Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 302942 | 11 pages | https://doi.org/10.1155/2011/302942

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

Academic Editor: Stefan Siegmund
Received21 Jan 2011
Revised29 Mar 2011
Accepted06 May 2011
Published28 Jun 2011

Abstract

We consider a nonlinear equation 𝐹(πœ€,πœ†,𝑒)=0, where the parameter πœ€ is a perturbation parameter, 𝐹 is a differentiable mapping from RΓ—R×𝑋 to π‘Œ, and 𝑋, π‘Œ are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation theorem.

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 𝐹(πœ€,πœ†,𝑒)=0,(1.1) where πœ€ indicates the perturbation. Fix πœ€=πœ€0; let (πœ†0,𝑒0) be a solution of 𝐹(πœ€0,β‹…,β‹…)=0. From the implicit function theorem, a necessary condition for bifurcation is that 𝐹𝑒(πœ€0,πœ†0,𝑒0) is not invertible; we call (πœ€0,πœ†0,𝑒0) a degenerate solution. In [12], Shi shows the persistence and the bifurcation of degenerate solutions when πœ€ varies near πœ€0 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 β€–β‹…β€– as the norm of Banach space 𝑋 and βŸ¨β‹…,β‹…βŸ© as the duality pair of a Banach space 𝑋 and its dual space π‘‹βˆ—. For a nonlinear operator 𝐹, we use 𝐹𝑒 as the partial derivative of 𝐹 with respect to argument 𝑒. 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]). Let 𝑋, π‘Œ be Banach spaces, and let π΄βˆˆβ„’(𝑋,π‘Œ) be a linear operator. Then, 𝐴+βˆˆβ„’(π‘Œ,𝑋) is called the generalized inverse of 𝐴 if it satisfies(i)𝐴𝐴+𝐴=𝐴,(ii)𝐴+𝐴𝐴+=𝐴+.

Definition 2.2 (see [13]). Let 𝑋,π‘Œ, and 𝐴 be the same as in Definition 2.1. If π΄βˆˆβ„’(𝑋,π‘Œ) has the bounded linear generalized inverse 𝐴+, then 𝐴 is called a generalized regular operator.

Lemma 2.3 (see [13]). Let π΄βˆˆβ„’(𝑋,π‘Œ), then 𝐴 is a generalized regular operator if and only if 𝑁(𝐴),𝑅(𝐴) are topologically complemented in 𝑋,π‘Œ, respectively. In this case, πΌβˆ’π΄+𝐴, 𝐴𝐴+ are bounded linear projectors from 𝑋, π‘Œ 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). Let π‘‰βŠ‚π‘Γ—π‘‹ be a neighborhood of (πœ†0,𝑒0),𝐹∈𝐢1(𝑉,π‘Œ). Suppose that (i)𝐹(πœ†0,𝑒0)=0;(ii)𝐹𝑒(πœ†0,𝑒0)βˆΆπ‘‹β†’π‘Œ is a generalized regular operator, and 𝐹dimπ‘π‘’ξ€·πœ†0,𝑒0𝐹β‰₯codimπ‘…π‘’ξ€·πœ†0,𝑒0ξ€Έξ€Έ=1,(2.1)(iii)πΉπœ†(πœ†0,𝑒0)βˆ‰π‘…(𝐹𝑒(πœ†0,𝑒0)).
Let 𝑍=𝑅((𝐹𝑒(πœ†0,𝑒0))+), then the subset {(πœ†,𝑒)|𝐹(πœ†,𝑒)=0} contains the curve (πœ†(𝑠),𝑒(𝑠))=(πœ†(𝑠),𝑒0+𝑠𝑀0+𝑧(𝑠)) near (πœ†0,𝑒0), where 𝑀0βˆˆπ‘(𝐹𝑒(πœ†0,𝑒0))⧡{πœƒ}, the mapping 𝑧(𝑠) is continuously differentiable near 𝑠=0, and πœ†(0)=πœ†0,πœ†ξ…ž(0)=0,π‘§ξ…ž(0)=𝑧(0)=πœƒ.

