Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions
© R. Ma and T. Chen. 2011
Received: 5 May 2010
Accepted: 7 July 2010
Published: 3 August 2010
where is continuous; see Gupta [1, 2]. In the past twenty more years, the existence of solutions and positive solutions of these kinds of problems and the Lidstone problem has been extensively studied; see [3–9] and the references therein. In , Ma was concerned with the existence of positive solutions of (1.1) and (1.2) under the assumptions:
Ma proved the following.
Theorem A (see [3, Theorem ]).
Then (1.1) and (1.2) have at least one positive solution.
under the assumption
Then (1.9) has at least one positive solution.
Theorem 1.1 generalizes [3, Theorem ] where the special case and was treated.
In this case, and the corresponding eigenfunction is . However, (1.15) and (1.16) has no positive solution. (In fact, suppose on the contrary that (1.15) and (1.16) has a positive solution . Multiplying (1.15) with and integrating from to , we get a desired contradiction!).
The following lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [14, Theorem ].
Since is a strongly positive compact endomorphism of and has nonempty interior, we have from Amann [15, Theorem ] that the set in [14, Theorem ] reduces to a single point . Now the desired result is a consequence of Dancer [14, Theorem ].
The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results about the spectrum of (1.12)–(1.14). Finally, in Section 3, we proved our main result.
2. Generalized Eigenvalues
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
if (2.8) and (2.9) have nontrivial solutions.
Assume that (H4) and (H5) hold. Let be the spectral radius of . Then (2.8) and (2.9) has an algebraically simple eigenvalue, , with a positive eigenfunction . Moreover, there is no other eigenvalue with a positive eigenfunction.
Proof of Theorem 2.6.
Now, by the Krein-Rutman theorem ([16, Theorem C]; [17, Theorem ]), has an algebraically simple eigenvalue with an eigenfunction . Moreover, there is no other eigenvalue with a positive eigenfunction. Correspondingly, with a positive eigenfunction of , is a simple eigenvalue of (2.8) and (2.9). Moreover, for (2.8) and (2.9), there is no other eigenvalue with a positive eigenfunction.
3. The Proof of the Main Result
Proof of Theorem 1.1.
We divide the proof into two steps.
The authors are very grateful to the anonymous referees for their valuable suggestions. This paper was supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.
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