# Global Bifurcation in2m-Order Generic Systems of Nonlinear Boundary Value Problems

Authors: Xiaoling Han; Jia Xu; Guowei Dai
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the systems of(-1)mu(2m)=λu+λv+uf(t,u,v),t∈(0,1),u(2i)(0)=u(2i)(1)=0, and 0≤i≤m-1,(-1)mv(2m)=μu+μv+vg(t,u,v),t∈(0,1),v(2i)(0)=v(2i)(1)=0,0≤i≤m-1, where λ,μ∈R are real parameters. f,g:[0,1]×R2→R are Ck,k≥3 functions and f(t,0,0)=g(t,0,0)=0,t∈[0,1]. It will be shown that if the functions, f and g are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, Ck manifolds.


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## Body

## 1. Introduction

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 [1], Rynne considered the global bifurcation in generic systems of coupled nonlinear Sturm-Liouville boundary value problem:
(1.1)L1u≔-(p1u′)′+q1u=μu+uf(x,u,v),x∈(0,1),a10u(0)+b10u′(0)=0,a11u(1)+b11u′(1)=0,L2v:=-(p2v′)′+q2v=νv+vg(x,u,v),x∈(0,1),a20v(0)+b20v′(0)=0,a21v(1)+b21v′(1)=0,
where pi∈C1[0,1], qi∈C[0,1], i=1,2, and qi are positive on [0,1]; μ,ν∈R are parameter; f,g:[0,1]×R2→R are Ck, k≥3 functions. The interesting results are that if the functions f and g are “generic” then for all integers m,n≥0, there are smooth 2-dimensional manifolds 𝒮m1,𝒮n2, of semitrivial solutions of the system which bifurcate from the eigenvalues λm0,μn0, of L1,L2, respectively. Furthermore, there are smooth curves ℬmn1⊂Sm1, ℬmn2⊂Sn2, 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, 𝒩mn, which “links” the curves ℬmn1,ℬmn2. Nodal properties of solutions on 𝒩mn and global properties of 𝒩mn are also discussed.Inspired by [1], in this paper, we consider the 2m-order systems of coupled nonlinear boundary value problems:
(1.2)(-1)mu(2m)=λu+λv+uf(t,u,v),t∈(0,1),u(2i)(0)=u(2i)(1)=0,0≤i≤m-1,(-1)mv(2m)=μu+μv+vg(t,u,v),t∈(0,1),v(2i)(0)=v(2i)(1)=0,0≤i≤m-1,
where λ,μ∈R are real parameters. f,g:[0,1]×R2→R are Ck, k≥3 functions and f(t,0,0)=g(t,0,0)=0, t∈[0,1]. It will be shown that if the functions f and g are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, Ck manifolds. The paper is organized as follows. In Section 2, by computing algebraic multiplicity of eigenvalue (λ,μ), 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 [1], 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 (λ,μ), we get the existence of nontrivial solution by first bifurcation.


