Abstract
In this paper, we consider the existence of positive solutions for secondorder differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundary value problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments α and also in the case when , . In all cases, the argument β can change the character on , see problem (1). It means that β can be delayed in some set and advanced in . An example is added to illustrate the results.
MSC: 34B10.
Keywords:
boundary value problems with delayed and advanced arguments; nonlocal boundary conditions; cone; existence of positive solutions; a fixed point theorem1 Introduction
Put , . Let us consider the following boundary value problem:
where , denote linear functionals on given by
involving Stieltjes integrals with suitable functions A and B of bounded variation on J. It is not assumed that , are positive to all positive x. As we see later, the measures dA, dB can be signed measures.
We introduce the following assumptions:
H_{1}: , , A and B are functions of bounded variation;
H_{2}: and h does not vanish identically on any subinterval;
Recently, the existence of multiple positive solutions for differential equations has been studied extensively; for details, see, for example, [131]. However, many works about positive solutions have been done under the assumption that the firstorder derivative is not involved explicitly in nonlinear terms; see, for example, [3,6,814,17,20,2527,30]. From this list, only papers [912,14,20,30] concern positive solutions to problems with deviating arguments. On the other hand, there are some papers considering the multiplicity of positive solutions with dependence on the firstorder derivative; see, for example, [2,4,5,7,15,16,18,19,2124,28,29,31]. Note that boundary conditions (BCs) in differential problems have important influence on the existence of the results obtained. In this paper, we consider problem (1) which is a problem with dependence on the firstorder derivative with BCs involving Stieltjes integrals with signed measures of dA, dB appearing in functionals , ; moreover, problem (1) depends on deviating arguments.
For example, in papers [2,4,15,18,22,24], the existence of positive solutions to secondorder differential equations with dependence on the firstorder derivative (but without deviating arguments) has been studied with various BCs including the following:
by fixed point theorems in a cone (such as AveryPeterson, an extension of Krasnoselskii’s fixed point theorem or monotone iterative method) with corresponding assumptions:
For example, in papers [811,20,22,30], the existence of positive solutions to secondorder differential equations including impulsive problems, but without dependence on the firstorder derivative, has been studied with various BCs including the following:
under corresponding assumptions by fixed point theorems in a cone (such as AveryPeterson, LeggettWilliams, Krasnoselskii or fixed point index theorem). See also paper [13], where positive solutions have been discussed for secondorder impulsive problems with boundary conditions
here has the same form as in problem (1) with signed measure dA appearing in functional .
Positive solutions to secondorder differential equations with boundary conditions that involve Stieltjes integrals have been studied in the case of signed measures in papers [25,26] with BCs including, for example, the following:
The main results of papers [25,26] have been obtained by the fixed point index theory for problems without deviating arguments. The study of positive solutions to boundary value problems with Stieltjes integrals in the case of signed measures has also been done in papers [3,7,13,14,27] for secondorder differential equations (also impulsive) or thirdorder differential equations by using the fixed point index theory, the AveryPeterson fixed point theorem or fixed point index theory involving eigenvalues.
Note that BCs in problem (1) with functionals , cover some nonlocal BCs, for example,
for some constants , and some functions , . In our paper, the assumption that the measures dA, dB in the definitions of , are positive is not needed. More precisely, one needs to choose the above functions , in such a way that the assumption H_{4} holds. It means that , can change sign on J.
A standard approach (see, for example, [2527]) to studying positive solutions of boundary value problems such as (1) is to translate problem (1) to a Hammerstein integral equation
to find a solution as a fixed point of the operator by using a fixed point theorem in a cone. , , are corresponding continuous functions while and have the same form as in problem (1). G denotes a Green function connected with our problem, so in our case it is given by
In our paper, we eliminate and from problem (2) to obtain the equation with a corresponding operator , and then we seek solutions as fixed points of this operator .
Note that if we put in the BCs of problem (1), then this new problem is more general than the previous one because in this case someone, for example, can take , . In this paper, we try to explain why for some cases we have to discuss problem (1) with constants or .
To apply such a fixed point theorem in a cone to problem (1), we have to construct a suitable cone K. Usually, we need to find a nonnegative function κ and a constant such that for and for and (see, for example, [2527]) to work with the inequality
Indeed, for problems without deviating arguments, someone can use any interval . It means that when on J, then we can take in the boundary conditions of problem (1) to work with the inequality
for ζ, ϱ such that , with ; see Section 5.