Proof. Since 𝐴=𝐹𝑒(πœ†0,𝑒0) is a generalized regular operator, there exist closed subspaces 𝑍 in 𝑋, π‘Œ1 in π‘Œ satisfing 𝑋=π‘βŠ•π‘(𝐴), π‘Œ=𝑅(𝐴)βŠ•π‘Œ1.
Taking an arbitrary 𝑀0βˆˆπ‘(𝐴)⧡{πœƒ}, from Lemma 2.3, 𝐹(πœ†,𝑒)=0 is equivalent to ξ€·πΌβˆ’π΄π΄+ξ€ΈπΉξ€·πœ†,𝑒0+𝑠𝑀0ξ€Έ+𝑧=0,𝐴𝐴+πΉξ€·πœ†,𝑒0+𝑠𝑀0ξ€Έ+𝑧=0,(2.2) where π‘ βˆˆπ‘, π‘§βˆˆπ‘.
Define πΊβˆΆπ‘Γ—π‘Γ—π‘β†’π‘…(𝐴) as 𝐺(𝑠,πœ†,𝑧)=𝐴𝐴+πΉξ€·πœ†,𝑒0+𝑠𝑀0ξ€Έ,𝐺+𝑧(πœ†,𝑧)ξ€·0,πœ†0ξ€Έ[(],0𝜏,πœ“)=𝐴𝐴+ξ€·πœπΉπœ†ξ€·πœ†0,𝑒0ξ€Έ+πΉπ‘’ξ€·πœ†0,𝑒0ξ€Έ[πœ“]ξ€Έ,=𝐴𝐴+𝐴[πœ“][πœ“],=𝐴(2.3) because of (iii), then 𝐺(πœ†,𝑧)(0,πœ†0,0)βˆΆπ‘…Γ—π‘β†’π‘…(𝐴) is an isomorphism.
For the equation 𝐺(𝑠,πœ†,𝑧)=0, by the implicit function theorem, there exist πœ€>0 and (πœ†(𝑠),𝑧(𝑠))∈𝐢1(βˆ’πœ€,πœ€), with πœ†(0)=πœ†0, 𝑧(0)=0 satisfying 𝐺(𝑠,πœ†(𝑠),𝑧(𝑠))=0.(2.4) From (2.2), we have πΉξ€·πœ†(𝑠),𝑒0+𝑠𝑀0ξ€Έ+𝑧(𝑠)=0,π‘ βˆˆ(βˆ’πœ€,πœ€).(2.5) Differentiating (2.5) with respect to 𝑠, we have πΉπœ†ξ€·πœ†(𝑠),𝑒0+𝑠𝑀0ξ€Έπœ†+𝑧(𝑠)ξ…ž(𝑠)+πΉπ‘’ξ€·πœ†(𝑠),𝑒0+𝑠𝑀0𝑀+𝑧(𝑠)ξ€Έξ€Ί0+π‘§ξ…žξ€»(𝑠)=0.(2.6) Setting 𝑠=0, πΉπœ†ξ€·πœ†0,𝑒0ξ€Έπœ†ξ…ž(0)+πΉπ‘’ξ€·πœ†0,𝑒0𝑀0+π‘§ξ…žξ€»(0)=0.(2.7) Thus, πœ†ξ…ž(0)=0 since (iii) and we have πΉπ‘’ξ€·πœ†0,𝑒0π‘§ξ€Έξ€Ίξ…žξ€»(0)=0,(2.8) that is, π‘§ξ…ž(0)βˆˆπ‘(𝐴)βˆ©π‘, we have π‘§ξ…ž(0)=0.

Corollary 2.5. Assume the conditions in Theorem 2.4 are satisfied and 𝐹dimπ‘π‘’ξ€·πœ†0,𝑒0𝐹=𝑛,π‘π‘’ξ€·πœ†0,𝑒0𝑀=span1,𝑀2,…,𝑀𝑛,(2.9) then the direction of the solution curves is determined by πœ†π‘–ξ…žξ…žξ«(0)=βˆ’π‘™,πΉπ‘’π‘’ξ€·πœ†0,𝑒0𝑀𝑖,𝑀𝑖𝑙,πΉπœ†ξ€·πœ†0,𝑒0,(2.10) where π‘™βˆˆπ‘…(𝐹𝑒(πœ†0,𝑒0))βŸ‚, 𝑖=1,2,…,𝑛. Furthermore, when πΉπ‘’π‘’ξ€·πœ†0,𝑒0𝑀𝑖,π‘€π‘–ξ€»ξ€·πΉβˆ‰π‘…π‘’ξ€·πœ†0,𝑒0ξ€Έξ€Έ(2.11) is satisfied, πœ†π‘–ξ…žξ…ž(0)β‰ 0, and the solution curve {(πœ†π‘–(𝑠),𝑒𝑖(𝑠))∢|𝑠|<𝛿} is a parabola-like curve which reaches an extreme point at (πœ†0,𝑒0).

