- Research Article
- Open Access
Positive Solutions of Singular Complementary Lidstone Boundary Value Problems
© Ravi P. Agarwal et al. 2010
- Received: 7 October 2010
- Accepted: 21 November 2010
- Published: 2 December 2010
- Differential Equation
- Positive Constant
- Green Function
- Point Theorem
- Existence Result
was discussed in . Here, is continuous at least in the interior of the domain of interest. Existence and uniqueness criteria for problem (1.5) are proved by the complementary Lidstone interpolating polynomial of degree . No contributions exist, as far as we know, concerning the existence of positive solutions of singular complementary Lidstone problems.
The aim of this paper is to give the conditions on the function in (1.1) which guarantee that the singular problem (1.1), (1.2) has a solution. The existence results are proved by regularization and sequential techniques, and in limit processes, the Vitali convergence theorem [16, 17] is applied.
The paper is organized as follows. In Section 2, we construct a sequence of auxiliary regular differential equations associated with (1.1). Section 3 is devoted to the study of auxiliary regular complementary Lidstone problems. We show that the solvability of these problems is reduced to the existence of a fixed point of an operator . The existence of a fixed point of is proved by a fixed point theorem of cone compression type according to Guo-Krasnosel'skii [18, 19]. The properties of solutions to auxiliary problems are also investigated here. In Section 4, applying the results of Section 3, the existence of a positive solution of the singular problem (1.1), (1.2) is proved.
Lemma 3.1 (see [10, Lemmas 2.1 and 2.3]).
We are now in the position to prove that problem (2.8), (1.2) has a solution.
Then, is a cone in and since for by (3.4) and satisfies (2.5), we see that . The fact that is a completely continuous operator follows from , from Lebesgue dominated convergence theorem, and from the Arzelà-Ascoli theorem.
The properties of solutions to problem (2.8), (1.2) are collected in the following lemma.
holds for and . Now, using (1.2), (3.4), (3.30), and (3.31), we see that assertion (i) is true. Hence, is decreasing on for and is increasing on this interval. Due to for , there exists a unique such that for . Consequently, assertion (ii) holds.
Then estimate (3.29) follows from relations (3.32)–(3.37).
are uniformly integrable on . Due to and for by ( ), this fact follows from [13, Criterion 11.10 (with and )].
The following theorem is the existence result for the singular problem (1.1), (1.2).
This work was supported by the Council of Czech Government MSM no. 6198959214.
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