In what follows, we will show the existence of a solution to the equation
(3.1)
satisfying the boundary conditions (1.3). Here, is a non-decreasing function, , and . Together with (3.1), for every , consider the auxiliary equation
(3.2)
where
(3.3)
Obviously,
(3.4)
and
(3.5)
The following three results can be found in [10].
Lemma 3.1 (see [[10], Corollary 2.17])
Let
and
be such that
(3.6)
Let, moreover, there exist a sequence of positive numbers such that
(3.7)
and let there exist
and
such that
where for almost every . Then there exists an upper function β to the problem (3.1), (1.3) satisfying
(3.8)
Lemma 3.2 (see [[10], Corollary 2.18])
Let and be such that (3.6) holds. If is a non-increasing function such that
where for almost every , then there exists an upper function β to the problem (3.1), (1.3) satisfying (3.8).
Lemma 3.3 (see [[10], Corollary 2.11])
Let
be such that
Then there exists a lower function α to the problem (2.17), (1.3) with
(3.9)
Lemma 3.4 Let and be such that (2.2) holds. Let, moreover, there exist a sequence of positive numbers such that (3.7) is fulfilled, and let there exist and such that
(3.10)
where for almost every . Then there exist and an upper function β to the problems (3.2), (1.3) for satisfying (3.8).
Proof
Put
(3.11)
(3.12)
Then, obviously, in view of (3.3), we have
(3.13)
and, consequently, on account of (2.2), (3.5), (3.10), and (3.13), there exists such that
(3.14)
(3.15)
Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of , it follows that β is also an upper function to (3.2), (1.3) for . □
Lemma 3.5 Let and be such that (2.2) holds. If is a non-increasing function such that
(3.16)
where for almost every , then there exist and an upper function β to the problems (3.2), (1.3) for satisfying (3.8).
Proof Define , , and by (3.11) and (3.12). Then, obviously, in view of (3.3), we have that (3.13) holds and, consequently, on account of (2.2), (3.5), (3.13), and (3.16), there exists such that (3.14) is valid and
(3.17)
Therefore, according to Lemma 3.2, there exists an upper function β to (3.2), (1.3) with satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of , it follows that β is also an upper function to (3.2), (1.3) for . □
Lemma 3.6
Let
(3.18)
and let either
(3.19)
or
(3.20)
Then, for every , there exists a constant such that for any and any positive solution u of (3.2), (1.3) with
we have the estimate
Proof Assume that (3.20) is fulfilled. Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Then there exist such that
(3.23)
Define the operator ϑ of ω-periodic prolongation by
(3.24)
Then, obviously, from (3.2) and (1.3) it follows that
(3.25)
The integration of (3.25) from to t, on account of (3.23), yields
(3.26)
From (3.21), (3.23), and (3.24) it follows that
(3.27)
Put
(3.28)
According to (3.18), we have
Thus, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.26), we arrive at
(3.30)
Put
Then, on account of (3.24) and (3.30), we have
(3.31)
On the other hand, the integration of (3.25) from t to , with respect to (3.23), results in
(3.32)
Now, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.32), we obtain
(3.33)
Therefore, in view of (3.24), from (3.33) we get
(3.34)
Consequently, (3.31) and (3.34) result in (3.22).
Now, suppose that (3.19) is fulfilled. Put
(3.35)
Then, according to (3.2), we have
(3.36)
where
Analogously to the above-proved, using (3.19) instead of (3.20), we obtain
with
Thus, (3.35) and (3.37) yield (3.22). □
Lemma 3.7
Let
(3.38)
and let either
(3.39)
or
(3.40)
Then, for every , there exists a constant such that for any and any positive solution u of (3.2), (1.3) satisfying (3.21), we have the estimate
(3.41)
Proof Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Thus, the integration of (3.2) from 0 to ω, in view of (1.3) and (3.4), yields
(3.42)
On the other hand, (3.38) implies the existence of such that
(3.43)
Let be such that
(3.44)
Obviously, either
or
Obviously, it is sufficient to show the estimate (3.41) is valid just in the case when (3.45) is fulfilled. Let, therefore, (3.45) hold.
