# A Fold Bifurcation Theorem of Degenerate Solutions in a Perturbed Nonlinear Equation

Authors: Ping Liu; Yuwen Wang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We consider a nonlinear equationF(ε,λ,u)=0, where the parameter ε is a perturbation
parameter, F is a differentiable mapping from R×R×X to Y, and X, Y are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation
theorem.


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## Body

## 1. Introduction

In [1, 2], Crandall and Rabinowitz proved two celebrated theorems which are now regarded as foundation of the analytical bifurcation theory in infinite-dimensional spaces and both results are based on the implicit function theorem. In [3], we obtained the generalized saddle-node bifurcation theorem by the generalized inverse. In [4], we proved a perturbed problem using Morse Lemma. For a more general introduction to bifurcation theory and other related methods in nonlinear analysis, see, for example, [5–7]. On the other hand, [8–11] provide a more detailed introduction to mathematical models in some recent new results in the application of bifurcation theory including chemical reactions, population ecology, and nonautonomous differential equations.In this paper, we continue the work of [3] and obtain an abstract bifurcation theorem under the opposite condition in [4]. We consider the solution set of
(1.1)F(ε,λ,u)=0,
where ε indicates the perturbation. Fix ε=ε0; let (λ0,u0) be a solution of F(ε0,·,·)=0. From the implicit function theorem, a necessary condition for bifurcation is that Fu(ε0,λ0,u0) is not invertible; we call (ε0,λ0,u0) a degenerate solution. In [12],  Shi shows the persistence and the bifurcation of degenerate solutions when ε varies near ε0 by the implicit function theorem and the saddle-node bifurcation theorem. In this paper, we prove a new perturbed bifurcation theorem by the generalized saddle-node bifurcation theorem.In the paper, we use∥·∥ as the norm of Banach space X and 〈·,·〉 as the duality pair of a Banach space X and its dual space X*. For a nonlinear operator F, we use Fu as the partial derivative of F with respect to argument u. For a linear operator L, we use N(L) as the null space of L and R(L) as the range of L.


