Abstract
Consider the singular Dirichlet problem
where
then the Fredholm alternative remains true.
MSC: 34B05.
Keywords:
singular Dirichlet problem; Fredholm alternative1 Introduction
Consider the boundary value problem
where
guarantees the validity of the Fredholm alternative. More precisely, if (3) holds, then problem (1), (2) is uniquely solvable for any q satisfying
iff the corresponding homogeneous equation
has no nontrivial solution satisfying (2). The above statement plays an important role in the theory of singular problems; however, it does not cover many interesting, even rather simple, equations. For example, consider the Dirichlet problem for the Euler equation
where α and β are real constants. By direct calculations, one can easily verify that if
has only the trivial solution, while problem (5) is uniquely solvable. However, in
this case
The aim of this paper is to show that the Fredholm alternative remains true even in the case when instead of (3) only the condition
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
For
By
Under a solution of equation (1) we understand a function
We say that a certain property holds in
Recall that we consider problem (1), (2), where
Theorem 1.1Let condition (6) hold. Then problem (1), (2) is uniquely solvable for anyqsatisfying (4) iff homogeneous problem (1_{0}), (2) has no nontrivial solution.
Remark 1.1 In Theorem 1.1, condition (4) is essential and cannot be omitted. Indeed, let
Evidently, (6) holds and problem (1_{0}), (2) has no nontrivial solution. On the other hand, a general solution of (1) is of the form
However, for
Hence,
Therefore, in view of (7), we get
Remark 1.2 Theorem 1.1 concerns half homogeneous problem (1), (2) and does not remain true for the fully nonhomogeneous problem
Let, for example,
Theorem 1.2Let (6) hold and problem (1_{0}), (2) have no nontrivial solution. Then there exists
Consider now a sequence of equations
where
Let, moreover,
Corollary 1.1Let (4), (6) hold and problem (1_{0}), (2) have no nontrivial solution. Let, moreover, (11) and (12) be fulfilled. Then the problems (1), (2) and (
and
2 Auxiliary statements
In this section, we consider the equation
where
Below we state some known results in a suitable for us form.
Proposition 2.1Let (15) hold. Then the problem
is uniquely solvable for any
has no nontrivial solution.
Proof See, e.g., [[1], Theorem 3.1] or [[2], Theorem 1.1]. □
Proposition 2.2Let (15) hold. Then there exist
has no nontrivial solution. Moreover, for any
the inequality
holds.
Proof In view of (15), there exist
Hence, the inequalities
hold as well. The latter inequalities, by virtue of [[2], Lemma 4.1], imply that for any
The second part of the proposition follows easily from the aboveproved part and [[2], Lemma 1.3]. □
Proposition 2.3Let (15) hold. Let, moreover,
admits the estimate
for
admits the estimate
for
Proof By virtue of (15) and [[1], Lemma 2.2], the initial value problems
and
have unique solutions
are fulfilled, where
On the other hand, by virtue of Proposition 2.2,
In view of Propositions 2.1 and 2.2, problem (17) has a unique solution v. By direct calculations, one can easily verify that
for
for
and
respectively,
are fulfilled with
Now, it follows from (22) and (23), in view of (21) and (24), that the estimates (18) and (20) hold with
□
3 Lemmas on a priori estimates
Lemma 3.1Let (4) and (6) hold. Then, for any
admits the estimate
Proof Let
or
or
Assume that (27) (resp., (28)) holds. Then, in view of (25), there is
Multiplying both sides of (1) by
Hence, in view of (30), we obtain
Multiplying both parts of the latter inequality by
Suppose now that (29) holds. Then either there is
or there is a sequence
If (32) holds, then evidently
are fulfilled, and therefore, in view of (33), inequality (31) holds as well. Thus, estimate (26) is fulfilled. □
Lemma 3.2Let (6) hold. Then there exist
admits the estimate
while every solutionuof equation (1) satisfying
admits the estimate
Proof Let
has a unique solution v and, moreover, for any
holds. Let us show that
Assume the contrary, let (41) be violated. Define
Then there exist
In view of (1), (39), and (42), it is clear that
Hence, by virtue of (43) and Proposition 2.2, we get
Lemma 3.3Let (6) hold and problem (1_{0}), (2) have no nontrivial solution. Then there exist
admits the estimate
Proof Suppose to the contrary that the lemma is not true. Then there exist sequences
and
Introduce the notation
Then it is clear that
and
Moreover, it follows from (45) that
and, consequently,
By virtue of Lemma 3.1, (46), and (47),
Hence, in view of (44) and (48), the sequence
where
By a direct calculation, one can easily verify that
whence, in view of (49)(51), we get
Thus
Now let
are fulfilled. Moreover, in view of (48), we have
and
Taking, moreover, into account (50), we get from (52) that
and thus
On account of (44) and (48), there exist
and
Then it follows from (52) that
Hence, in view of (46),
4 Proofs of the main results
Proof of Theorem 1.1 To prove the theorem, it is sufficient to show that if problem (1_{0}), (2) has no nontrivial solution, then problem (1), (2) has at least one solution.
Let
By virtue of Lemma 3.3, the problem
has no nontrivial solution. Hence, by virtue of Proposition 2.1, the problem
has a unique solution
holds, where
On the other hand, on account of Lemma 3.1 and (55), we have
where
In view of (53), (55), and (56), the sequence
where
Taking into account (54), one can easily verify, by a direct calculation, that
Hence, in view of (57) and (58), we get
Thus
Further, by virtue of Lemma 3.2 and (55), the inequalities
and
are fulfilled. Hence, on account of (57), we get
and thus
Proof of Theorem 1.2 According to Theorem 1.1, problem (1), (2) has a unique solution u. By virtue of Lemma 3.3, the estimate
holds. On the other hand, it follows from Lemma 3.1 that
The latter two inequalities imply (9) with
□
Proof of Corollary 1.1 By virtue of Theorem 1.1, problems (1), (2) and (
Then it is clear that
where
Hence, by virtue of Theorem 1.2,
Taking now into account (12), (59), and (60), we get (13) and (14). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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. FSIS113 ‘Modern methods of mathematical problem modelling in engineering’. Research was also supported by RVO: 67985840.
References

Kiguradze, IT, Lomtatidze, AG: On certain boundaryvalue problems for secondorder linear ordinary differential equations with singularities. J. Math. Anal. Appl.. 101(2), 325–347 (1984). Publisher Full Text

Kiguradze, IT, Shekhter, BL: Singular boundary value problems for second order ordinary differential equations. J. Sov. Math.. 43(2), 2340–2417 (1988). Publisher Full Text