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
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
where
involving Stieltjes integrals with suitable functions A and B of bounded variation on J. It is not assumed that
We introduce the following assumptions:
H_{1}:
H_{2}:
H_{3}:
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
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:
or
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
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
for some constants
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
In our paper, we eliminate
Note that if we put
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
Indeed, for problems without deviating arguments, someone can use any interval
for ζ, ϱ such that
Note that for problems with delayed or advanced arguments, we have to use interval
see Section 3. For the case
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
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
2 Some lemmas
Let us introduce the following notations:
Lemma 1Let
with
(i)
(ii)
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
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
with
Proof Integrating the differential equation in (3) two times, we have
Put
Now, finding from this
Next, putting
Now, we have to eliminate
Solving this system with respect to
Define the operator T by
with
We consider the Banach space
with
Let us introduce the following assumption.
H_{4}: A and B are functions of bounded variation and
(i)
(ii)
Lemma 3Let the assumptions H_{1}H_{4}hold. Then
Proof Clearly,
Note that
Note that
Hence, Tu is concave and
We next show that
Finally, we show that
To do it, we consider two steps. Let
Step 1. Let
Let
so
It yields
Let
so
It yields
Step 2. Let
so
Hence,
It shows
Remark 2 Take
If we assume that
Remark 3 Take
If we assume that
Remark 4 Let
(i)
(ii)
(iii)
We consider only case (i). First of all, we see that dA, dB change the sign and are increasing. Indeed, for
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
(i)
(ii)
Note that every cone
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
for all
Similarly, we say the map φ is a nonnegative continuous convex functional on a cone P of a real Banach space E if
for all
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 onPsatisfying
for all
is completely continuous and there exist positive numbersa, b, cwith
(S_{1}):
(S_{2}):
(S_{3}):
ThenThas at least three fixed points
and
We apply Theorem 1 with the cone K instead of P and let
Note that
Now, we can formulate the main result of this section.
Theorem 2Let the assumptions H_{1}H_{4}hold with
with
and
(A_{1}):
(A_{2}):
(A_{3}):
Then problem (1) has at least three nonnegative solutions
and
Proof Basing on the definitions of T, we see that
Let
Moreover, in view of (7),
Combining it, we have
This proves that
Now, we need to show that condition (S_{1}) is satisfied. Take
Then
for
This proves that
Let
Moreover,
It yields
This proves that condition (S_{1}) holds.
Now, we need to prove that condition (S_{2}) is satisfied. Take
so condition (S_{2}) holds.
Indeed,
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
This ends the proof. □
Example Consider the following problem:
where
with
Note that
so the assumption H_{4} holds; see Remark 4. Next,
Put
and
for
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,
4 Positive solutions to problem (1) with advanced arguments
In this section, we consider the case when
with
Now
Theorem 3Let the assumptions H_{1}H_{4}hold with
with
and
(B_{1}):
(B_{2}):
(B_{3}):
Then problem (1) has at least three nonnegative solutions
and
5 Positive solutions to problem (1) for the case when
α
(
t
)
=
t
on J
In this section, we consider problem (1) when
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
with
and
(C_{1}):
(C_{2}):
(C_{3}):
Then problem (1) has at least three nonnegative solutions
and
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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