## 2. Preliminaries

Definition 2.1 (see [13]).
LetX,Y be Banach spaces, and let A∈ℒ(X,Y) be a linear operator. Then, A+∈ℒ(Y,X) is called the generalized inverse of A if it satisfies(i)
AA+A=A,(ii)
A+AA+=A+.Definition 2.2 (see [13]).
LetX,Y,and A be the same as in Definition 2.1. If A∈ℒ(X,Y) has the bounded linear generalized inverse A+,then A is called a generalized regular operator.Lemma 2.3 (see [13]).
LetA∈ℒ(X,Y), then  A is a generalized regular operator if and only if  N(A),R(A) are topologically complemented in  X,Y, respectively. In this case,  I-A+A,  AA+ are bounded linear projectors from  X,  Y into  N(A),  R(A), respectively.We recall the generalized saddle-node bifurcation in [3] and give an alternate proof here using the generalized Lyapunov-Schmidt reduction.Theorem 2.4 (generalized saddle-node bifurcation).
LetV⊂R×X be a neighborhood of (λ0,u0),F∈C1(V,Y). Suppose that(i)
F(λ0,u0)=0;(ii)
Fu(λ0,u0):X→Y is a generalized regular operator, and
(2.1)dim⁡N(Fu(λ0,u0))≥codimR(Fu(λ0,u0))=1,(iii)
Fλ(λ0,u0)∉R(Fu(λ0,u0)).
LetZ=R((Fu(λ0,u0))+), then the subset {(λ,u)|F(λ,u)=0} contains the curve (λ(s),u(s))=(λ(s),u0+sw0+z(s)) near (λ0,u0), where w0∈N(Fu(λ0,u0))∖{θ}, the mapping z(s) is continuously differentiable near s=0, and λ(0)=λ0,λ′(0)=0,z′(0)=z(0)=θ.Proof.
SinceA=Fu(λ0,u0) is a generalized regular operator, there exist closed subspaces Z in X, Y1 in Y satisfing X=Z⊕N(A), Y=R(A)⊕Y1.
Taking an arbitraryw0∈N(A)∖{θ}, from Lemma 2.3, F(λ,u)=0 is equivalent to
(2.2)(I-AA+)F(λ,u0+sw0+z)=0,AA+F(λ,u0+sw0+z)=0,
where s∈R, z∈Z.
DefineG:R×R×Z→R(A) as
(2.3)G(s,λ,z)=AA+F(λ,u0+sw0+z),G(λ,z)(0,λ0,0)[(τ,ψ)]=AA+(τFλ(λ0,u0)+Fu(λ0,u0)[ψ]),=AA+A[ψ]=A[ψ],
because of (iii), then G(λ,z)(0,λ0,0):R×Z→R(A) is an isomorphism.
For the equationG(s,λ,z)=0, by the implicit function theorem, there exist ε>0 and (λ(s),z(s))∈C1(-ε,ε), with λ(0)=λ0, z(0)=0 satisfying
(2.4)G(s,λ(s),z(s))=0.
From (2.2), we have
(2.5)F(λ(s),u0+sw0+z(s))=0,s∈(-ε,ε).
Differentiating (2.5) with respect to s, we have
(2.6)Fλ(λ(s),u0+sw0+z(s))λ′(s)+Fu(λ(s),u0+sw0+z(s))[w0+z′(s)]=0.
Setting s=0,
(2.7)Fλ(λ0,u0)λ′(0)+Fu(λ0,u0)[w0+z′(0)]=0.
Thus, λ′(0)=0 since (iii) and we have
(2.8)Fu(λ0,u0)[z′(0)]=0,
that is, z′(0)∈N(A)∩Z, we have z′(0)=0.Corollary 2.5.
Assume the conditions in Theorem2.4 are satisfied and
(2.9)dim⁡N(Fu(λ0,u0))=n,N(Fu(λ0,u0))=span⁡{w1,w2,…,wn},
then the direction of the solution curves is determined by
(2.10)λi′′(0)=-〈l,Fuu(λ0,u0)[wi,wi]〉〈l,Fλ(λ0,u0)〉,
where l∈R(Fu(λ0,u0))⊥, i=1,2,…,n. Furthermore, when
(2.11)Fuu(λ0,u0)[wi,wi]∉R(Fu(λ0,u0))
is satisfied, λi′′(0)≠0, and the solution curve {(λi(s),ui(s)):|s|<δ} is a parabola-like curve which reaches an extreme point at (λ0,u0).We illustrate our result by a simple example.Example 2.6.
Define(2.12)F(λ,(xy))=λ-x2-y2=0,
where U=(xy)∈R2, λ∈R. From simple calculations, we obtain
(2.13)FU=(-2x,-2y),Fλ=1,FUU=(-200-2).
We analyze the bifurcation at (0,(00)). It is easy to see that N(FU)=span{w1,w2}, where w1=(10), w2=(01), R(FU)={0}. So, obviously, dimN(FU)=2, codimR(FU)=1, and Fλ∉R(FU). From the above calculation,
(2.14)FUU[w1,w1]=-2,FUU[w2,w2]=-2.
Obviously, FUU(0,(00))[wi,wi]∉R(FU(0,(00))) and λi′′(0)=-2, i=1,2. Thus, we can apply Corollary 2.5 to (2.12). In fact, all solution curves for all wi∈N(FU) form a surface (see Figure 1).Figure 1
Bifurcation diagram of the equationλ-x2-y2=0 in Example 2.6.


