# Fredholm alternative for the second-order singular Dirichlet problem

Alexander Lomtatidze12 and Zdeněk Opluštil2*

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Žižkova 22, Brno, 616 62, Czech Republic

2 Faculty of Mechanical Engineering, Institute of Mathematics, Brno University of Technology, Technická 2, Brno, 616 69, Czech Republic

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Boundary Value Problems 2014, 2014:13  doi:10.1186/1687-2770-2014-13

 Received: 13 September 2013 Accepted: 19 November 2013 Published: 13 January 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Consider the singular Dirichlet problem

where are locally Lebesgue integrable functions. It is proved that if

then the Fredholm alternative remains true.

MSC: 34B05.

##### Keywords:
singular Dirichlet problem; Fredholm alternative

### 1 Introduction

Consider the boundary value problem

(1)

(2)

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

(3)

guarantees the validity of the Fredholm alternative. More precisely, if (3) holds, then problem (1), (2) is uniquely solvable for any q satisfying

(4)

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

(5)

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

(6)

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).

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 (10), (2) has no nontrivial solution.

Remark 1.1 In Theorem 1.1, condition (4) is essential and cannot be omitted. Indeed, let , , for , and

(7)

Evidently, (6) holds and problem (10), (2) has no nontrivial solution. On the other hand, a general solution of (1) is of the form

However, for , we have

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

(8)

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.

Theorem 1.2Let (6) hold and problem (10), (2) have no nontrivial solution. Then there existssuch that for anyqsatisfying (4), the solutionuof problem (1), (2) admits the estimate

(9)

Consider now a sequence of equations

where are such that

(10)

Let, moreover, satisfy (4) and

(11)

Corollary 1.1Let (4), (6) hold and problem (10), (2) have no nontrivial solution. Let, moreover, (11) and (12) be fulfilled. Then the problems (1), (2) and (), (2) have unique solutionsuand, respectively,

(12)

and

(13)

### 2 Auxiliary statements

In this section, we consider the equation

where , q satisfies (4), and

(14)

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

(15)

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 above-proved 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

(16)

(17)

for, while the solutionvof the problem

(18)

(19)

for.

Proof By virtue of (15) and [[1], Lemma 2.2], the initial value problems

and

have unique solutions and , respectively, and the estimates

(20)

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

(21)

for . Analogously, the (unique) solution v of problem (19) is of the form

(22)

for , where and are solutions of the problems

and

respectively, , , and the estimates

(23)

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

(24)

(25)

Proof Let . Then it is clear that either

(26)

or

(27)

or

(28)

Assume that (27) (resp., (28)) holds. Then, in view of (25), there is (resp., ) such that

(29)

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

(30)

Suppose now that (29) holds. Then either there is such that

(31)

or there is a sequence such that

(32)

(33)

If (32) holds, then evidently for and, consequently, (31) is fulfilled. On the other hand, if (34) holds, then, by virtue of the above-proved, 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

(34)

(35)

while every solutionuof equation (1) satisfying

(36)

(37)

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

(38)

has a unique solution v and, moreover, for any (resp., ), the estimate

(39)

holds. Let us show that

(40)

Assume the contrary, let (41) be violated. Define

Then there exist and (resp., and ) such that

(41)

(42)

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 (10), (2) have no nontrivial solution. Then there exist, , andsuch that for anyandand anyqsatisfying (4), every solutionuof equation (1) satisfying

Proof Suppose to the contrary that the lemma is not true. Then there exist sequences , , , and such that (11) holds,

(43)

and

(44)

Introduce the notation

Then it is clear that

(45)

and

(46)

Moreover, it follows from (45) that

(47)

and, consequently,

(48)

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 Arzelá-Ascoli lemma, we can assume, without loss of generality, that

(49)

where and, moreover,

(50)

By a direct calculation, one can easily verify that

whence, in view of (49)-(51), we get

Thus and is a solution of equation (10).

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

(51)

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 (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.

Let , , , , ϱ, and be from the assertions of Lemmas 3.2 and 3.3. Let, moreover, the sequences and be such that

(52)

By virtue of Lemma 3.3, the problem

has no nontrivial solution. Hence, by virtue of Proposition 2.1, the problem

(53)

has a unique solution . Moreover, by virtue of Lemma 3.3, the estimate

(54)

holds, where

On the other hand, on account of Lemma 3.1 and (55), we have

(55)

where

In view of (53), (55), and (56), the sequence is uniformly bounded and equicontinuous in . Hence, by virtue of the Arzelá-Ascoli lemma, we can suppose, without loss of generality, that

(56)

where and, moreover,

(57)

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

(58)

Then it is clear that

where

(59)

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. FSI-S-11-3 ‘Modern methods of mathematical problem modelling in engineering’. Research was also supported by RVO: 67985840.

### References

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2. 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