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Proof. Since £"(t) > 0, t > 0, £'(t) increases. Thus,

is well defined. Suppose, L G (—oo, 0]. Then, for each 0 < s < t, s s s s which is impossible, because limsj0 = oo. This shows the validity of the first limit of (32). Now, for each integer n > 2, there exists tn G [n — 1 ,n] such that and, hence, there exists a sequence tn, n > 1, such that in f oo and (.'{tn) —* 0 as n —» oo, since i(n) —» 0. Therefore, by the monotonicity of £', we obtain that £'(t) I 0 as 11 oo, which concludes the proof of the second limit. Finally, note that

We already know that limtT00(£'i) = 0. Moreover, I't is increasing, because

{£'£)' = £"£ + {£')2 > 0. Thus, limtToo(^)' = 0 and the third limit of (32) falls down from (33). □

3.2. Global lower estimates for i{t) The main result of this section reads as follows.

Proposition 3.1. Suppose p > 1 and f G C[0, oo) Pi C^O, oo) satisfies f(t) > 0 and f'(t) > 0 for each t > 0. Then, for each t > 0,

where £ is the unique solution of (21) and

Proof. Multiplying I" = f£? by I' and rearranging terms gives Moreover, according to Lemma 3.1,

Thus, integrating (37) in [i, oo) and using /' > 0 gives

and, hence,

for each i>0. Going back to the differential equation, (39) implies and, hence, for each t > 0,

Note that the change of variable

transforms (40) into

Moreover, by Lemma 3.1, rj(0) = 0, because limt|0 I' — —oo. Hence, integrating (42) in [0, i] shows that, for each t > 0,

The estimate (34) follows readily by substituting (43) into (41). Finally, the estimate (35) follows from (34) integrating it in [t, oo). Note that, due to Lemma 3.1, limtfoo £'(t) =0. □

In the special case when f(t) = ¡317, t > 0, for some positive constants /3 and 7, the function F(t) introduced in (36) is given by

Therefore, it has the same behaviour as the solution of (21), given by (31). Consequently, the general global lower estimate (35) seems really sharp. Actually, if / is assumed to be constant for sufficiently large t, then (38) becomes

for sufficiently large i, and, consequently, it is indeed modulated by F(t).

3.3. Global properties of the function F(t)

In this section we shall study some general properties satisfied by the function F(t) introduced in (36). It will be useful to express it in the form

The following result shows that it is well defined if / G C{0, oo) satisfies (1), and it establishes some of its main properties.

Lemma 3.2. Suppose p > 1 and f G C[0, oo) satisfies (1). Then, F G C2(0, oo) and, for each t > 0,

Moreover, lim Fit) = oo, lim F'it) = -oo, lim Fit) = lim F'it) = 0, (44)

t|0 t|o v ' tToo «Too and tio F(t) no -F'(t) t|o F(t) v '

Proof. As / is non-decreasing, for each t> e > 0, we have that

and, hence,

Therefore, F(t) is well defined for each t > 0 and lim F(t) = 0.

tfoo

Moreover, F € C2(0,oo), since A € C1(0,oo) satisfies A(t) > 0, t > 0, and (46) implies that lim F'it) = lim = 0. tfoo Uoo A(t)

m-jt 3>0, #•(.)—^<0, F»(i) = -iii> 0. (48)

The second limit of (44) follows from the second identity of (48), because >1(0) = 0. Now, since / is non-decreasing, we find that t E±i

0 0
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