## 3. Main Theorems

Applying Theorem2.4, we discuss the bifurcation of solutions of the perturbed problem. We consider the solution set of
(3.1)F(ε,λ,u)=0,
where the parameter ε indicates the perturbation, F∈C1(M,Y), M≡R×R×X, and X, Y are Banach spaces. Let
(3.2)H(ε,λ,u,w)=(F(ε,λ,u)Fu(ε,λ,u)[w]).Suppose that(ε0,λ0,u0,w0) is a solution of H(ε,λ,u,w)=0. For (ε0,λ0,u0)∈M and (3.3)w0∈X1≡{x∈X:‖x‖=1},
by Hahn-Banach theorem, there exists a closed subspace X3 of X with codimension 1 such that X=L(w0)⊕X3, where L(w0)=span{w0} and d(w0,X3)=inf{||w0-x||:x∈X3}>0. Let X2=w0+X3={w0+x:x∈X3}. Then, X2 is a closed hyperplane of X with codimension 1. Since X3 is a closed subspace of X and X3 is also a Banach space in the subspace topology, Hence we can regard M1=M×X2 as a Banach space with product topology. Moreover, the tangent space of M1 is homeomorphic to M×X3 (see [12] for more on the setting).In the following, we will still use the conditions (Fi) on F defined in [12].(F1)
dimN(Fu(ε0,λ0,u0))=codimR(Fu(ε0,λ0,u0))=1, and N(Fu(ε0,λ0,u0))=span{w0};(F2)
Fλ(ε0,λ0,u0)∉R(Fu(ε0,λ0,u0));(F3)
Fλu(ε0,λ0,u0)[w0]∉R(Fu(ε0,λ0,u0));(F4)
Fuu(ε0,λ0,u0)[w0,w0]∉R(Fu(ε0,λ0,u0));(F5)
Fε(ε0,λ0,u0)∉R(Fu(ε0,λ0,u0)).We use the convention that (Fi′) means that the condition defined in (Fi) does not hold.Theorem 3.1.
LetF∈C2(M,Y),  T0=(ε0,λ0,u0,w0)∈M1 such that  H(T0)=(0,0). Suppose that the operator  F satisfies  (F1),  (F2′),  (F3),  (F4′), and  (F5) at  T0. One also assumes that
(3.4)Fuu(ε0,λ0,u0)[v1,w0]+Fλu(ε0,λ0,u0)[w0]∈R(Fu(ε0,λ0,u0)),
where v1∈X3∖{0} is the unique solution of
(3.5)Fλ(ε0,λ0,u0)+Fu(ε0,λ0,u0)[v]=0.
Then, there exists δ>0 such that all the solutions of H(ε,λ,u,w)=(0,0) near T0 form two C2 curves:
(3.6){T1(s)=(ε1(s),λ1(s),u1(s),w1(s)),s∈I=(-δ,δ)},{T2(s)=(ε2(s),λ2(s),u2(s),w2(s)),s∈I=(-δ,δ)},
where εi(s)=ε0+τi(s), s∈I; τi(·)∈C2(I,R); τi(0)=τi′(0)=0, and
(3.7)λ1(s)=λ0+z11(s),λ2(s)=λ0+s+z21(s),s∈I,u1(s)=u0+sw0+z12(s),u2(s)=u0+sv1+z22(s),s∈I,w1(s)=w0+sψ0+z13(s),w2(s)=w0+sψ1+z23(s),s∈I,
where zij(·)∈C2(I,Z), zij(0)=zij′(0)=0  (i=1,2, j=1,2,3), ψ0∈X3, ψ1∈X3 are the unique solution of
(3.8)Fuu(ε0,λ0,u0)[w0,w0]+Fu(ε0,λ0,u0)[ψ]=0,(3.9)Fuu(ε0,λ0,u0)[v1,w0]+Fλu(ε0,λ0,u0)[w0]+Fu(ε0,λ0,u0)[ψ]=0,
respectively.Remark 3.2.
Theorem2.4 complements Theorem  3.2 in [4], where the opposite condition (3.4) is imposed.Proof.
