- Open Access
Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions
© Jankowski; licensee Springer. 2013
- Received: 20 September 2012
- Accepted: 15 January 2013
- Published: 17 January 2013
In this paper, we consider the existence of positive solutions for second-order 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.
- boundary value problems with delayed and advanced arguments
- nonlocal boundary conditions
- existence of positive solutions
- a fixed point theorem
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:
H1: , , A and B are functions of bounded variation;
H2: and h does not vanish identically on any subinterval;
H3: or for , , .
Recently, the existence of multiple positive solutions for differential equations has been studied extensively; for details, see, for example, [1–31]. However, many works about positive solutions have been done under the assumption that the first-order derivative is not involved explicitly in nonlinear terms; see, for example, [3, 6, 8–14, 17, 20, 25–27, 30]. From this list, only papers [9–12, 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 first-order derivative; see, for example, [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 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 first-order derivative with BCs involving Stieltjes integrals with signed measures of dA, dB appearing in functionals , ; moreover, problem (1) depends on deviating arguments.
or , respectively.
here has the same form as in problem (1) with signed measure dA appearing in functional .
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 second-order differential equations (also impulsive) or third-order differential equations by using the fixed point index theory, the Avery-Peterson fixed point theorem or fixed point index theory involving eigenvalues.
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 H4 holds. It means that , can change sign on J.
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 .
for ζ, ϱ such that , with ; see Section 5.
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 first-order derivative have been investigated only for problems without deviating arguments, see [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 28, 29, 31]. Moreover, BCs in problem (1) cover some nonlocal BCs discussed earlier.
Motivated by [25–27], in this paper, we apply the fixed point theorem due to Avery-Peterson 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 first-order 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 Avery-Peterson 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.
Here, VarA denotes the variation of a function A on J.
This proves case (i).
we get the result in case (ii). This ends the proof. □
Let us introduce the assumption.
We require the following result.
Solving this system with respect to , and then substituting to (6), we have the assertion of this lemma. This ends the proof. □
Let us introduce the following assumption.
, , , , for where , , , δ, Δ are defined as in the assumption H0,
, , , , , , , , , .
Lemma 3 Let the assumptions H1-H4 hold. Then .
Note that , in view of the assumptions H1, H2, H4 and the positivity of Green’s function G.
Hence, Tu is concave and on J.
To do it, we consider two steps. Let .
Step 1. Let . Then or and .
It shows . This ends the proof. □
If we assume that , then , , , .
If we assume that , then , , , .
It proves that the assumption H4 holds.
By a similar way, we prove the assertion in case (ii) or (iii).
Now, we present the necessary definitions from the theory of cones in Banach spaces.
for all and all , and
Note that every cone induces an ordering in E given by if .
for all and .
for all and .
Definition 3 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
We will use the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1).
Theorem 1 (see )
is completely continuous and there exist positive numbers a, b, c with such that
(S1): and for ,
(S2): for with ,
(S3): and for with .
Note that . Put .
Now, we can formulate the main result of this section.
(A1): for ,
(A2): for ,
(A3): for .
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 (A1) implies .
This proves that .
This proves that condition (S1) holds.
so condition (S2) holds.
This shows that condition (S3) is satisfied.
This ends the proof. □
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.
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.
(B1): for ,
(B2): for ,
(B3): for .
Functionals Ψ, Θ, φ are defined as in Section 3; the cone K is now replaced by .
(C1): for ,
(C2): for ,
(C3): for .
In this paper, we have discussed boundary value problems for second-order differential equations with deviating arguments and with dependence on the first-order 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.
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