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Solvability for system of nonlinear singular differential equations with integral boundary conditions
Boundary Value Problems volume 2014, Article number: 158 (2014)
By using the Banach contraction principle and the Leggett-Williams fixed point theorem, this paper investigates the uniqueness and existence of at least three positive solutions for a system of mixed higher-order nonlinear singular differential equations with integral boundary conditions:
where the nonlinear terms , satisfy some growth conditions, are linear functionals given by , involving Stieltjes integrals with positive measures, and . We give an example to illustrate our result.
MSC: 34B16, 34B18.
The purpose of this paper is to establish the uniqueness and existence of at least three positive solutions for a system of mixed higher-order nonlinear singular differential equations with integral boundary conditions,
where , , are allowed to be singular at and/or , , , do not vanish identically on any subinterval of , the functionals are linear functionals given by , involving Stieltjes integrals with positive measures, and .
The theory of boundary value problems with integral conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, and plasma physics can be reduced to boundary value problems with integral conditions, which included, as special cases, two-point, three-point and multi-point boundary value problems considered by many authors (see –).
In recent years, to the best of our knowledge, although there are many papers concerning the existence of positive solutions for th order boundary value problems with different kinds of boundary conditions for system (see – and the references therein), results for the system (1.1) are rarely seen. Moreover, the methods mainly depend on the Krasonsel’skii fixed point theorem, fixed point index theory, the upper and lower solution technique, some new fixed point theorem for cones, etc. For example, in , by applying the Krasonsel’skii fixed point theorem, Henderson and Ntouyas studied the existence of at least one positive solution for the following system:
where and are nonsingular. In , , of the system (1.3) are replaced by , , and in , , of the system (1.3) are replaced by , , where is singular. By using fixed point index theory and the Krasonsel’skii fixed point theorem, the existence of one and/or two positive solutions is established.
On the other hand, Webb  gave a unified method of tackling many nonlocal boundary value problems, which have been applied to the study of the problem with Stieltjes integrals,
We mention that Stieltjes integrals are also used in the framework of nonlinear boundary conditions in several papers (see – and the references therein). In particular, Yang  studied the existence of positive solutions for the following system by using fixed point index theory in a cone:
Infante and Pietramala  studied the following system as a special case to illustrate the obtained theory:
By constructing a special cone and using fixed point index theory, Cui and Sun  studied the existence of at least one positive solution for the system with Stieltjes integrals,
By using fixed point index theory and a priori estimates achieved by utilizing some properties of concave functions, Xu and Yang  showed the existence and multiplicity positive solutions for the system of the generalized Lidstone problems, where the system are mixed higher-order differential equations.
Motivated by the work of the above papers, we aim to investigate the solvability for the system (1.1). The main features are as follows: Firstly, the method we adopt, which has been widely used, is different from –, –. Secondly, the nonlinear terms we considered here satisfy some growth conditions. In –, , , , , the sublinear or superlinear conditions are used for . Moreover, the form of the Stieltjes integrals we consider here is quite general, which involves that of the Stieltjes integrals in –, ,  and is different from . This implies that the case of boundary conditions (1.1) covers the multi-point boundary conditions and also the integral boundary conditions in a single framework.
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and several lemmas. In Section 3, by applying the fixed-point theorem, we obtain the uniqueness and existence of at least three positive solutions for the system (1.1). In Section 4, we give an example to illustrate our result.
2 Preliminaries and lemmas
Let be a real Banach space. A nonempty, closed, convex set is said to be a cone, which satisfies the following conditions:
Let be a real Banach space with cone . A map is said to be a non-negative continuous concave functional on if is continuous and
for all and .
Let , be two numbers such that and be a non-negative continuous concave functional on . We define the following convex sets:
Letbe completely continuous operator andbe a non-negative continuous concave functional onsuch thatfor. Suppose there existsuch that
Thenhas at least three fixed points, , insuch that
is said to be a positive solution of the system (1.1) if and only if satisfies the system (1.1) and , , for any .
Let, then the boundary value problem
has the integral representation
By Taylor’s formula, we have
so, we reduce the equation of problem (2.1) to an equivalent integral equation,
By (2.4) and (2.5), combining with the conditions , , and letting , we have
Substituting and into (2.4) and (2.5), we have
which is equivalent to the boundary value problem (2.1). □
The function, defined by (2.3) has the following properties:
, for ;
, for ,
Throughout this paper, we assume that the following condition is satisfied.