We apply Theorem2.4 to the operator H, so we need to verify all the conditions. We define a differential operator K:R×X×X3→Y×Y,
(3.10)K[τ,v,ψ]=H(λ,u,w)(ε0,λ0,u0,w0)[τ,v,ψ]=(τFλ(ε0,λ0,u0)+Fu(ε0,λ0,u0)[v]τFλu(ε0,λ0,u0)[w0]+Fuu(ε0,λ0,u0)[v,w0]+Fu(ε0,λ0,u0)[ψ]).
(1)dimN(K)=2. Suppose that (τ,v,ψ)∈N(K) and (τ,v,ψ)≠0. If τ=0, from Fu(ε0,λ0,u0)[v]=0 and (F1), then we have v=kw0 and
(3.11)kFuu(ε0,λ0,u0)[w0,w0]+Fu(ε0,λ0,u0)[ψ]=0.
From (F4′), we can define ψ0∈X3 is the unique solution of (3.8). Thus, (0,w0,ψ0)∈N(K) and (τ,v,ψ)=k(0,w0,ψ0).
Next, we considerτ≠0. Without loss of generality, we assume that τ=1. Notice that Fλ(ε0,λ0,u0)∈R(Fu(ε0,λ0,u0)) from (F2′), we can define that v1∈X3∖{0} is unique solution of (3.5). Substituting τ=1, v=v1 into (3.10), we have
(3.12)Fλu(ε0,λ0,u0)[w0]+Fuu(ε0,λ0,u0)[v1,w0]+Fu(ε0,λ0,u0)[ψ]=0.
From (3.4), there exists a unique ψ1∈X3 satisfies (3.9). Then,
(3.13)N(K)=span⁡{(0,w0,ψ0),(1,v1,ψ1)},
that is, dimN(K)=2.
(2)codimR(K)=1. We only claim that
(3.14)R(K)=R(Fu(ε0,λ0,u0))×Y.
Let (h,g)∈R(K) and (τ,v,ψ)∈R×X×X3 satisfy
(3.15)τFλ(ε0,λ0,u0)+Fu(ε0,λ0,u0)[v]=h,(3.16)τFλu(ε0,λ0,u0)[w0]+Fuu(ε0,λ0,u0)[v,w0]+Fu(ε0,λ0,u0)[ψ]=g.
Using (3.15) and (F2′),  then (h,g)∈R(Fu(ε0,λ0,u0))×Y and R(K)⊂R(Fu(ε0,λ0,u0))×Y.
Conversely, for any(h,g)∈R(Fu(ε0,λ0,u0))×Y, from (F3), set
(3.17)τ1=〈l,g〉〈l,Fλu(ε0,λ0,u0)[w0]〉,
where l∈R(Fu(ε0,λ0,u0))⊥⊂Y*. From (F2′), we have
(3.18)h-τ1Fλ(ε0,λ0,u0)∈R(Fu(ε0,λ0,u0)).
Set v2=[Fu|X3]-1[h-τ1Fλ(ε0,λ0,u0)]∈X3, we obtain that
(3.19)τ1Fλ(ε0,λ0,u0)+Fu(ε0,λ0,u0)[v2]=h.
Substituting τ=τ1, v=v2 into (3.16), we have
(3.20)τ1Fλu(ε0,λ0,u0)[w0]+Fuu(ε0,λ0,u0)[v2,w0]+Fu(ε0,λ0,u0)[ψ]=g.
Using (F1), (F3), then there exists v3∈X3, τ2∈R satisfies
(3.21)Fuu(ε0,λ0,u0)[v2,w0]=τ2Fλu(ε0,λ0,u0)[w0]+Fu(ε0,λ0,u0)[v3].
Substituting (3.21) into (3.20), we have
(3.22)(τ1+τ2)Fλu(ε0,λ0,u0)[w0]+Fu(ε0,λ0,u0)[ψ+v3]=g.
Applying l to (3.22), we have τ2=0 because of the definition of τ1 and
(3.23)g-τ1Fλu(ε0,λ0,u0)[w0]∈R(Fu(ε0,λ0,u0)).
Thus we can define
(3.24)ψ2=[Fu∣X3]-1{g-τ1Fλu(ε0,λ0,u0)[w0]}-v3∈X3.
Therefore, K(τ1,v2,ψ2)=(h,g), that is, R(Fu(ε0,λ0,u0))×Y⊂R(K). Hence, R(K)=R(Fu(ε0,λ0,u0))×Y. That is, codimR(K)=1.
(3)Hε(ε0,λ0,u0,w0)∉R(K). Since R(K)=R(Fu(ε0,λ0,u0))×Y, we need only to show that Fε(ε0,λ0,u0)∉R(Fu(ε0,λ0,u0)) but that is exactly assumed in (F5). So, the statement of the theorem follows from Theorem 2.4.