If for , then applying (3.43) in (3.42) we obtain a contradiction. Thus, there exist points such that
(3.46)
(3.47)
where ϑ is the operator defined by (3.24). Obviously, (3.25) holds.
Assume that (3.39) holds. Then, according to Lemma 3.6, there exists such that (3.22) holds. The integration of (3.25) from to , in view of (3.4), (3.21), (3.22), (3.39), (3.43), (3.44), and (3.46), results in
(3.48)
Note that in view of (3.46), we have . Consequently, from (3.48) we obtain
(3.49)
where
Note that does not depend on k. Therefore, if we apply (3.39) in (3.49), it can be easily seen, with respect to (3.44), that there exists a constant such that (3.41) holds.
If (3.40) holds, we integrate (3.25) from to and apply similar steps as above, just using (3.47) instead of (3.46). Finally, we arrive at
with
Therefore, also in this case, there exists a constant such that (3.41) holds. □
Lemma 3.8 Let and be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, there exist a sequence of positive numbers such that (3.7) holds, and let there exist and such that (3.10) is fulfilled, where for almost every . Then there exists a positive solution u to (3.1), (1.3).
Proof According to Lemma 3.4, there exist and an upper function β to the problems (3.2), (1.3) for satisfying (3.8). On the other hand, in view of (3.4) and (3.38), there exists for such that
Thus, if we put for , according to Theorem 1.1, there exists a solution to (3.2), (1.3) for satisfying
(3.50)
Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants , , and , not depending on k, such that
(3.51)
(3.52)
(3.53)
where
Therefore, according to the Arzelà-Ascoli theorem, there exist and such that
(3.54)
Moreover, since are solutions to (3.2), (1.3), in view of (3.3), (3.52), and (3.54), we have , , and is a positive solution to (3.1), (1.3). □
The following assertion can be proved analogously to Lemma 3.8, just Lemma 3.5 is used instead of Lemma 3.4.
Lemma 3.9 Let and be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, be a non-increasing function and let (3.16) be fulfilled, where for almost every . Then there exists a positive solution u to (3.1), (1.3).
Lemma 3.10 Let be non-decreasing, , and be such that (2.2) holds. Let, moreover, there exist such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, there exist a sequence of positive numbers such that (2.8) holds and let there exist , , and such that (2.9) and (2.10) are fulfilled, where for almost every and σ is given by (2.11). Then there exists a lower function α to the problem (3.1), (1.3).
Proof Because is a positive function, from (2.4) and (2.11) we obtain that σ is a positive increasing function. Therefore, there exists an inverse function to σ which is also increasing. Moreover, in view of (2.4) and (2.11), it follows that
(3.55)
Consider the auxiliary equation
(3.56)
Put , . Then from (2.2) we get
(3.57)
and, in view of (3.55), from (2.5) we have
(3.58)
Furthermore, the substitution in (2.6), resp (2.7), with respect to (2.11) yields
(3.59)
resp.
(3.60)
Moreover, put for . Then from (2.8), in view of (3.55), we get
(3.61)
Finally, (2.10) results in
and so, since is a non-decreasing function, from (2.9) we obtain
(3.62)
Therefore, applying Lemma 3.8, according to (3.57)-(3.62), there exists a positive solution u to the problem (3.56), (1.3).
Now, we put for , i.e., in view of (2.11),
Obviously, is a positive function and
Thus, it can be easily seen that α is a lower function to the problem (3.1), (1.3). □
Analogously to the proof of Lemma 3.10, one can prove the following assertion applying Lemma 3.9 instead of Lemma 3.8.
Lemma 3.11 Let be non-decreasing, , and be such that (2.2) holds. Let, moreover, there exist such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, be a non-increasing function and let (2.12) be fulfilled, where for almost every and σ is given by (2.11). Then there exists a lower function α to the problem (3.1), (1.3).