(H1): does not vanish identically on any subinterval of , , , where is defined by Lemma 2.6 and there exists such that .
By (H1), we can choose a subinterval such that . Let ; it is easy to see that . By Lemma 2.6, we have , .
By Lemma 2.5, it is easy to prove that is a positive solution of the system (1.1) if and only if is a positive solution of the following integral system:
Let be a Banach space endowed with the norm , where , and define the cone by
It is easy to prove that is a Banach space and is a cone in .
Define the operator by
For any , considering , , we have , , for . From (2.6) and Lemma 2.6, we have
It follows from (2.8) and Lemma 2.6 that we have
Similarly, it follows from (2.7) and Lemma 2.6 that we have
From the above, we conclude that , that is, . □
3 Main result
For convenience, we use the following notation:
Then , .
Suppose that the condition (H1) is satisfied and there exist non-negative numbers, , , , such that for alland:
where, . Then the system (1.1) has a unique positive solution in.
By Lemma 2.5, the system (1.1) has a unique positive solution if and only if the operator has a unique fixed point in .
Define , and , such that
First we show that , where . For , we have
In the same way, we obtain
Now we shall prove that is a contraction. Let ; applying (2.6) we get
With the help of (3.1) and (3.2) we obtain
this together with (3.3) implies
Similarly, applying (2.7), with the help of (3.1) and (3.2) we have
Taking (3.4) and (3.5) into account we have
where . So, is a contraction, hence it has a unique point fixed in which is the unique positive solution of the system (1.1). The proof is completed. □
Define the non-negative continuous concave functional on by
We observe here that , for each .
Throughout this section, we assume that , , are four positive numbers satisfying .
Suppose that the condition (H1) is satisfied and there exist non-negative numbers: , , such thatand, satisfy the following growth conditions:
(H2): , , ;
(H3): , , , ;
(H4): , , , ;
(H5): , , , .
Then the system (1.1) has at least three positive solutions, , such that, andwith.
It is clear that the existence of positive solutions for the system (1.1) is equivalent to the existence of fixed points of in .
We first prove that is a completely continuous operator. In fact, if , then and by condition (H2), we have
Thus, by condition (H3), we have
Therefore, , that is, . Standard applications of the Arzelà-Ascoli theorem imply that is a completely continuous operator.
Now, we show that conditions (A1)-(A3) of Lemma 2.3 are satisfied.
Firstly, let , , it follows that , , which shows that , and, for , we have , . By condition (H4) of Theorem 3.2, we obtain
Similarly, by condition (H4) of Theorem 3.2, we can obtain
Therefore, condition (A1) of Lemma 2.3 is satisfied.
Secondly, in a completely analogous argument to the proof of , by condition (H5) of Theorem 3.2, condition (A2) of Lemma 2.3 is satisfied.
Finally, we show that condition (A3) of Lemma 2.3 is satisfied. If and , then
Therefore, condition (A3) of Lemma 2.3 is satisfied.
Thus, all conditions of Lemma 2.3 are satisfied. By Lemma 2.3, the system (1.1) has at least three positive solutions , , such that , and , with . The proof is completed. □
Consider the following system of nonlinear mixed-order ordinary differential equations:
Then the system (4.1) is equivalent to the following system of nonlinear integral equations:
We choose , , , , , , and
By Lemma 2.6, we have
Choose ; by Remark 2.7, we obtain . Then by direct calculation we obtain
It is easy to verify that the condition (H1) holds. Let , , , , , , . Also, it is easy to verify that , , , satisfy conditions (H2)-(H5).
Thus, by Theorem 3.2, the system (4.1) has at least three positive solutions , , such that , and with .
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The authors are grateful to the referees for their careful reading. This research is supported by the Nature Science Foundation of Anhui Provincial Education Department (Grant Nos. KJ2014A252 and KJ2013A248) and Professors (Doctors) Scientific Research Foundation of Suzhou University (Grant No. 2013jb04).
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. All authors read and approved the final manuscript.
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Li, Y., Zhang, H. Solvability for system of nonlinear singular differential equations with integral boundary conditions. Bound Value Probl 2014, 158 (2014). https://doi.org/10.1186/s13661-014-0158-7
- positive solutions
- integral boundary conditions
- higher-order differential equations
- fixed point theorem