Note that for problems with delayed or advanced arguments, we have to use interval or , respectively. We see that if , then for problem (1) with deviated arguments. It shows that the approach from papers [2527] needs a little modification to problems with delayed or advanced arguments. Consider the situation on J. In this case, we can put in the boundary conditions of problem (1) to find a constant to work with the inequality
see Section 3. For the case on J, we can put to work similarly as in Section 3; see Section 4. Note that in the above three cases for the argument β, we need only the assumption , which means that β can change the character in J.
Note that in cited papers, positive solutions to differential equations with dependence on the firstorder derivative have been investigated only for problems without deviating arguments, see [2,4,5,7,15,16,18,19,2124,28,29,31]. Moreover, BCs in problem (1) cover some nonlocal BCs discussed earlier.
Motivated by [2527], in this paper, we apply the fixed point theorem due to AveryPeterson to obtain sufficient conditions for the existence of multiple positive solutions to problems of type (1). In problem (1), an unknown x depends on deviating arguments which can be both of advanced or delayed type. To the author’s knowledge, it is the first paper when positive solutions have been investigated for such general boundary value problems with functionals , and with deviating arguments α, β in differential equations in which f depends also on the firstorder derivative. It is important to indicate that problems of type (1) have been discussed with signed measures of dA, dB appearing in Stieltjes integrals of functionals , .
The organization of this paper is as follows. In Section 2, we present some necessary lemmas connected with our main results. In Section 3, we first present some definitions and a theorem of Avery and Peterson which is useful in our research. Also in Section 3, we discuss the existence of multiple positive solutions to problems with delayed argument α, by using the above mentioned AveryPeterson theorem. At the end of this section, an example is added to verify theoretical results. In Section 4, we formulate sufficient conditions under which problems with advanced argument α have positive solutions. In the last section, we discuss problems of type (1) when on J.
2 Some lemmas
Let us introduce the following notations:
Lemma 1Let, , . Assume thatAandBare functions of bounded variation and, moreover,
with
Then
Here, VarAdenotes the variation of a functionAonJ.
Proof Note that in case (i), we have
so
Hence,
Combining this with the relation
we obtain
This proves case (i).
In case (ii), similarly,
so
Hence,
Adding to this the relation
we get the result in case (ii). This ends the proof. □
Remark 1 If we assume that A and B are increasing functions, then there exists such that
Hence,
Similarly, we can show that
Now, the constant M from Lemma 1 has the form
Consider the following problem:
Let us introduce the assumption.
H_{0}: A and B are functions of bounded variation and
for
We require the following result.
Lemma 2Let the assumption H_{0}hold and let. Then problem (3) has a unique solution given by
with
Proof Integrating the differential equation in (3) two times, we have
Put and use the boundary conditions from problem (3) to obtain
Now, finding from this and then substituting it to formula (4), we have
Next, putting , we can find , and then substitute it to formula (5) to obtain
Now, we have to eliminate and from (6). If u is a solution of (6), then
Solving this system with respect to , and then substituting to (6), we have the assertion of this lemma. This ends the proof. □
Define the operator T by
with
We consider the Banach space with the maximum norm . Define the cone by
with
Let us introduce the following assumption.
H_{4}: A and B are functions of bounded variation and
(i) , , , , for where , , , δ, Δ are defined as in the assumption H_{0},
Lemma 3Let the assumptions H_{1}H_{4}hold. Then.
Proof Clearly, is a positive solution of problem (1) if and only if solves the operator equation . Then
Note that , in view of the assumptions H_{1}, H_{2}, H_{4} and the positivity of Green’s function G.
Hence, Tu is concave and on J.
Finally, we show that
To do it, we consider two steps. Let .
so
It yields
so
It yields
so
Hence,
It shows . This ends the proof. □
Remark 2 Take , . Note that the measure changes the sign and is increasing. It is easy to show that
If we assume that , then , , , .
Remark 3 Take , . Note that the measure changes the sign and is increasing. It is easy to show that
If we assume that , then , , , .
Remark 4 Let , , . Then the assumptions H_{3}, H_{4} hold if one of the following conditions is satisfied:
We consider only case (i). First of all, we see that dA, dB change the sign and are increasing. Indeed, for , , we have
It means that the assumption H_{3} holds. Moreover,
It proves that the assumption H_{4} holds.
By a similar way, we prove the assertion in case (ii) or (iii).