## 4. Calculations of Bifurcation Directions

In Theorem3.1, we have ε1(0)=ε2(0)=ε0, ε1′(0)=ε2′(0)=0, λ1(0)=λ2(0)=λ0, u1(0)=u2(0)=u0, w1(0)=w2(0)=w0, λ1′(0)=0, u1′(0)=w0, w1′(0)=ψ0, λ2′(0)=1, u2′(0)=v1, w2′(0)=ψ1.To completely determine the turning direction of curve of degenerate solutions, we need some calculations.Let{Ti(s)=(εi(s),λi(s),ui(s),wi(s)):s∈(-δ,δ)} be a curve of degenerate solutions which we obtain in Theorem 3.1. Differentiating H(εi(s),λi(s),ui(s),wi(s))=0 with respect to s, we obtain
(4.1)Fεεi′(s)+Fλλi′(s)+Fu[ui′(s)]=0,Fεu[wi(s)]εi′(s)+Fλu[wi(s)]λi′(s)+Fuu[wi(s),ui′(s)]+Fu[wi′(s)]=0.
Setting s=0 in (4.1), we get exactly Fu[w0]=0, (3.5), (3.8), and (3.9). We differentiate (4.1) again, and we have (omit the subscript i in the equation)
(4.2)Fεε[ε′(s)]2+Fεε′′(s)+Fλλ[λ′(s)]2+Fλλ′′(s)+Fuu[u′(s),u′(s)]+Fu[u′′(s)]+2Fελε′(s)λ′(s)+2Fεu[u′(s)]ε′(s)+2Fλu[u′(s)]λ′(s)=0,(4.3)Fεεu[w(s)][ε′(s)]2+Fελu[w(s)]ε′(s)λ′(s)+Fεuu[u′(s),w(s)]ε′(s)+Fεu[w′(s)]ε′(s)+Fεu[w(s)]ε′′(s)+Fελu[w(s)]ε′(s)λ′(s)+Fλλu[w(s)][λ′(s)]2+Fλuu[u′(s),w(s)]λ′(s)+Fλu[w′(s)]λ′(s)+Fλu[w(s)]λ′′(s)+Fεuu[u′(s),w(s)]ε′(s)+Fλuu[u′(s),w(s)]λ′(s)+Fuuu[u′(s),u′(s),w(s)]+Fuu[w′(s),u′(s)]+Fuu[w(s),u′′(s)]+Fεu[w′(s)]ε′(s)+Fλu[w′(s)]λ′(s)+Fuu[w′(s),u′(s)]+Fu[w′′(s)]=0,(4.4)Fεεu[w(s)][ε′(s)]2+Fεu[w(s)]ε′′(s)+Fλu[w(s)]λ′′(s)+Fλλu[w(s)][λ′(s)]2+Fuuu[u′(s),u′(s),w(s)]+Fuu[w(s),u′′(s)]+Fu[w′′(s)]+2Fελu[w(s)]ε′(s)λ′(s)+2Fεuu[u′(s),w(s)]ε′(s)+2Fλuu[u′(s),w(s)]λ′(s)+2Fεu[w′(s)]ε′(s)+2Fλu[w′(s)]λ′(s)+2Fuu[w′(s),u′(s)]=0.Settings=0 in (4.2), we obtain
(4.5)Fεε1′′(0)+Fλλ1′′(0)+Fuu[w0,w0]+Fu[u1′′(0)]=0,(4.6)Fεε2′′(0)+Fλλ+Fλλ2′′(0)+Fuu[v1,v1]+Fu[u2′′(0)]+2Fλu[v1]=0.
And applying l to it, we have
