The Generalized Saddle-Node Bifurcation of Degenerate Solution

Seria I: PRACE MATEMATYCZNE XLV (2) (2005), 145-150

Ping Liu, Yu-Wen Wang^∗

The Generalized Saddle-Node Bifurcation of Degenerate Solution

Abstract. In this paper we discuss the bifurcation problem for the abstract operator equation of the form F (u, λ) = θ with a parameter λ, where F : X × R → Y is a C¹mapping, X, Y are Banach spaces. By the bounded linear generalized inverse A⁺ of A = Fu(u0, λ0), an abstract bifurcation theorem for the case dimN (Fu(u0, λ0)) ≥ codimR(Fu(u0, λ0)) = 1 has been obtained.

2000 Mathematics Subject Classification: 35J25, 35J65.

Key words and phrases: Operator equation, bifurcation, generalized inverse of oper- ator.

1. Introduction. Let F : X × R → Y be a nonlinear differential mapping, and (u₀, λ₀) a solution of F (u, λ) = θ. If F_u(u₀, λ₀) is a linear homomorphism (so (u₀, λ₀) is a non-degenerate solution), then by the implicit function theorem, for λ close to λ₀, F (u, λ) = 0 has a unique solution (u(λ), λ) and u(λ) is continuously differentiable with respect to λ. Thus λ₀ is a bifurcation point only if F_u(u₀, λ₀) is singular. If (u₀, λ₀) is a solution of F (u, λ) = 0 with N (F_u(u₀, λ₀)) 6= {θ}, then we call (u₀, λ₀) a degenerate solution. Very commonly at a degenerate solution (u₀, λ₀), λ₀is a simple eigenvalue of F_u(u₀, λ₀) that is

(F₁) dimN (F_u(u₀, λ₀)) = codimR(F_u(u₀, λ₀)) = 1

and N (F_u(u₀, λ₀)) = span{w₀}, where N (F_u(u₀, λ₀)) and R(F_u(u₀, λ₀)) are the null space and the range of F_u(u₀, λ₀).

M. G. Crandall and P. H. Rabinowitz, under the condition (F1) proved two bifurcation theorems (Theorem 1.7 in [1], Theorem 3.2 in [2]), but they also gave

∗The research was supported in part by the National Science Foundation Grant (19971023,10471032) of China.

an example for which dim(N (F_u(u₀, λ₀))) = 2. J. P. Shi. systematically studied the bifurcation of degenerate solution with the condition F₁and its applications [3]. In this paper, we assume that

(F₂) dimN (F_u(u₀, λ₀)) ≥ codimR(F_u(u₀, λ₀)) = 1

and extend the results of Crandall and Rabinowitz for (F₂) holding. The tools to prove the results are the generalized inverse and the implicit function theorem.

Definition 1.1 ([13]) Let X, Y be Banach spaces, A ∈ L(X, Y ) be a linear op- erator, A⁺∈ L(Y, X) is called generalized inverse, if it satisfies

(i) AA⁺A = A, (ii) A⁺AA⁺= A.

Definition 1.2 Let X, Y, A be the same as Definition 1.1, if A ∈ L(X, Y ) has bounded linear generalized inverse A⁺, then A is called generalized regular operator.

Definition 1.3 ([5]) Let X, Y be Banach spaces, V ⊂ X × R an open set, and F ∈ C^p(V, Y ), p ≥ 0. Suppose that there exists a trivial solution (u(λ), λ) of equation F (u, λ) = θ in a neighborhood of (u₀, λ₀) such that (u(λ), λ) → (u₀, λ₀).(u₀, λ₀) is called a bifurcation solution of equation F (u, λ) = 0 if (u_n, λ_n) → (u₀, λ₀) such that u_n6= u(λn) and F (u_n, λ_n) = θ (n = 1, 2, · · · ).

Lemma 1.4 ([6]) Let A ∈ L(X, Y ), then A is a generalized regular operator iff N (A), R(A) are topologically complemented in X, Y respectively. In this case, P_{N (A)}= I − A⁺A, P_R(A)= AA⁺ and X = N (A) ⊕ R(A⁺).

2. Abstract Bifurcation Theorem.

Theorem 2.1 (Generalized Saddle–Node Bifurcation) Let V ⊂ X × R be a neighborhood of (u₀, λ₀), F ∈ C¹(V, Y ). Suppose that

(i) F_u(u₀, λ₀) : X → Y is a generalized regular operator, and dimN (F_u(u₀, λ₀)) ≥ codimR(F_u(u₀, λ₀)) = 1.

