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Solvability for system of nonlinear singular differential equations with integral boundary conditions
© Li and Zhang; licensee Springer. 2014
Received: 4 April 2014
Accepted: 13 June 2014
Published: 23 September 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.
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 –).
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.
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
for all and .
Letbe completely continuous operator andbe a non-negative continuous concave functional onsuch thatfor. Suppose there existsuch 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 .
which is equivalent to the boundary value problem (2.1). □
, 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 , .
It is easy to prove that is a Banach space and is a cone in .
From the above, we conclude that , that is, . □
3 Main result
Then , .
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 .
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. □
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 .
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.
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.
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. □
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 .
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).
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