## 2. Existence of Nontrivial Solution

LetX={w∈C2m[0,1]:w(2i)(0)=w(2i)(1)=0,0≤i≤m-1}, X0=X∖{0}, Y={w∈C2m-1[0,1]:w(2i)(0)=w(2i)(1)=0,0≤i≤m-1}, Z=C[0,1]; these spaces are endowed with their usual supnorms. We also use the space E=R2×X2 with norm ∥(λ,μ,u,v)∥2=|λ|2+|μ|2+∥u∥2+∥v∥2. For u,v∈C[0,1], let 〈u,v〉=∫01‍uv.DefineL:X→Z by
(2.1)Lu:=(-1)mu(2m),u∈X,
then L is invertible and 〈Lu,v〉=〈u,Lv〉.LetGm(t,s) be the Green’s function of the problem
(2.2)(-1)mw(2m)(t)=0,t∈(0,1),w(2i)(0)=w(2i)(1)=0,0≤i≤m-1,
then
(2.3)Gi(t,s)=∫01G(t,ξ)Gi-1(ξ,s)dξ,2≤i≤m,
where
(2.4)G1(t,s)=G(t,s)={t(1-s),0≤t≤s≤1,s(1-t),0≤s≤t≤1.
So problem (1.2) is equivalent to
(2.5)u(t)=λ∫01‍Gm(t,s)u(s)ds+λ∫01‍Gm(t,s)v(s)ds+∫01‍Gm(t,s)u(s)f(s,u(s),v(s))ds,v(t)=μ∫01‍Gm(t,s)u(s)ds+μ∫01‍Gm(t,s)v(s)ds+∫01‍Gm(t,s)v(s)g(s,u(s),v(s))ds.
In fact, problem (2.5) can be expressed in the form
(2.6)(uv)=λ(AA00)(uv)+μ(00AA)(uv)+(N(u,v)M(u,v)),
where A:Y→Y is given by
(2.7)Ax(t)=∫01‍Gm(t,s)x(s)ds,N(u,v)(t):=∫01‍Gm(t,s)u(s)f(s,u(s),v(s))ds,M(u,v)(t):=∫01‍Gm(t,s)v(s)g(s,u(s),v(s))ds,
note that N(u,v)/∥(u,v)∥→0, M(u,v)/∥(u,v)∥→0 for (u,v) near (0,0) in Y2.We discuss the bifurcation phenomena for problem (2.6).From [2] we know that the set {(λ,μ)∈R2:(λ,μ,0,0)  is a bifurcation point for (2.6)} that is contained in the set
(2.8)ΣA{(λ,μ)∈R2:(uv)=[λ(AA00)+μ(00AA)(uv)hasanontrivialsolution(uv)inY2]}.Note that the equation definingΣA is equivalent to the system
(2.9)Lu=λ(u+v),Lv=μ(u+v).
If (2.9) has a nontrivial solution (uv), then u and v can be shown to solve
(2.10)L(L-(λ+μ))x=0.
It is known in [3] that L has a discrete sequence of simple eigenvalues
(2.11)0<λ1<λ2<⋯<λn→+∞.
It can thus be shown that ΣA={(λ,μ):λ+μ=λn,n=1,2,…}.Theorem  2.1 of [2] guarantees that elements of ΣA of odd algebraic multiplicity are bifurcation points. Suppose (λ,μ) satisfies λ+μ=λn, and Lxn=λnxn. It is easy to show that (λ,μ) has geometric multiplicity 1. In fact, since u and v solve (2.10), it must be the case that u=αxn, v=βxn. Then (2.9) yields
(2.12)(λn00λn)(αβ)xn=(λλμμ)(αβ)xn.
Then (2.12) has a nontrivial solution with β=(μ/λ)α if λ≠0; α=0 if (λ,μ)=(0,λn). So (λ,μ) has geometric multiplicity 1. We will shown that the algebraic multiplicity at (λ,μ) is also 1.Assume for the moment thatλ=0, μ=λn. In this case, the equation
(2.13)(L-λ-λ-μL-μ)2(uv)=(00)
implies
(2.14)(L-λ-λ-μL-μ)(uv)=(0βxn),
where β∈R. We have u=0 from Lu=0, so Lv=μv+βxn=λnv+βxn; then 〈Lv,xn〉=〈λnv+βxn,xn〉, so β=0. It follows that the algebraic multiplicity at (0,λn) is 1. By the homotopy invariance of Leray-Schauder degree, the algebraic multiplicity at (λ,μ) is odd for each (λ,μ) such that λ+μ=λn.We have established the following result.Theorem 2.1.
Ifλn is an eigenvalue of L, then the points belonging to the set {(λ,μ):λ+μ=λn} are all bifurcation points.From Theorem  3.3 of [2] one has the following.Theorem 2.2.
Suppose that(λ0,μ0)∈ΣA. Then there is a two-dimensional continuum 𝒞n emerging in R2×Y2 from (λ0,μ0,0,0). If (λ0,μ0)≠(0,λn) or (λn,0), then at least locally, all nontrivial solutions (λ,μ,u,v) in 𝒞n are such that u and v have n-1 simple zeros in (0,1).