(ii) Fλ(u₀, λ₀) 6∈ R(F_u(u₀, λ₀));

Let Z = R((F_u(u₀, λ₀))⁺), then

(I) The subset {(u, λ)|F (u, λ) = θ} contains the curve (u(s), λ(s)) = (u₀+ sw₀+ z(s), λ(s)) near (u₀, λ₀), where w₀ ∈ N (F_u(u₀, λ₀)) \ {θ}, the mapping s 7→

(z(s), λ(s)) ∈ Z × R is continuously differential near s = 0, and λ(0) = 0, λ⁰(0) = 0, z⁰(0) = z(0) = θ.

(II) If F ∈ C^k(V, Y ), then λ(s), z(s) are k-differential, and λ⁰⁰(0) = −hl, F_uu(u₀, λ₀)[w₀, w₀]i

hl, F_λ(u₀, λ₀)i , where l ∈ N ((F_u(u₀, λ₀))^∗) \ {θ}.

Proof (I) Taking an arbitrary w0∈ N ((F_u(u₀, λ₀))^∗) \ {θ}, define G : R × (R × Z) → Y as

(1) G(s, λ, z) = F (u₀+ sw₀+ z, λ) − F (u₀, λ₀), then G and F have the same smoothness and G(0, λ₀, θ) = θ.

Next to prove that G_(λ,z)(0, λ₀, θ) : Z × R → Y is a regular operator. First of all, we assert that

(2) G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = τ F_λ(u₀, λ₀) + F_u(u₀, λ₀)[ψ].

Indeed, by the definition of Fréchet derivative operator and (1), we have G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = lim

t→0

G(0, λ₀+ tτ, θ + tψ) − G(0, λ₀, θ) t

= lim

t→0

F (u₀+ tψ, λ₀+ tτ ) − F (u₀, λ₀) t

= lim

t→0

 F (u0+ tψ, λ₀+ tτ ) − F (u₀+ tψ, λ₀)

t +

F (u₀+ tψ, λ₀) − F (u₀, λ₀) t



= τ F_λ(u₀, λ₀) + F_u(u₀, λ₀)[ψ].

This is (2). Next to prove that G_(λ,z)(0, λ₀, θ) is an injective. Suppose that there exists (τ, ψ) ∈ R × Z such that

G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = 0.

By (2), we have

(3) τ F_λ(u₀, λ₀) + F_u(u₀, λ₀)[ψ] = θ.

Since A⁺is bounded linear, then R(F_u(u₀, λ₀)) is closed, hence Banach Closed Range Theorem implies that

(4) R(F_u(u₀, λ₀))^⊥= N (F_u(u₀, λ₀)^∗).

Taking l ∈ N (F_u(u₀, λ₀)^∗) and l 6= θ. Applying l to (3), we get from (4) that τ < l, F_λ(u₀, λ₀) >= 0.

Because of Fλ(u₀, λ₀) 6∈ R(F_u(u₀, λ₀)), it follows that from (4) that

< l, F_λ(u₀, λ₀) >6= 0

and so τ = 0 and hence

F_u(u₀, λ₀)[ψ] = θ.

and therefore ψ ∈ N (F_u(u₀, λ₀)), note that

ψ ∈ Z = R((F_u(u₀, λ₀))⁺)

and R((F_u(u₀, λ₀))⁺) is the topological complement of N (F_u(u₀, λ₀)) from Lemma 1.4, so ψ = θ, i.e. G_(λ,z)(θ, λ₀, 0) is an injective.

To prove that G_(λ,z)(0, λ₀, 0) is surjective.

For any y ∈ Y , set τ = < l, y >

< l, F_λ(u₀, λ₀) >, ψ = (F_u(u₀, λ₀))⁺(y − τ F_λ(u₀, λ₀)),

where (F_u(u₀, λ₀))⁺ is bounded linear generalized inverse of F_u(u₀, λ₀). By the condition (i), it exists. By Lemma 1.4 and condition (i), the complement subspace Z of N (F_u(u₀, λ₀)) exists and

Z = R(F_u⁺(u₀, λ₀)), thus ψ ∈ Z i.e. (ψ, τ ) ∈ Z × R. In view of (2)

G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = τ F_λ(u₀, λ₀) + F_u(u₀, λ₀)[ψ]

= τ F_λ(u₀, λ₀) + F_u(u₀, λ₀)(F_u(u₀, λ₀))⁺(y − τ F_λ(u₀, λ₀)) (5)

Since

F_λ(u₀, λ₀) 6∈ R(F_u(u₀, λ₀)), codim(R(F_u(u₀, λ₀)) = 1, then

Y = R(F_u(u₀, λ₀)) ⊕ span{F_λ(u₀, λ₀)}

and hence, y has a unique decomposition

(6) y = y₁+ tF_λ(u₀, λ₀), where y₁∈ R(F_u(u₀, λ₀)).

