Some consequences of an existence result by Kiguradze and Partsvania for singular Dirichlet problems
© Figueroa and Lópaez Pouso; licensee Springer 2014
Received: 13 December 2013
Accepted: 13 June 2014
Published: 24 September 2014
A sharp theorem by Kiguradze and Partsvania ensures the existence of extremal solutions between given lower and upper solutions for singular Dirichlet problems. This paper has a twofold purpose: first, we present a new sufficient condition for one of Kiguradze and Partsvania’s assumptions, and we illustrate its applicability in the study of a new family of examples; second, we combine Kiguradze and Partsvania’s theorem with Heikkilä’s iterative technique to obtain a new result on the existence of extremal solutions for a more general class of discontinuous and singular functional boundary value problems. In particular, our framework includes classical equations with delay (or advance), singularities with respect to the independent variable, and implicit functional boundary conditions.
MSC: 34A12, 34A36.
1 Introduction and first results
We are going to review the results in a paper by Kiguradze and Partsvania  and then we are going to use them in the proof of a new more general existence result of extremal solutions for functional and singular second-order problems.
Our revision of the results in  is not merely a reproduction, as it includes some contributions of our own. Specifically, Proposition 1.1 provides us with a new sufficient condition for a technical assumption in , and we will use it in the analysis of a new family of examples.
for all the mapping is measurable;
for almost all (a.a.) the mapping is continuous;
- (iii)for every the function defined almost everywhere by
is integrable on for every sufficiently small .
Let denote the set of all real valued functions which are absolutely continuous over each compact subinterval of .
- (a)and admits the representation
- (b)the inequality
- (c)the limits and exist and
We say that is a solution of (1.1) if it is both a lower and an upper solution, and we are interested in solutions of (1.1) which are extremal in the following sense.
We say that a solution of (1.1) is the greatest (smallest) solution of (1.1) in if and () on for any other solution .
When both the smallest and the greatest solutions of (1.1) in exist we call them the extremal solutions of (1.1) in .
The main result in  reads as follows.
, Theorem 1]
The condition ‘’ is hard to check in practice. For that reason, the authors included some sufficient hypotheses which imply it. One of them is a one-sided Nagumo condition, in the spirit of . The following is a slight modification of Corollary 11 in .
Assume that there exist, which are, respectively, a lower and an upper solution of problem (1.1) such thatfor alland that the following conditions hold:
(H1):For eachand allthe mappingis measurable.
(H2):For almost all, alland allthe functionsandare continuous.
is integrable onfor every sufficiently small.
Then problem (1.1) has the extremal solutions in. Moreover, ifis a solution of (1.1) such that, thenfor all.
there exists such that .
The previous two inequalities contradict (1.5), thus proving that is possible for no .
One can prove in an analogous way that for all . □
Finding a function to check (1.5) with is not easy, but the following proposition quite simplifies that task.
Let, , , , and.
The assumptions guarantee that is continuous and integrable.
The proof is similar if . □
We close this section with a new example of the applicability of Kiguradze and Partsvania’s results. Notice how our Proposition 1.1 simplifies the verification of condition (H4) in Corollary 1.1.
has the extremal solutions between the lower solution and the upper solution .
Conditions (H1) through (H3) in Corollary 1.1 are obviously satisfied. The hardest part concerns showing that (H4) is satisfied with, for instance, the following choice of the functions and the constants: let us define for all , for all , , , and, according to (1.4), .
and this implies (1.2) and (1.3).
Notice that the relation between and implies that for all , so our problem reduces to proving that is integrable.
which is integrable on because .
Similar computations show that is integrable on , and then it is integrable on the whole of because it is continuous. Hence, Proposition 1.1 ensures that condition (1.5) holds with .
2 Singular functional problems
where and .
where and represent, respectively, the initial and the final state of the solution. Problem (2.1) also includes multipoint boundary conditions, which have received a lot of attention in the last few years; see , .
We begin by introducing the concepts of lower and upper solutions for problem (2.1). We do this simply by extending Definition 1.1 in the obvious way.
- (a)and admits the representation
- (b)the inequality
- (c)for all we have
Our existence result for (2.1) will be proven by means of Kiguradze and Partsvania’s results described in the previous section along with Heikkilä’s generalized iterative technique. The following result will be essential.
, Theorem 1.2.2]
We are at last in a position to introduce and prove our main result.
and assume that the following conditions hold:
(F1):For eachthe mappingsatisfies conditions (i), (ii), and (iii), anduniformly in, in the sense that the functiondoes not depend on.
(F2):For a.a. , alland allthe mappingis nonincreasing on.
for alland all.
(B3):For alland all, the mappingis nondecreasing in.
Then problem (2.1) has extremal solutions in.
where for each (respectively, each ), (respectively, ) is the unique fixed point of the contractive mapping (respectively, ).
and therefore is continuous by virtue of (B1).
A similar argument shows that is well defined and continuous too.
Now, condition (F1) guarantees by application of Theorem 1.1 that the extremal solutions in of problem exist and, in particular, is well defined.
Moreover, condition (F1) also ensures that there exists a continuous and integrable function such that for all and all .
which implies that has its fixed point, namely , inside the interval . Hence, on .
A similar argument shows that for all .
Summing up, is a lower solution of problem , whose greatest solution between and is precisely . Therefore on the whole of .
Claim 3: Operatorhas the extremal fixed points. Let be a monotone sequence. As is nondecreasing, the sequence is monotone and bounded and therefore it has a pointwise limit, say .
We are going to prove that tends to uniformly on and that .
Since the are continuous at and , we deduce from (2.4), (2.5), (2.6), and condition (B1) that is uniformly equicontinuous on . Hence tends to uniformly on and is continuous on .
Moreover, (2.4) implies that is absolutely continuous on , thus proving that .
Now we can apply Lemma 2.1 to ensure that operator has the extremal fixed points in .
Claim 4: The greatest fixed point ofcorresponds with the greatest solution of problem (2.1) in. Let be the greatest fixed point of . As , it is clear that is a solution of problem (2.1). Now, let be another solution of (2.1). In this case, both and solve , and so taking into account that is the greatest of such solutions. Now, condition (2.2) implies that .
The existence of the least solution of (2.1) in follows from a proper redefinition of the operator . □
The next less general version of Theorem 2.1 is easier to use in practice.
Under the conditions of Theorem 2.1, replace (F1) by
:For eachthe functionsatisfies (H1)-(H4) with the same functionin (H4) for all.
Then the conclusion of Theorem 2.1holds.
3 An example of application
We finish this paper with a corollary of our main result which applies to a large family of boundary conditions satisfying stronger conditions that (B1)-(B3). This corollary is illustrated later with a concrete example.
Then the following conditions are sufficient to guarantee the conclusion of Theorem 2.1:
():The functionis uniformly continuous in.
():For alland allthe functionis Lipschitzian with Lipschitz constant, and the functionis nondecreasing. Moreover, there exists.
We will show that these conditions imply (B1)-(B3).
and now (B1) follows from () and the fact that is bounded in the compact set .
Then (B2) follows from (3.1).
Finally, condition () follows from the facts that is nondecreasing and is nonnegative. □
We claim that for sufficiently small value of problem (3.2) has the extremal solutions between the lower solution and the upper solution . Notice that and are solutions of the differential equation in (3.2) but not solutions of the whole problem.
and then the condition (H4) is satisfied as in Example 1.1 (with the same function for all ), thanks to condition (3.3).
The remaining conditions over are easily verified, and therefore we omit further explanations.
and so condition () holds.
and so condition () holds.
This work was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.
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