Applications of Fixed Point and Approximate AlgorithmsView this Special Issue
On Six Solutions for m-Point Differential Equations System with Two Coupled Parallel Sub-Super Solutions
Under the assumption of two coupled parallel subsuper solutions, the existence of at least six solutions for a kind of second-order m-point differential equations system is obtained using the fixed point index theory. As an application, an example to demonstrate our result is given.
In this paper, we consider the following second-order -point boundary value problems of nonlinear equations system where are continuous and satisfying with for and .
Multipoint boundary value problems arise in many applied sciences for example, the vibrations of a guy wire composed of parts with a uniform cross-section throughout, but different densities in different parts can be set up as a multipoint boundary value problems (see ). Many problems in the theory of elastic stability can be modelled by multipoint boundary value problems (see ). The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev . Subsequently, Gupta  studied certain three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, the solvability of more general nonlinear multipoint boundary value problems has been discussed by several authors using various methods. We refer the readers to [5–12] and the references therein.
In the recent years, many authors have studied existence and multiplicity results for solutions of multipoint boundary value problems via the well-ordered upper and lower solutions method, see [8, 13, 14] and the references therein. However, only in very recent years, some authors considered the multiplicity of solutions under conditions of non-well-ordered upper and lower solutions. For some abstract results concerning conditions of non-well-ordered upper and lower solutions, the readers are referred to recent works [15–18].
In , Xu et al. considered the following second-order three-point boundary value problem where . He obtained the following result. First, let us give the following condition to be used later.
There exists such that Let the function be for .
Theorem 1.1. Suppose that holds, and are two strict lower solutions of (1.2), and are two strict upper solutions of (1.2), and . Moreover, assume for some . Then, the three-point boundary value problem (1.2) has at least six solutions.
We would also like to mention the result of Yang , in . Yang studied the following integral boundary value problem where and , and denote the Riemann-Stieltjes integrals of with respect to and , respectively. Some sufficient conditions for the existence of either none, or one, or more positive solutions of the problem (1.5) were established. The main tool used in the proofs of existence results is a fixed point theorem in a cone, due to Krasnoselskii and Zabreiko.
At the same time, we note that Webb and Lan  have considered the first eigenvalue of the following linear problem they also investigated the existence and multiplicity of positive solutions of several related nonlinear multipoint boundary value problems. Furthermore, Ma and O'Regan  studied the spectrum structure of the problem (1.6), and the authors obtained the concrete computational method and the corresponding properties of real eigenvalue of (1.6) by constructing an auxiliary function. Their work is very fundamental to further study for multipoint boundary value problems. By extending and improving the work in , Rynne  showed that the associated Sturm-Liouville problem consisting of (1.6) has a strictly increasing sequence of simple eigenvalues with eigenfunctions .
Very recently, Kong et al.  were concerned with the general boundary value problem with a variable By relating (1.7) to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, the existence and nonexistence of nodal solutions of (1.7) were obtained. We also point out that Webb  made the excellent remark on some existence results of symmetric positive solutions obtained in some recent papers and the author also corrected the values of the principle eigenvalue previously given in some examples.
In this paper, by means of two coupled parallel subsuper solutions, we obtain some sufficient conditions for the existence of six solutions for (1.1) and our main tool is based on the fixed point index theory. At the end of this paper, we will give an example which illustrates that our work is true. Our method stems from the paper .
2. Preliminaries and a Lemma
In the section, we shall give some preliminaries and a lemma which are fundamental to prove our main result.
Let be an ordered Banach space in which the partial ordering ≤ is induced by a cone . A cone is said to be normal if there exists a constant , such that implies , the smallest is called the normal constant of . is called solid, if int, that is, has nonempty interior. Every cone in defines a partial ordering in given by if and only if . If and , we write ; if cone is solid and int, we write . is called total if . Let be a bounded linear operator. is said to be positive if . An operator is strongly increasing, that is, implies . If is a linear operator, is strongly increasing implying is strongly positive.
Let be an ordered Banach space, a total cone in , the partial ordering ≤ induced by . is a positive completely continuous linear operator. Let the spectral radius of , the conjugated operator of , and the conjugated cone of . Since is a total cone (i.e., ), according to the famous Krein-Rutman theorem (see ), we infer that if , then there exist and , such that Fixed such that (2.1) holds. For , let then is also a cone in . One can refer [26–28] for definition and properties about the cones.