We illustrate our result by a simple example.

Example 2.6. Define 𝐹π‘₯π‘¦πœ†,ξƒͺξƒͺ=πœ†βˆ’π‘₯2βˆ’π‘¦2=0,(2.12) where ξ€·π‘ˆ=π‘₯π‘¦ξ€Έβˆˆπ‘2, πœ†βˆˆπ‘. From simple calculations, we obtain πΉπ‘ˆ=ξ€·ξ€Έβˆ’2π‘₯,βˆ’2𝑦,πΉπœ†=1,πΉπ‘ˆπ‘ˆ=ξƒͺβˆ’200βˆ’2.(2.13) We analyze the bifurcation at ξ€·(0,00ξ€Έ). It is easy to see that 𝑁(πΉπ‘ˆ)=span{𝑀1,𝑀2}, where 𝑀1=ξ€·10ξ€Έ, 𝑀2=ξ€·01ξ€Έ, 𝑅(πΉπ‘ˆ)={0}. So, obviously, dim𝑁(πΉπ‘ˆ)=2, codim𝑅(πΉπ‘ˆ)=1, and πΉπœ†βˆ‰π‘…(πΉπ‘ˆ). From the above calculation, πΉπ‘ˆπ‘ˆξ€Ίπ‘€1,𝑀1ξ€»=βˆ’2,πΉπ‘ˆπ‘ˆξ€Ίπ‘€2,𝑀2ξ€»=βˆ’2.(2.14) Obviously, πΉπ‘ˆπ‘ˆξ€·(0,00ξ€Έ)[𝑀𝑖,𝑀𝑖]βˆ‰π‘…(πΉπ‘ˆξ€·(0,00ξ€Έ)) and πœ†π‘–ξ…žξ…ž(0)=βˆ’2, 𝑖=1,2. Thus, we can apply Corollary 2.5 to (2.12). In fact, all solution curves for all π‘€π‘–βˆˆπ‘(πΉπ‘ˆ) form a surface (see Figure 1).

3. Main Theorems

Applying Theorem 2.4, we discuss the bifurcation of solutions of the perturbed problem. We consider the solution set of 𝐹(πœ€,πœ†,𝑒)=0,(3.1) where the parameter πœ€ indicates the perturbation, 𝐹∈𝐢1(𝑀,π‘Œ), 𝑀≑𝐑×𝐑×𝑋, and 𝑋, π‘Œ are Banach spaces. Let 𝐹𝐻(πœ€,πœ†,𝑒,𝑀)=𝐹(πœ€,πœ†,𝑒)𝑒[𝑀]ξƒͺ(πœ€,πœ†,𝑒).(3.2)

Suppose that (πœ€0,πœ†0,𝑒0,𝑀0) is a solution of 𝐻(πœ€,πœ†,𝑒,𝑀)=0. For (πœ€0,πœ†0,𝑒0)βˆˆπ‘€ and 𝑀0βˆˆπ‘‹1≑{π‘₯βˆˆπ‘‹βˆΆβ€–π‘₯β€–=1},(3.3) by Hahn-Banach theorem, there exists a closed subspace 𝑋3 of 𝑋 with codimension 1 such that 𝑋=𝐿(𝑀0)βŠ•π‘‹3, where 𝐿(𝑀0)=span{𝑀0} and 𝑑(𝑀0,𝑋3)=inf{||𝑀0βˆ’π‘₯||∢π‘₯βˆˆπ‘‹3}>0. Let 𝑋2=𝑀0+𝑋3={𝑀0+π‘₯∢π‘₯βˆˆπ‘‹3}. 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(𝐹𝑒(πœ€0,πœ†0,𝑒0))=codim𝑅(𝐹𝑒(πœ€0,πœ†0,𝑒0))=1, and 𝑁(𝐹𝑒(πœ€0,πœ†0,𝑒0))=span{𝑀0};(F2)πΉπœ†(πœ€0,πœ†0,𝑒0)βˆ‰π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0));(F3)πΉπœ†π‘’(πœ€0,πœ†0,𝑒0)[𝑀0]βˆ‰π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0));(F4)𝐹𝑒𝑒(πœ€0,πœ†0,𝑒0)[𝑀0,𝑀0]βˆ‰π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0));(F5)πΉπœ€(πœ€0,πœ†0,𝑒0)βˆ‰π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0)).

We use the convention that (πΉπ‘–ξ…ž) means that the condition defined in (𝐹𝑖) does not hold.