(4.7)ε1′′(0)=0,(4.8)ε2′′(0)=-〈l,Fλλ+Fuu[v1,v1]+2Fλu[v1]〉〈l,Fε〉,
Using (F2′), (F4′), (F5). From (4.7), (4.5) implies u1′′(0)=λ1′′(0)v1+ψ0+kw0. Setting s=0 in (4.4),
(4.9)Fλu[w0]λ1′′(0)+Fuuu[w0,w0,w0]+Fuu[w0,u1′′(0)]+Fu[w1′′(0)]+2Fuu[ψ0,w0]=0,(4.10)Fεu[w0]ε2′′(0)+Fλu[w0]λ2′′(0)+Fλλu[w0]+Fuuu[v1,v1,w0]+Fuu[w0,u2′′(0)]+Fu[w2′′(0)]+2Fλuu[v1,w0]+2Fλu[ψ1]+2Fuu[ψ1,v1]=0.
Substituting the expression of u1′′(0) into (4.9), we have
(4.11)λ1′′(0)(Fλu[w0]+Fuu[v1,w0])+3Fuu[ψ0,w0]+Fuuu[w0,w0,w0]+kFuu[w0,w0]+Fu[w1′′(0)]=0.
And applying l to it, we obtain 〈l,Fuuu[w0,w0,w0]+3Fuu[ψ0,w0]〉=0, that is,
(4.12)Fuuu[w0,w0,w0]+3Fuu[ψ0,w0]∈R(Fu(ε0,λ0,u0)).
Assume Fεu[w0]∈R(Fu(ε0,λ0,u0)) and applying l to (4.10),
(4.13)λ2′′(0)=-〈l,Fλλu[w0]+Fuuu[v1,v1,w0]+Fuu[w0,u2′′(0)]+2Fλuu[v1,w0]+2Fλu[ψ1]+2Fuu[ψ1,v1]〉〈l,Fλu[w0]〉.
We differentiate (4.2) again:
(4.14)Fεεε[ε′(s)]3+Fεε′′′(s)+Fλλλ[λ′(s)]3+Fλλ′′′(s)+Fuuu[u′(s),u′(s),u′(s)]+Fu[u′′′(s)]+3Fεεε′(s)ε′′(s)+3Fλλλ′(s)λ′′(s)+3Fuu[u′′(s),u′(s)]+3Fελε′′(s)λ′(s)+3Fελε′(s)λ′′(s)+3Fεu[u′(s)]ε′′(s)+3Fεu[u′′(s)]ε′(s)+3Fλu[u′′(s)]λ′(s)+3Fλu[u′(s)]λ′′(s)+3Fελλε′(s)(λ′(s))2+3Fεελ(ε′(s))2λ′(s)+3Fεεu[u′(s)](ε′(s))2+3Fεuu[u′(s),u′(s)]ε′(s)+3Fλλu[u′(s)](λ′(s))2+3Fλuu[u′(s),u′(s)]λ′(s)+6Fελu[u′(s)]ε′(s)λ′(s)=0.
Setting s=0 in (4.14), we obtain
(4.15)Fεε1′′′(0)+Fλλ1′′′(0)+Fuuu[w0,w0,w0]+Fu[u1′′′(0)]+3Fuu[u1′′(0),w0]+3Fλu[w0]λ1′′(0)=0,(4.16)Fλλ2′′′(0)+Fεε2′′′(0)+3λ2′′(0)(Fλλ+Fλu[v1])+3ε2′′(0)(Fελ+Fεu[v1])+Fλλλ+Fuuu[v1,v1,v1]+Fu[u2′′′(0)]+3Fuu[u2′′(0),v1]+3Fλu[u2′′(0)]+3Fλλu[v1]+3Fλuu[v1,v1]=0.
Substituting the expression of u1′′(0) into (4.15) and applying l to it, we have ε1′′′(0)=0 using (3.4), (4.12), and (F5).



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*Source: 302942-2011-06-28.xml*