3 Positive solutions to problem (1) with delayed arguments
Now, we present the necessary definitions from the theory of cones in Banach spaces.
Definition 1 Let E be a real Banach space. A nonempty convex closed set is said to be a cone provided that
Note that every cone induces an ordering in E given by if .
Definition 2 A map Φ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E if is continuous and
Similarly, we say the map φ is a nonnegative continuous convex functional on a cone P of a real Banach space E if is continuous and
Definition 3 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Let φ and Θ be nonnegative continuous convex functionals on P, let Φ be a nonnegative continuous concave functional on P, and let Ψ be a nonnegative continuous functional on P. Then, for positive numbers a, b, c, d, we define the following sets:
We will use the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1).
Theorem 1 (see [1])
LetPbe a cone in a real Banach spaceE. Letφand Θ be nonnegative continuous convex functionals onP, let Φ be a nonnegative continuous concave functional onP, and let Ψ be a nonnegative continuous functional onPsatisfyingforsuch that for some positive numbersandd,
is completely continuous and there exist positive numbersa, b, cwithsuch that
ThenThas at least three fixed pointssuch that
and
We apply Theorem 1 with the cone K instead of P and let . Now, we define the nonnegative continuous concave functional Φ on K by
Now, we can formulate the main result of this section.
Theorem 2Let the assumptions H_{1}H_{4}hold with, . Let, . In addition, we assume that there exist positive constantsa, b, c, d, M, and such that
with
and
Then problem (1) has at least three nonnegative solutions, , satisfying, ,
Proof Basing on the definitions of T, we see that is equicontinuous on J, so T is completely continuous.
Let , so . By Lemma 1, , so , . Assumption (A_{1}) implies .
Moreover, in view of (7),
Combining it, we have
Now, we need to show that condition (S_{1}) is satisfied. Take
This proves that
Let for . Then for , so , . Assumption (A_{2}) implies . Hence,
Moreover,
It yields
This proves that condition (S_{1}) holds.
Now, we need to prove that condition (S_{2}) is satisfied. Take and . Then
so condition (S_{2}) holds.
Indeed, , so . Suppose that with . Note that , . Then
and finally,
This shows that condition (S_{3}) is satisfied.
Since all the conditions of Theorem 1 are satisfied, problem (1) has at least three nonnegative solutions , , such that for , and
This ends the proof. □
Example Consider the following problem:
where
with . For example, we can take , on J with fixed . Indeed, , , , , and
Note that , so the measure changes the sign on J. Moreover,
so the assumption H_{4} holds; see Remark 4. Next,
Put , , , then , , . Let , . Then
and
All the assumptions of Theorem 2 hold, so problem (8) has at least three positive solutions.
Remark 5 We can also construct an example in which, for example, to use the results of Remark 4. Note that also this measure changes the sign.
4 Positive solutions to problem (1) with advanced arguments
In this section, we consider the case when on J, so the interval is now replaced by . It means that we can put with in the boundary conditions of problem (1) because someone can take as an example. Let us introduce the cone by
with
Now . Functionals Ψ, Θ, φ are defined as in Section 3. We formulate only the main result using the cone instead of K (see Theorem 2); the proof is similar to the previous one.
Theorem 3Let the assumptions H_{1}H_{4}hold with, . Let, . In addition, we assume that there exist positive constantsa, b, c, d, M, and such that
with
and
Then problem (1) has at least three nonnegative solutions, , satisfying, ,
5 Positive solutions to problem (1) for the case when on J
In this section, we consider problem (1) when on J and . It means that now for some fixed constants ζ, ϱ such that . For we can show that , , . Now, for , we introduce the cone by
Functionals Ψ, Θ, φ are defined as in Section 3; the cone K is now replaced by .
Theorem 4Let the assumptions H_{1}H_{4}hold with. Let, , . In addition, we assume that there exist positive constantsa, b, c, d, M, and such that
with
and
Then problem (1) has at least three nonnegative solutions, , satisfying, ,
6 Conclusions
In this paper, we have discussed boundary value problems for secondorder differential equations with deviating arguments and with dependence on the firstorder derivative. In our research, the deviating arguments can be both delayed and advanced. By using the fixed point theorem of Avery and Peterson, new sufficient conditions for the existence of positive solutions to such boundary problems have been derived. An example is provided for illustration.
Competing interests
The author declares that he has no competing interests.
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