## 3. Genericity Result

We begin by stating the basic transversality theorems as given in [4] (see Theorems 1.1, 1.2 and Remark A1.1 in the appendix of [4]). Let X,Y,Z be real, separable Banach spaces. Let V⊂X,W⊂Y be open sets, and let F:V×W→Z be a Ck, k≥1 mapping such that for every w0∈W, F(·,w0) is a Fredholm mapping of index l<k, that is, for all (v0,w0)∈V×W; the linear operator DvF(v0,w0) is Fredholm with index l<k (where DF:X×Y→Z,DvF:X→Z, DwF:Y→Z, will denote, resp., the Fréchet derivative of F). We say that 0 is a regular value of F if the operator DF(v0,w0) is onto every point (v0,w0) such that F(v0,w0)=0. 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 [4]).
If 0 is a regular value ofF, then the set
(3.1)Θ={w0∈W:0isaregularvalueofF(·,w0)}
is a residual subset of W. For every w0∈Θ, the set
(3.2){v∈V:F(v,w0)=0}
is the disjoint union of a finite, or countable, collection of connected Ck submanifolds of X of dimension l.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 f and g, we need an appropriate space of functions and a topology on this space. Let ℱ be the set of all real valued Ck functions defined on [0,1]×R2. We define a topology on ℱ as follows. For any f∈ℱ and any continuous, positive function ϵ:R2→R+, we define an ϵ-neighbourhood of f by
(3.3)Uϵ(f)={h∈ℱ:∑|α|≤k‍|Dαf(t,η,ζ)-Dαh(t,η,ζ)|<ε(η,ζ),(t,η,ζ)∈[0,1]×R2}
(here Dα is the usual multi-index notation for partial derivatives); a subset ℰ⊂ℱ is defined to be open if and only if for every f∈ℰ there is a function ϵ such that the Uϵ(f) of f lies in ℰ.Letℱ1⊂ℱ denote the set of functions such that f(·,0,0)=0. Let 𝒩(f,g) denote the set of nontrivial solutions of (1.2), we get the following result.Theorem 3.2.
There is a residual set𝒢⊂ℱ12 such that if (f,g)∈𝒢, then the set 𝒩 consists of a countable collection of 2-dimensional, Ck manifolds.Proof .
LetE0=R2×X02 and, for any (f,g)∈ℱ12, define a Ck function G(f,g):E0→Z2 by
(3.4)G(f,g)(λ,μ,u,v)=(Lu-λu-λv-f(·,u,v)u,Lv-μu-μv-g(·,u,v)v,),(λ,μ,u,v)∈E0.
Clearly, 𝒩=G(f,g)-1(0). For any ρ>1, let
(3.5)Aρ={(λ,μ,u,v)∈E0:∥(λ,μ,u,v)∥<ρ,∥u∥>ρ-1,∥v∥>ρ-1},
and let A¯ρ be the closure of Aρ. For any integer r≥1, let 𝒯r be the set of functions (f,g)∈ℱ12 such that G(f,g) is transverse on Ar+δ for some sufficiently small δ>0.We first prove that each set𝒯r is open in ℱ12. For any (f,g)∈ℱ12, ρ>1, the set 𝒩(f,g)∩A¯ρ is compact. Thus (f,g)∉𝒯r if and only if there exists z∈𝒩(f,g)∩A¯r such that the operator DG(f,g)(z) is not surjective (here the derivative is with respect to (λ,μ,u,v)). Thus, in deciding whether (f,g)∈𝒯r, the values of f and g on |u|>r, |v|>r are irrelevant. Therefore, it suffices to show that if (fi,gi)∈ℱ12∖𝒯r, i=1,2,…, is a sequence converging to (f0,g0)∈ℱ12 with respect to the Ck topology on [0,1]×[-r,r]2, then (f0,g0)∉𝒯r. Now, for each i, there exists zi∈𝒩(fi,gi)∩A¯r such that DG(fi,gi)(zi) is not surjective. Furthermore, it can be shown that the sequence zi converges (in E) to a point z0 (using standard Sobolev embedding and regularity arguments; see for instance, the proof of Theorem  2.1 in [4] or the proof of Theorem  3.a.1 in [5]). Then, by continuity, z0∈𝒩(f0,g0)∩A¯r, and the operator DG(f0,g0)(z0)=limi→∞DG(fi,gi)(zi) cannot be surjective since this would contradict standard perturbation results for Fredholm operators (see for instance, Theorem  13.6 in [6]). Thus (f0,g0)∉𝒯r, which completes the proof that 𝒯r is open in ℱ12.We will now show that the sets𝒯r are dense in ℱ12. Choose an arbitrary, fixed (f,g)∈ℱ12 and r≥1. Let θr:R→R be a decreasing C∞ function such that θr(s)=1 if s≤r+1 and θr(s)=0 if s≥r+2. For any set of functions γ=(γij)i,j=12∈Γ:=Ck([0,1])4 we define fγ:[0,1]×R2→R by