Lemma 2.2 (see ). Suppose that . Then, the BVP has Green's function where
For convenience, we list the following hypotheses which will be used in our main result. are strictly increasing; there exist constants and such that ,, , and (iv) , where ; there exist constants , such that, for all , we have , ,, and; uniformly for , uniformly for , where is the first eigenvalue of the following boundary value problem:
It is well known that , where linear operator is defined as .
3. Main Results
Theorem 3.1. Assume , hold, then BVP (1.1) has at least six distinct continuous solutions.
Proof. It is easy to check that BVP (1.1) is equivalent to the following integral equation systems:
where is defined as in Lemma 2.2. By , we know that .
Let , define the norm in as . Then, is a Banach space with this norm. Let , . Then, is a normal and solid cone. Set , such that it is clear that the solutions of (1.1) are equivalent to the fixed points of .
Set , let where , then . It is easy to see that is a strongly positive completely continuous operator, and it follows from and the continuity of that is a strongly positive completely continuous operator. Since are strictly increasing continuous functions, we know that is a strictly increasing continuous bounded operator. By , we can prove that is completely continuous. We infer from the increasing properties of and that is increasing.
Let , then satisfy By (iii) and (i), we have It follows from (ii),(iv), and the increasing property of that Equations (3.2), (3.5), and (3.6) imply that Similarly, by , we obtain
By [20, Lemma 3], we get that satisfies condition H. Therefore, there exist , such that By the definition of spectral radius of completely continuous operator, we have , and combining (3.9), we infer that Let , then . According to the proof in , we can get that satisfies condition .
By condition , we obtain that there exists , such that Equations (3.12)–(3.15) imply Since are continuous in , so they are bounded, then there exists such that By virtue of (3.18) and the increasing properties of and , one shows
In addition, if satisfy , then it follows from (3.12), (3.18), and the increasing property of that where Similarly, if satisfy , then combining the increasing property of with (3.14) and (3.18), we know that where . Let . By (3.21) and (3.22), we get that if or , it is obvious that It follows from (3.16), (3.19), and (3.23) that In a similar way, from (3.12) and (3.14), we can show that there exists such that Let . It follows from (3.24) and (3.25) that if or , then In a similar way, from (3.13) and (3.15), we can prove that there exists constant such that if or , then Let , , , , , then or , ; therefore, in virtue of expression of , and (3.26), we have where . Since , one can show
This implies that there exist and such that Similarly, we get by (3.27) that there exist and such that We get by (3.7) that Let . Since is normal, then is bounded (see ). Choose such that Let , then and is a bounded open set. By the proof of Theorem 2.1 in , we can show that where satisfies .
Equation (3.34) implies that has no fixed point on . It is easy to prove that is a retract of , which together with (3.32) implies that the fixed point index over with respect to is well defined, and a standard proof yields Set , then for any , we have . It follows from (3.34) that , and by (3.35) and the homotopy invariance of the fixed point index, we get Let . By means of usual method (see ), we get that It is evident that has no fixed point on , by (3.36), (3.37), and the additivity of the fixed point index, we have Set , and choose such that Let Similarly to the proof of (3.37) and (3.38), we get that Choose such that Set By virtue of (3.31) and the same method as that for (3.34), we have By (3.44), similarly to the proof of (3.38), we can prove that Equations (3.37)–(3.41), (3.45) imply that has at least six distinct fixed points, that is, the system of differential equations (1.1) has at least six solution in .
4. An Example
In this section, we present a simple example to explain our results.
Consider the following second-order three-point BVP for nonlinear equations system: where , , , , , are strictly increasing continuous functions, and condition is satisfied. Choose . Some direct calculations show Therefore, condition is satisfied.
Choosing , it is easy to check that Therefore, condition is satisfied. At last, we will check condition , by the method of [9, 31], and we consider the linear eigenvalue problem Let . By the paper , we know that the sequence of positive eigenvalue of (4.4) is exactly given by , where is the sequence of positive solutions of . In , Han obtained that , moreover . It is easy to know that uniformly for , uniformly for . Therefore, condition is also satisfied. Consequently, all conditions of Theorem 3.1 are satisfied, and we get the system of differential equations (4.1) has at least six solutions in .
The authors are very grateful to the anonymous referees for their valuable suggestions. This project is supported by the National Natural Science Foundation of China (10971046), the University Science and Technology Foundation of Shandong Provincial Education Department (J10LA62), the Natural Science Foundation of Shandong Province (ZR2009AM004, ZR2010AL014), and the Doctor of Scientific Startup Foundation for Shandong University of Finance (08BSJJ32).
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