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Global Bifurcation in -Order Generic Systems of Nonlinear Boundary Value Problems
We consider the systems of , and , where are real parameters. are functions and . It will be shown that if the functions, and are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, manifolds.
Many scientific and technological problems that are modeled mathematically by systems of ODEs, for example, mathematical models of series circuits and mechanical systems involving several springs attached in series can lead to a system of differential equations. Furthermore, such systems are often encountered in chemical, ecological, biological, and engineering applications, thereby attracting constant interest of researchers in recent years, on several aspects of the problems; we focus on one aspect here, namely, the existence of solutions, from the point of view of bifurcations.
In , Rynne considered the global bifurcation in generic systems of coupled nonlinear Sturm-Liouville boundary value problem: where , , , and are positive on ; are parameter; are , functions. The interesting results are that if the functions and are “generic” then for all integers , there are smooth 2-dimensional manifolds , of semitrivial solutions of the system which bifurcate from the eigenvalues , of , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of nontrivial solutions. It is shown that there is a single such manifold, , which “links” the curves . Nodal properties of solutions on and global properties of are also discussed.
Inspired by , in this paper, we consider the -order systems of coupled nonlinear boundary value problems: where are real parameters. are , functions and , . It will be shown that if the functions and are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, manifolds. The paper is organized as follows. In Section 2, by computing algebraic multiplicity of eigenvalue , we get the set of bifurcation points of problem (1.2) and obtain the existence of nontrivial solution. In Section 3, we get some genericity result.
Note that problem (1.2) is different from the problem (1.1). In , the manifolds of nontrivial solution come from the secondary bifurcations which take place along the smooth curves in the manifolds of semitrivial solutions. In this paper, by computing algebraic multiplicity of eigenvalue , we get the existence of nontrivial solution by first bifurcation.
2. Existence of Nontrivial Solution
Let , , , ; these spaces are endowed with their usual supnorms. We also use the space with norm . For , let .
Define by then is invertible and .
We discuss the bifurcation phenomena for problem (2.6).
Note that the equation defining is equivalent to the system If (2.9) has a nontrivial solution , then and can be shown to solve It is known in  that has a discrete sequence of simple eigenvalues It can thus be shown that .
Theorem 2.1 of  guarantees that elements of of odd algebraic multiplicity are bifurcation points. Suppose satisfies , and . It is easy to show that has geometric multiplicity 1. In fact, since and solve (2.10), it must be the case that , . Then (2.9) yields Then (2.12) has a nontrivial solution with if ; if . So has geometric multiplicity 1. We will shown that the algebraic multiplicity at is also 1.
Assume for the moment that , . In this case, the equation implies where . We have from , so ; then , so . It follows that the algebraic multiplicity at is 1. By the homotopy invariance of Leray-Schauder degree, the algebraic multiplicity at is odd for each such that .
We have established the following result.
Theorem 2.1. If is an eigenvalue of , then the points belonging to the set are all bifurcation points.
From Theorem 3.3 of  one has the following.
Theorem 2.2. Suppose that . Then there is a two-dimensional continuum emerging in from . If or , then at least locally, all nontrivial solutions in are such that and have simple zeros in .
3. Genericity Result
We begin by stating the basic transversality theorems as given in  (see Theorems 1.1, 1.2 and Remark A1.1 in the appendix of ). Let be real, separable Banach spaces. Let be open sets, and let be a , mapping such that for every , is a Fredholm mapping of index , that is, for all ; the linear operator is Fredholm with index (where , , will denote, resp., the Fréchet derivative of ). We say that is a regular value of if the operator is onto every point such that . Also, a subset of a topological space is said to be residual if it contains the intersection of a countable collection of open dense sets. Note that the intersection of a countable collection of residual sets is also residual.
Theorem 3.1 (see ). If 0 is a regular value of , then the set is a residual subset of . For every , the set is the disjoint union of a finite, or countable, collection of connected submanifolds of of dimension .
A property of the elements of a topological space is said to be generic if it holds for all elements in a residual subset of the space. Since we wish to discuss properties of the systems (1.2) which hold for “generic” functions and , we need an appropriate space of functions and a topology on this space. Let be the set of all real valued functions defined on . We define a topology on as follows. For any and any continuous, positive function , we define an -neighbourhood of by (here is the usual multi-index notation for partial derivatives); a subset is defined to be open if and only if for every there is a function such that the of lies in .
Let denote the set of functions such that . Let denote the set of nontrivial solutions of (1.2), we get the following result.
Theorem 3.2. There is a residual set such that if , then the set consists of a countable collection of 2-dimensional, manifolds.
Proof . Let and, for any , define a function by Clearly, . For any , let and let be the closure of . For any integer , let be the set of functions such that is transverse on for some sufficiently small .
We first prove that each set is open in . For any , , the set is compact. Thus if and only if there exists such that the operator is not surjective (here the derivative is with respect to . Thus, in deciding whether , the values of and on , are irrelevant. Therefore, it suffices to show that if , , is a sequence converging to with respect to the topology on , then . Now, for each , there exists such that is not surjective. Furthermore, it can be shown that the sequence converges (in ) to a point (using standard Sobolev embedding and regularity arguments; see for instance, the proof of Theorem 2.1 in  or the proof of Theorem 3.a.1 in ). Then, by continuity, , and the operator cannot be surjective since this would contradict standard perturbation results for Fredholm operators (see for instance, Theorem 13.6 in ). Thus , which completes the proof that is open in .
We will now show that the sets are dense in . Choose an arbitrary, fixed and . Let be a decreasing function such that if and if . For any set of functions we define by for ; we define similarly, using . Clearly, for any given positive functions (see the definition of the topology on ) we have , , for all sufficiently small . We now define a function by Since is a Banach space (with norm ), we can differentiate with respect to and apply transversality results. Clearly, is .
Lemma 3.3. The derivative is surjective at any point .
Proof. We must show that for any , the following equation can be solved for :
where denote the derivatives at with respect to , and , respectively; these operators have the form
(where denotes , etc.). From [4, Page 301] we know that the operator is Fredholm with index and, if , then is a solution of a homogeneous coupled pair of linear ordinary differential equations. Let . If there is nothing further to prove, so suppose that and let be a basis for . Since the functions , and , are all nonzero and can be regarded as solutions of homogeneous linear ordinary differential equations (or systems), it follows from the uniqueness of the solution of the initial value problem for such equations that these functions cannot be identically zero on any open set. Thus there must exist an open interval and a number , such that for we have , , and the functions , are linearly independent on .
Now, since and are bounded away from 0 on , for any , the equation has a solution . Thus, we can choose , such that , ( is the Kronecker delta); thus the set spans a complement of in , and so the operator is surjective. This proves Lemma 3.3.
It now follows from Lemma 3.3 and Theorem 3.1 (it can readily be verified that satisfies the necessary Fredholm conditions) that there exists a residual set such that if then the mapping is transverse, that is, . Since was arbitrary, it follows from this that the set is dense in . Now let . By construction, is a residual subset of and, for any , the mapping is transverse. It follows from , and the results in , that is Fredholm with index 2. So from Theorem 3.1 the zero set of this mapping is a 2-dimensional, manifold. Thus we have proved Theorem 3.2.
This paper is supported by the NNSF of China (nos. 11101335, 11201378, and 11261052), Gansu Provincial Department of Education Fund (no. 1101-02), and Project of Science and Technology Bureau of Lanzhou (no. 2011-2-72).
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