(3.6)fγ(t,η,ζ)=f(t,η,ζ)+θr((η2+ζ2)1/2)(γ11(x)η+γ12(x)ζ),
for (t,η,ζ)∈[0,1]×R2; we define gγ:[0,1]×R2→R similarly, using γ21,γ22. Clearly, for any given positive functions ε1,ε2 (see the definition of the topology on ℱ) we have fγ∈Uϵ1(f), gγ∈Uϵ2(g), for all sufficiently small γ∈Γ. We now define a function H:Ar+1×Γ→Z×Z by
(3.7)H(λ,μ,u,v,γ)=G(fγ,gγ)(λ,μ,u,v),(λ,μ,u,v,γ)∈Ar+1×Γ.
Since Γ is a Banach space (with norm ∥γ∥2=∑i,j=12∥γi,j∥k2), we can differentiate H with respect to γ and apply transversality results. Clearly, H is Ck.Lemma 3.3.
The derivativeDH(z0) is surjective at any point z0∈H-1(0).Proof.
We must show that for anyζ=(ζ1,ζ2)∈Z2, the following equation can be solved for (λ-,μ-,u-,v-,γ-)∈E×Γ:
(3.8)H(u,v)0(u-,v-)+H(λ,μ)0(λ-,μ-)+Hγ0γ-=ζ,
where H(u,v)0,H(λ,μ)0,Hγ0 denote the derivatives at z0=(λ0,μ0,u0,v0,γ0) with respect to (u,v), (λ,μ) and γ, respectively; these operators have the form
(3.9)H(u,v)0(u-,v-)=(Lu-Lv-)-((λ0+f0+u0fu0)u-+(λ0+u0fv0)v-(μ0+g0+v0gv0)v-+(μ0+v0gu0)u-),H(λ,μ)0(λ-,μ-)=(-λ-(u0+v0)-μ-(u0+v0)),Hγ0γ-=(γ-11u0u0+γ-12u0v0γ-21u0v0+γ-22v0v0),
(where f0 denotes fγ0(·,u0,v0), etc.). From [4, Page 301] we know that the operator H(u,v)0 is Fredholm with index 0 and, if ξ=(ξ1,ξ2)∈R(H(u,v)0)⊥⊂Z2, then ξ is a solution of a homogeneous coupled pair of linear ordinary differential equations. Let d0=dimR(H(u,v)0)⊥. If d0=0 there is nothing further to prove, so suppose that d0≥1 and let {ξi∈Z2:i=1,…,d0} be a basis for R(H(u,v)0)⊥. Since the functions u0, v0 and ξi, i=1,…,d0 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 I0⊂(0,1) and a number δ>0, such that for x∈I0 we have |u0(x)|>δ, |v0(x)|>δ, and the functions ξi, i=1,…,d0 are linearly independent on I0.
Now, sinceu0 and v0 are bounded away from 0 on I0, for any ψ=(ψ1,ψ2)∈C0∞(I0)2, the equation Hγ0γ-=ψ has a solution γ-∈C0k(I0)4. Thus, we can choose γj∈C0k(I0)4, j=1,…,d0 such that 〈Hγ0γj,ξi〉=δij, i,j=1,…,d0 (δij is the Kronecker delta); thus the set {Hγ0γj:j=1,…,d0} spans a complement of R(H(u,v)0)⊥ in Z2, and so the operator (u-,v-,γ-)→H(u,v)0(u-,v-)+Hγ0γ-∈Z2 is surjective. This proves Lemma 3.3.
It now follows from Lemma3.3 and Theorem 3.1 (it can readily be verified that H satisfies the necessary Fredholm conditions) that there exists a residual set Θ⊂Γ such that if γ∈Θ then the mapping H(·,γ):Ar+1→Z2 is transverse, that is, (fγ,gγ)∈𝒯r. Since (f,g)∈ℱ12 was arbitrary, it follows from this that the set 𝒯r is dense in ℱ12. Now let 𝒢=⋂r=1∞‍𝒯r. By construction, 𝒢 is a residual subset of ℱ12 and, for any (f,g)∈𝒢, the mapping G(f,g):E0→Z2 is transverse. It follows from H(u,v)0, and the results in [7], that DG(f,g)(z) is Fredholm with index 2. So from Theorem 3.1 the zero set of this mapping is a 2-dimensional, Ck manifold. Thus we have proved Theorem 3.2.Remark 3.4.
For more information about generic result or nodal solutions, see [8–12].



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