Our existence theory is based on using the unbounded lower and upper solution technique. Here we list some assumptions for convenience.
H1: BVP (1)-(2) has a pair of upper and lower solutions β, α in X with
and satisfies the Nagumo condition with respect to α and β.
H2: For any fixed , , when , , the following inequality holds:
H3: There exists a constant such that
where ψ is the function in Nagumo’s condition of f.
Lemma 3.1 Suppose conditions (H1) and (H3) hold. Then there exists a constant such that every solution u of (1)-(2) with
satisfies .
Proof Set
where is any arbitrary constant. Choose where C is the nonhomogeneous boundary data, and satisfies
If holds for any , then the result follows immediately. If not, we claim that does not hold for all . Otherwise, without loss of generality, we suppose
But then, for any , it follows that
which is a contraction. So there must exist such that . Furthermore, if
just take and then the proof is completed. Finally, there exist such that , , or , , . Suppose that , , . Obviously,
from which one concludes that . Since and are arbitrary, we have if for . In a similarly way, we can show that , if for .
Therefore there exists a , just related with α, β, and ψ, h under the Nagumo condition of f, such that . □
Remark 3.1 Similarly, we can prove that
Remark 3.2 The Nagumo condition plays a key role in estimating the prior bound for the th derivative of the solution of BVP (1)-(2). Since the upper and lower solutions are in X, and may be asymptotic linearly at infinity.
Theorem 3.2 Suppose the conditions (H1)-(H3) hold. Then BVP (1)-(2) has at least one solution satisfying
Moreover, there exists a such that .
Proof Let be the same as in Lemma 3.1. Define the auxiliary functions , and as
for , and
Consider the modified differential equation with the truncated function
(16)
with the boundary conditions (2). To complete the proof, it suffices to show that problem (16)-(2) has at least one solution u satisfying
(17)
and
(18)
We divide the proof into the following two steps.
Step 1: By contradiction we shall show that every solution u of problem (16)-(2) satisfy (17) and (18).
Suppose the right hand inequality in (17) does not hold for . Set , then
Case I. .
Obviously, we have . By the boundary conditions, it follows that
which is a contradiction.
Case II. There exists a such that .
Clearly, we have
(19)
On the other hand,
Subcase i. If , from the definition of , we obtain
Subcase ii. If , from the conditions (H2), we find
Similarly following the above argument, we could discuss the other two cases or , , and we have the following inequality:
Thus,
which contradicts (19).
Thus, , . Similarly, we can show that , . Integrating this inequality and using the boundary conditions in (2), (13), and (14), we obtain the inequality (17). Inequality (18) then follows from Lemma 3.1. In conclusion, u is the required solution to BVP (1)-(2).
Step 2. Problem (16)-(2) has a solution u.
Consider the operator defined by
(20)
Lemma 2.1 shows that the fixed points of T are the solutions of BVP (16)-(2). Next we shall prove that T has at least one fixed point by using the Schäuder fixed point theorem. For this, it is enough to show that is completely continuous.
(1) is well defined.
For any , by direct calculation, we find
Obviously, . Further, because
(21)
where , the Lebesgue dominated convergent theorem implies that
Thus, .
(2) is continuous.
For any convergent sequence in X, there exists such that . Thus, as in (21), we have
and hence is continuous.
(3) is compact.
For this it suffices to show that T maps bounded subsets of X into relatively compact sets. Let B be any bounded subset of X, then there exists such that , . For any , , we have
where
where , and thus TB is uniformly bounded. Further, for any , if , we have
that is, TB is equi-continuous. From Lemma 2.3, it follows that if TB is equi-convergent at infinity, then TB is relatively compact. In fact, we have
and, therefore, is completely continuous. The Schäuder fixed point theorem now ensures that the operator T has a fixed point, which is a solution of the BVP (1)-(2). □
Remark 3.3 The upper and lower solutions are more strict at infinity than usually assumed. For such right boundary conditions, it is not easy to estimate the sign of even though we have , , where or . It remains unsettled whether this strict inequality can be weakened.