Fredholm alternative for the second-order singular Dirichlet problem
© Lomtatidze and Opluštil; licensee Springer. 2014
Received: 13 September 2013
Accepted: 19 November 2013
Published: 13 January 2014
Consider the singular Dirichlet problem
where are locally Lebesgue integrable functions. It is proved that if
then the Fredholm alternative remains true.
has only the trivial solution, while problem (5) is uniquely solvable. However, in this case and therefore condition (3) is not satisfied.
holds. The paper is organized as follows. At the end of this section, we state our main results, the proofs of which one can find in Section 4. In Section 2, we recall some known results in a suitable for us form. Section 3 is devoted to a priori estimates and plays a crucial role in the proofs of the main results.
Throughout the paper we use the following notation.
ℝ is the set of real numbers.
For , we put .
, where is the set of continuous functions .
For , we put .
is the set of functions , which are absolutely continuous together with their first derivative on every closed subinterval of .
is the set of functions , which are Lebesgue integrable on every closed subinterval of .
By (resp., ) we denote the right (resp., left) limit of the function at the point a (resp., b).
We say that a certain property holds in if it takes place on every closed subinterval of .
Recall that we consider problem (1), (2), where . Our main results are the following.
Theorem 1.1 Let condition (6) hold. Then problem (1), (2) is uniquely solvable for any q satisfying (4) iff homogeneous problem (10), (2) has no nontrivial solution.
Therefore, in view of (7), we get and, consequently, problem (1), (2) has no solution.
Let, for example, , , , and . It is clear that (6) holds and the corresponding homogeneous problem (10), (2) has no nontrivial solution. On the other hand, a general solution of (1) is of the form for and, therefore, (8) has no solution.
2 Auxiliary statements
Below we state some known results in a suitable for us form.
has no nontrivial solution.
hold as well. The latter inequalities, by virtue of [, Lemma 4.1], imply that for any satisfying either or , homogeneous problem (16) has no nontrivial solution.
The second part of the proposition follows easily from the above-proved part and [, Lemma 1.3]. □
3 Lemmas on a priori estimates
are fulfilled, and therefore, in view of (33), inequality (31) holds as well. Thus, estimate (26) is fulfilled. □
Hence, by virtue of (43) and Proposition 2.2, we get for , which contradicts (42). Therefore, (41) is fulfilled. The estimate (36) (resp., (38)) now follows from (40) and (41). □
Thus and is a solution of equation (10).
Hence, in view of (46), for . Taking now into account (50), we get , and thus is a nontrivial solution of problem (10), (2). However, this contradicts an assumption of the lemma. □
4 Proofs of the main results
Proof of Theorem 1.1 To prove the theorem, it is sufficient to show that if problem (10), (2) has no nontrivial solution, then problem (1), (2) has at least one solution.
Thus and is a solution of equation (1).
and thus and . Consequently, is a solution of problem (1), (2). □
Taking now into account (12), (59), and (60), we get (13) and (14). □
Published results were supported by the project ‘Popularization of BUT R&D results and support systematic collaboration with Czech students’ CZ.1.07/2.3.00/35.0004 and by Grant No. FSI-S-11-3 ‘Modern methods of mathematical problem modelling in engineering’. Research was also supported by RVO: 67985840.
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