Abstract
We investigate in this paper the following second-order multipoint boundary value problem: , , , . Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameter by using global bifurcation method. We also obtain the infinite interval of parameter about the existence of positive solution.
1. Introduction
In this paper, we shall study the following second-order multipoint boundary value problem: where , , , and is a positive parameter.
The multipoint boundary value problems for ordinary differential equations play an important role in physics and applied mathematics, and so on. The existence and multiplicity of nontrivial solutions for multipoint boundary value problems have been extensively considered (including positive solutions, negative solutions, or sign-changing solutions) by using the fixed point theorem with lattice, fixed point index theory, coincidence degree theory, Leray-Schauder continuation theorems, upper and lower solution method, and so on (see [1–25] and references therein). On the other hand, some scholars have studied the global structure of nontrivial solutions for second-order multipoint boundary value problems (see [26–32] and references therein).
There are few papers about the global structure of nontrivial solutions for the boundary value problem (1). Motivated by [1, 26–32], we shall investigate the global structure of positive solutions of the boundary value problem (1). In [1], the authors only have studied the existence of positive solutions, but in this paper, we prove that the set of nontrivial positive solutions of the boundary value problem (1) possesses an unbounded connected component.
This paper is arranged as follows. In Section 2, some notation and lemmas are presented. In Section 3, we prove the main results of the boundary value problem (1). Finally, in Section 4, two examples are given to illustrate the main results obtained in Section 3.
2. Preliminaries
Let be a Banach space, be a cone, and be a completely continuous operator.
Definition 1 (see [33]). Let be an open set, , and . If, for any , there exists the solution of the equation , satisfying then is called a bifurcation point of the cone operator .
Definition 2 (see [33]). Let be an open set, , and . If, for any , there exists the solution of the equation , satisfying then is called an asymptotic bifurcation point of the cone operator .
Definition 3 (see [34]). Let be a linear operator and map into . The linear operator is -positive if there exists such that, for any , we can find an integer and real numbers such that
Lemma 4 (see [34]). Let be an open set of . Assume that the operator has no fixed points on . If there exists a linear operator and such that(i);(ii)for some ,then
Lemma 5 (see [34]). Let be completely continuous and be a completely continuous -bounded linear operator. If, for any , , , then , where is unique eigenvalue of corresponding to positive eigenfunction.
Lemma 6 (see [34]). Let be a compact metric space and and be disjoint, compact subsets of . If there does not exist connected subset of such that and , then there exist disjoint compact subsets and such that and .
3. Main Results
Let with the norm ; then is a Banach space. Let . Obviously, is a normal cone of .
In this paper, we always assume that.
Lemma 7 (see [1]). Suppose that holds. Let and be the solutions of respectively. Then(i) is increasing on and ;(ii) is decreasing on and .Let where by [1]. Let where
Define the operators , , and : where and is defined by (7).
Obviously, . It is easy to know that the solutions of the boundary value problem (1) are equivalent to the solutions of the equation Let be the closure of nontrivial positive solution set of (11). Then is also the closure of the nontrivial positive solution set of the boundary value problem (1).
We give the following assumptions:, where is the solution of (4). is continuous, ...
Lemma 8 (see [1]). Suppose that are satisfied. Then, for the operator defined by (9), the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Theorem 9. Suppose that are satisfied. Then(i)the operator defined by (8) has at least a bifurcation point corresponding to positive solution; the operator has no bifurcation points in corresponding to positive solution, where is defined by Lemma 8;(ii) possesses an unbounded connected component passing through , and , where is defined by Lemma 8.
Proof. By , it is easy to know that is completely continuous and is completely continuous; and . By Lemma 8, we have .
By , for any , there exists such that that is, Let . From (8) and (13), for any , we have where . Clearly, is completely continuous and
By Lemma 7 and (6) and (7), it follows that For any , by (14) and (15), we have Let . It follows from (16) that is a -bounded operator by Definition 3. By Krein-Rutman theorem, there exists such that By (14) and (17), we have So, by (18) and Lemma 4, we have In the following, we prove that the operator has at least one bifurcation point on and has no bifurcation points on .
We shall prove that, for any , there must exist and such that where .
Without loss of generality, we may assume that the equation has no solutions on . By (19), we get Obviously, Set By (21) and (22) and the homotopy invariance of fixed point index, there exists such that has a solution . Namely, , where .
Choose . Then there exist and with such that . And Assume that . Then is a bifurcation point of the cone operator .
By (14) and Lemma 5, for any , the equation has no solutions in , where . Hence, has no bifurcation points in corresponding to positive solution, and
Let By the above proof, we know that . If, for any , the connected component of is bounded, which passes through , then is a compact set.
Let be an open neighborhood of in . If , then is a compact metric space, and and are disjoint, compact subsets of . Since has the property of maximal connectivity, there exists no connected subset of such that and . By Lemma 6, we know that there exist two compact subsets of such that
Obviously, the distance . Let Then is an open neighborhood of . Let . Then . Let
Clearly, is a bounded open set of , and . Hence is an open covering of . Since is compact, there exist such that is also an open covering of . Let . Then is a bounded open set of , and
Take sufficiently large such that . For , let , where . Evidently, is an open set of , and . And has no nontrivial solutions on when is sufficiently small.
Let . Then . Since and , we know that has no solutions on . By the general homotopy invariance of topological degree, we get
Since , , so Therefore, by (20), we have which contradicts (26).
Hence, possesses an unbounded connected component passing through
By the above proof and the arbitrariness of , we obtain that (i) the cone operator has at least a bifurcation point (the cone operator has no bifurcation point in ) and (ii) possesses an unbounded connected component passing through , and
Theorem 10. Suppose that and are satisfied. Then the operator has no asymptotic bifurcation points in .
Proof. For any , there exists sufficiently small such that where is defined by Lemma 8.
By , for the above , there exists such that that is, Set ; then Let For any , there exists such that . By (32), we have where It follows from (33) that .
By (33), we get that So Therefore, is bounded. By the arbitrariness of , the operator has no asymptotic bifurcation point in .
By Theorems 9 and 10, we have the following theorem.
Theorem 11. Suppose that are satisfied. Then, for any , the boundary value problem (1) has at least one positive solution.
Furthermore, we take in , that is, the following condition .
Theorem 12. Suppose that and are satisfied. Then(i)the operator has no asymptotic bifurcation points in ;(ii) possesses an unbounded connected component passing through , and
Proof. Since and are satisfied, it follows from Theorem 10 that (i) holds.
By , for sufficiently large , there exists such that that is,Let . From (7) and (35), for any , we have where . Clearly, is completely continuous and
Similar to the proof of Theorem 9, we obtain that the operator has a bifurcation point corresponding to positive solution and possesses an unbounded connected component passing through , and Since can take sufficiently large value, we know that (i) and (ii) hold. The proof is completed.
It follows from Theorem 12 that we have the following theorem.
Theorem 13. Suppose that and are satisfied. Then, for any , the boundary value problem (1) has at least one positive solution.
4. Applications
In this section, two examples are given to illustrate our main results.
Example 14. Consider the following boundary value problem: whereBy simple calculations, . The nonlinear term satisfies the conditions of Theorem 11. Thus, for any , the boundary value problem (37) has at least one positive solution by Theorem 11.
Example 15. Consider the following boundary value problem: whereThe nonlinear term satisfies the conditions of Theorem 13. Thus, for any , the boundary value problem (39) has at least one positive solution by Theorem 13.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
The project is supported by the National Science Foundation of China (11571207).