Abstract
Keywords:
singular Dirichlet problem; Fredholm alternative1 Introduction
Consider the boundary value problem
where . We are mainly interested in the case when the functions p and q are not (in general) integrable on . In this case, equation (1) as well as problem (1), (2) are said to be singular. It is well known that for singular problem (1), (2), the condition
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 , then the homogeneous problem
has only the trivial solution, while problem (5) is uniquely solvable. However, in this case and therefore condition (3) is not satisfied.
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.
, where is the set of continuous functions .
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).
Under a solution of equation (1) we understand a function which satisfies it almost everywhere in . A solution of equation (1) satisfying (2) is said to be a solution of problem (1), (2).
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 1.1Let 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 , , for , and
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
Hence,
Therefore, in view of (7), we get and, consequently, problem (1), (2) has no solution.
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, , , , and . It is clear that (6) holds and the corresponding homogeneous problem (1_{0}), (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.
Theorem 1.2Let (6) hold and problem (1_{0}), (2) have no nontrivial solution. Then there existssuch that for anyqsatisfying (4), the solutionuof problem (1), (2) admits the estimate
Consider now a sequence of equations
Let, moreover, satisfy (4) and
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 (), (2) have unique solutionsuand, respectively,
and
2 Auxiliary statements
In this section, we consider the equation
Below we state some known results in a suitable for us form.
Proposition 2.1Let (15) hold. Then the problem
is uniquely solvable for anyandqsatisfying (4) iff the homogeneous problem
has no nontrivial solution.
Proof See, e.g., [[1], Theorem 3.1] or [[2], Theorem 1.1]. □
Proposition 2.2Let (15) hold. Then there existandsuch that, for anysatisfying eitheror, the homogeneous problem
has no nontrivial solution. Moreover, for any (whereare the same as above) satisfying
the inequality
holds.
Proof In view of (15), there exist and such that
Hence, the inequalities
hold as well. The latter inequalities, by virtue of [[2], 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 aboveproved part and [[2], Lemma 1.3]. □
Proposition 2.3Let (15) hold. Let, moreover, andbe from the assertion of Proposition 2.2. Then there existssuch that for anyand anyqsatisfying (4), the solutionvof the problem
admits the estimate
for, while the solutionvof the problem
admits the estimate
Proof By virtue of (15) and [[1], Lemma 2.2], the initial value problems
and
have unique solutions and , respectively, and the estimates
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 . Analogously, the (unique) solution v of problem (19) is of the form
for , where and are solutions of the problems
and
respectively, , , and the estimates
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 anyand, every solutionuof equation (1) satisfying
admits the estimate
Proof Let . Then it is clear that either
or
or
Assume that (27) (resp., (28)) holds. Then, in view of (25), there is (resp., ) such that
Multiplying both sides of (1) by (resp., by ) and integrating it from to (resp., from to ), we get
Hence, in view of (30), we obtain
Multiplying both parts of the latter inequality by (resp., by ), we get
Suppose now that (29) holds. Then either there is such that
or there is a sequence such that
If (32) holds, then evidently for and, consequently, (31) is fulfilled. On the other hand, if (34) holds, then, by virtue of the aboveproved, the inequalities
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, , andsuch that for any, and anyqsatisfying (4), every solutionuof equation (1) satisfying
admits the estimate
while every solutionuof equation (1) satisfying
admits the estimate
Proof Let , , and ϱ be from the assertion of Propositions 2.2 and 2.3 with for . Let, moreover, (resp., ) and u be a solution of problem (1), (35) (resp., (1), (37)). By virtue of Propositions 2.2 and 2.3, the problem
has a unique solution v and, moreover, for any (resp., ), the estimate
holds. Let us show that
Assume the contrary, let (41) be violated. Define
Then there exist and (resp., and ) such that
In view of (1), (39), and (42), it is clear that and
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). □
Lemma 3.3Let (6) hold and problem (1_{0}), (2) have no nontrivial solution. Then there exist, , andsuch that for anyandand anyqsatisfying (4), every solutionuof equation (1) satisfying
admits the estimate
Proof Suppose to the contrary that the lemma is not true. Then there exist sequences , , , and such that (11) holds,
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 is uniformly bounded in and, therefore, the sequence is equicontinuous in . Taking, moreover, into account (46), by virtue of the ArzeláAscoli lemma, we can assume, without loss of generality, that
By a direct calculation, one can easily verify that
whence, in view of (49)(51), we get
Thus and is a solution of equation (1_{0}).
Now let , , and be from the assertion of Lemma 3.2. Assume, without loss of generality, that and for any natural n. Then, by virtue of Lemma 3.2, (46), and (47), the estimates
are fulfilled. Moreover, in view of (48), we have
and
Taking, moreover, into account (50), we get from (52) that
and thus satisfies the conditions
On account of (44) and (48), there exist , , and such that
and
Then it follows from (52) that
Hence, in view of (46), for . Taking now into account (50), we get , and thus is a nontrivial solution of problem (1_{0}), (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 (1_{0}), (2) has no nontrivial solution, then problem (1), (2) has at least one solution.
Let , , , , ϱ, and be from the assertions of Lemmas 3.2 and 3.3. Let, moreover, the sequences and be such that
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 . Moreover, by virtue of Lemma 3.3, the estimate
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 is uniformly bounded and equicontinuous in . Hence, by virtue of the ArzeláAscoli lemma, we can suppose, without loss of generality, that
Taking into account (54), one can easily verify, by a direct calculation, that
Hence, in view of (57) and (58), we get
Thus and is a solution of equation (1).
Further, by virtue of Lemma 3.2 and (55), the inequalities
and
are fulfilled. Hence, on account of (57), we get
and thus and . Consequently, is a solution of problem (1), (2). □
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 (), (2) have unique solutions u and , respectively. Let
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