||’ <= ||u’ (t) ||.