Applying l to (6), we have

(7) t = < l, y >

< l, F_λ(u₀, λ₀) > = τ.

therefore, combining (6) with (7), we obtain

(8) y − τ F_λ(u₀, λ₀) ∈ R(F_u(u₀, λ₀)).

By Lemma 1.4, F_u(u₀, λ₀)(F_u(u₀, λ₀))⁺ is the bounded linear projector from Y to R(F_u(u₀, λ₀)), and hence, combining (5) with (8), we obtain that

G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = τ F_λ(u₀, λ₀) + y − τ F_λ(u₀, λ₀) = y, and so for each y ∈ Y , there exists (τ, ψ) ∈ R × Z, such that

G_(λ,z)(0, λ₀, θ)[(τ, ψ)] = y,

i.e. G_(λ,z)(0, λ₀, θ) : Z × R → Y is surjective, and hence G_(λ,z)(0, λ₀, θ) is one-to one and onto. It follows from Banach Inverse Operator Theorem that G_(λ,z)(0, λ₀, θ) is a regular operator.

For equation G(s, λ, z) = θ, by the implicit function theorem, there exists ε > 0 and s 7→ λ(s), |λ(s) − λ₀| < δ, s 7→ z(s), s ∈ I(−ε, ε) and (z(s), λ(s)) is continuously differential on I = (−ε, ε) with λ(0) = λ₀, z(0) = θ, and

G(s, λ(s), z(s)) = θ, s ∈ I and hence

F (u(s), λ(s)) = F (u₀+ sw₀+ z(s), λ(s)) = θ.

i.e. the subset {(u, λ)| F (u, λ) = θ} contains the curve

(u(s), λ(s)) = (u₀+ sw₀+ z(s), λ(s)), s ∈ I.

Differentiate G(s, λ, z) = 0 with respect to s ,

G_(λ,z)(s, λ, z)[λ⁰(s), z⁰(s)] + G_s(s, λ, z) = θ and hence

G_(λ,z)(0, λ₀, θ)[λ⁰(0), z⁰(0)] = −G_s(0, λ₀, θ) = −F_u(u₀, λ₀)w₀= θ, the inverse of G_(λ,z)(0, λ₀, θ) implies that λ⁰(0) = z⁰(0) = θ, then (I) follows.

(II) Differentiating two sides of F (u(s), λ(s)) = θ, we obtain that (9) F_u(u(s), λ(s))[u⁰(s)] + F_λ(u(s), λ(s))λ⁰(s) = θ.

Differentiating again two sides of (9), we have

F_λλ(λ⁰(s))²+ F_λu[u⁰(s)]λ⁰(s) + F_λλ⁰⁰(s)

+ F_uλλ⁰(s)[u⁰(s)] + F_uu[u⁰(s), u⁰(s)] + F_u[u⁰⁰(s)] = θ Taking s = 0, note that λ⁰(0) = 0, we obtain

(10) F_λλ⁰⁰(0) + F_uu⁰⁰(0) + F_uu[w₀, w₀] = θ.

Taking l ∈ N ((F_u(u₀, λ₀))^∗) = R(F_u(u₀, λ₀))^⊥, and acting two sides of (10), we have λ⁰⁰(0)hl, F_λ(u₀, λ₀)i + hl, F_uu(u₀, λ₀)[w₀, w₀]i = 0

and hence

λ⁰⁰(0) = −hl, F_uu(u₀, λ₀)[w₀, w₀]i hl, F_λ(u₀, λ₀)i .

as that (II) follows. _

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Ping Liu

Department of Mathematics, Northeast Normal University Changchun,130000, P. R. China

E-mail: wangyuwen2003@sohu.com Yu-Wen Wang

Department of Mathematics, Harbin Normal University Harbin,150080, P. R. China

E-mail: wangyuwen2003@sohu.com

(Received: 20.12.03)