Existence of Solutions to a Nonlocal Boundary Value Problem with Nonlinear Growth
© Xiaojie Lin. 2011
Received: 17 July 2010
Accepted: 17 October 2010
Published: 24 October 2010
This paper deals with the existence of solutions for the following differential equation: , , subject to the boundary conditions: , , where , , is a continuous function, is a nondecreasing function with . Under the resonance condition , some existence results are given for the boundary value problems. Our method is based upon the coincidence degree theory of Mawhin. We also give an example to illustrate our results.
where , , is a continuous function, is a nondecreasing function with . In boundary conditions (1.2), the integral is meant in the Riemann-Stieltjes sense.
with the boundary condition (1.2) has nontrivial solutions. Otherwise, we call them a problem at nonresonance.
However, if (i.e., resonance case), then the method in  is not valid.
The purpose of this paper is to study the existence of solutions for nonlocal BVP (1.1), (1.2) at resonance case (i.e., ) and establish some existence results under nonlinear growth restriction of . Our method is based upon the coincidence degree theory of Mawhin .
2. Main Results
We first recall some notation, and an abstract existence result.
Let , be real Banach spaces, let be a linear operator which is Fredholm map of index zero (i.e., , the image of , , the kernel of are finite dimensional with the same dimension as the ), and let , be continuous projectors such that = , = and , . It follows that is invertible; we denote the inverse by . Let be an open bounded, subset of such that , the map is said to be -compact on if is bounded, and is compact. Let be a linear isomorphism.
The theorem we use in the following is Theorem of .
Let be a Fredholm operator of index zero, and let be -compact on . Assume that the following conditions are satisfied:
(i) for every ,
(ii) for every ,
where is a projection with . Then the equation has at least one solution in .
Then BVP (1.1), (1.2) is .
We will establish existence theorems for BVP (1.1), (1.2) in the following two cases:
case (i): , ;
case (ii): , .
Let be a continuous function and assume that
Let be a continuous function. Assume that assumption (H1) of Theorem 2.2 is satisfied, and
3. Proof of Theorems 2.2 and 2.3
We first prove Theorem 2.2 via the following Lemmas.
Then . Hence (3.7) is valid.
Thus, is a Fredholm operator of index zero.
We define a projector by . Then we show that defined in (3.2) is a generalized inverse of .
then . The proof of Lemma 3.1 is finished.
If assumptions (H1), (H2) and , , and hold, then the set for some is a bounded subset of .
Then we show that is bounded.
If assumption (H2) holds, then the set is bounded.
From assumption (H2), , so , clearly is bounded.
where is the linear isomorphism given by , for all , . Then is bounded.
which contradicts . Then = and we obtain ; therefore, is bounded.
The proof of Theorem 2.2 is now an easy consequence of the above lemmas and Theorem 2.1.
Proof of Theorem 2.2.
Let such that . By the Ascoli-Arzela theorem, it can be shown that is compact; thus is -compact on . Then by the above Lemmas, we have the following.
(i) for every .
(ii) for every .
Then by Theorem 2.1, has at least one solution in , so that the BVP (1.1), (1.2) has at least one solution in . The proof is completed.
since . The remainder of the proof is the same.
By using the same method as in the proof of Theorem 2.2 and Lemmas 3.1–3.5, we can show Lemma 3.7 and Theorem 2.3.
Proof of Theorem 2.3.
thus, by using the same method as in the proof of Lemmas 3.2 and 3.3, we can prove that is bounded too. Similar to the other proof of Lemmas 3.4–3.7 and Theorem 2.2, we can verify Theorem 2.3.
Finally, we give two examples to demonstrate our results.
and has the same sign as when , we may choose , and then the conditions (H1)–(H3) of Theorem 2.2 are satisfied. Theorem 2.2 implies that BVP (3.61) has at least one solution, .
and has the same sign as when , we may choose , and then all conditions of Theorem 2.3 are satisfied. Theorem 2.3 implies that BVP (3.65) has at least one solution .
This work was sponsored by the National Natural Science Foundation of China (11071205), the NSF of Jiangsu Province Education Department, NFS of Xuzhou Normal University.
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