By Richard A. Silverman
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Extra info for Complex Analysis with Applications
Then, one obtains for x, y ∈ L2 ([t0 , T ], Rn ) t |(V x)(t) − (V y)(t)| = [f (s, xs ) − f (s, ys )] ds t0 ≤ t t0 ≤ λ(s)|xs − ys |2 ds t 1/2 t λ2 (s) ds t0 t0 1/2 |xs − ys |22 ds . Since both x and y are equal to x0 (s) on [t0 − h, t0 ], it is obvious that, for s ∈ [t0 , T ], |xs − ys |2 ≤ |x − y|2 , the last norm being that of L2 ([t0 , T ], Rn ). Hence, from the inequality above we derive for t ∈ [t0 , T ] |(V x)(t) − (V y)(t)| ≤ (T − t0 )1/2 t 1/2 λ2 (s) ds t0 |x − y|2 . (38) From (38), taking into account λ ∈ L2 ([t0 , T ], Rn ), one derives the continuity of the operator V : L2 ([t0 , T ], Rn ) → C([t0 , T ], Rn ).
Remark If t + h ∈ / [t0 , T ], then we extend x(t) outside [t0 , T ] by letting x(t) = θ. 22 Auxiliary concepts Kolmogorov’s criterion. Let M ⊂ Lp ([t0 , T ], Rn ), 1 ≤ p < ∞. Necessary and sufﬁcient conditions for the relative compactness of M are: 1 2 M is bounded in Lp ; xh → x as h → 0, uniformly with respect to x ∈ M, where t+h xh (t) = h−1 x(s) ds, h > 0. V. P. Akilov  or N. T. Schwarz . In a similar manner as in case of p spaces C([t0 , T ], Rn ) and C([t0 , T ), Rn ), one can reduce the compactness in Lloc to the Lp -compactness on each compact interval in [t0 , T ).
Gripenberg et al.  for the general case (any p ≥ 1), and equation (1). A second application of the general results in this section is concerned with the integrodifferential equation x(t) ˙ =f t t, x(t), k(t, s)x(s) ds , (32) t0 under the usual initial condition x(t0 ) = x 0 ∈ Rn . Such equations are often encountered in applications, sometimes as perturbed equations of ordinary differential equations, when it has the (special) form x(t) ˙ = f (t, x(t)) + t k(t, s)x(s) ds. 11), this kind of perturbation is usually related to the “memory” of the system.