Theorem 3.1. Let 𝐹∈𝐢2(𝑀,π‘Œ), 𝑇0=(πœ€0,πœ†0,𝑒0,𝑀0)βˆˆπ‘€1 such that 𝐻(𝑇0)=(0,0). Suppose that the operator 𝐹 satisfies (𝐹1), (𝐹2ξ…ž), (𝐹3), (𝐹4ξ…ž), and (𝐹5) at 𝑇0. One also assumes that πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣1,𝑀0ξ€»+πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»ξ€·πΉβˆˆπ‘…π‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€Έ,(3.4) where 𝑣1βˆˆπ‘‹3⧡{0} is the unique solution of πΉπœ†ξ€·πœ€0,πœ†0,𝑒0ξ€Έ+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[𝑣]=0.(3.5) Then, there exists 𝛿>0 such that all the solutions of 𝐻(πœ€,πœ†,𝑒,𝑀)=(0,0) near 𝑇0 form two 𝐢2 curves: 𝑇1ξ€·πœ€(𝑠)=1(𝑠),πœ†1(𝑠),𝑒1(𝑠),𝑀1ξ€Έξ€Ύ,𝑇(𝑠),π‘ βˆˆπΌ=(βˆ’π›Ώ,𝛿)2(ξ€·πœ€π‘ )=2(𝑠),πœ†2(𝑠),𝑒2(𝑠),𝑀2(ξ€Έξ€Ύ,𝑠),π‘ βˆˆπΌ=(βˆ’π›Ώ,𝛿)(3.6) where πœ€π‘–(𝑠)=πœ€0+πœπ‘–(𝑠), π‘ βˆˆπΌ; πœπ‘–(β‹…)∈𝐢2(𝐼,𝐑); πœπ‘–(0)=πœξ…žπ‘–(0)=0, and πœ†1(𝑠)=πœ†0+𝑧11(𝑠),πœ†2(𝑠)=πœ†0+𝑠+𝑧21𝑒(𝑠),π‘ βˆˆπΌ,1(𝑠)=𝑒0+𝑠𝑀0+𝑧12(𝑠),𝑒2(𝑠)=𝑒0+𝑠𝑣1+𝑧22𝑀(𝑠),π‘ βˆˆπΌ,1(𝑠)=𝑀0+π‘ πœ“0+𝑧13(𝑠),𝑀2(𝑠)=𝑀0+π‘ πœ“1+𝑧23(𝑠),π‘ βˆˆπΌ,(3.7) where 𝑧𝑖𝑗(β‹…)∈𝐢2(𝐼,𝑍), 𝑧𝑖𝑗(0)=π‘§ξ…žπ‘–π‘—(0)=0  (𝑖=1,2, 𝑗=1,2,3), πœ“0βˆˆπ‘‹3, πœ“1βˆˆπ‘‹3 are the unique solution of πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]𝐹=0,(3.8)π‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣1,𝑀0ξ€»+πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]=0,(3.9) respectively.

Remark 3.2. Theorem 2.4 complements Theorem  3.2 in [4], where the opposite condition (3.4) is imposed.

Proof. We apply Theorem 2.4 to the operator 𝐻, so we need to verify all the conditions. We define a differential operator πΎβˆΆπ‘Γ—π‘‹Γ—π‘‹3β†’π‘ŒΓ—π‘Œ,  𝐾[]𝜏,𝑣,πœ“=𝐻(πœ†,𝑒,𝑀)ξ€·πœ€0,πœ†0,𝑒0,𝑀0ξ€Έ[]=ξƒ©πœ,𝑣,πœ“πœπΉπœ†ξ€·πœ€0,πœ†0,𝑒0ξ€Έ+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[𝑣]πœπΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]ξƒͺ.(3.10)
(1) dim𝑁(𝐾)=2. Suppose that (𝜏,𝑣,πœ“)βˆˆπ‘(𝐾) and (𝜏,𝑣,πœ“)β‰ 0. If 𝜏=0, from 𝐹𝑒(πœ€0,πœ†0,𝑒0)[𝑣]=0 and (𝐹1), then we have 𝑣=π‘˜π‘€0 and π‘˜πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]=0.(3.11) From (𝐹4ξ…ž), we can define πœ“0βˆˆπ‘‹3 is the unique solution of (3.8). Thus, (0,𝑀0,πœ“0)βˆˆπ‘(𝐾) and (𝜏,𝑣,πœ“)=π‘˜(0,𝑀0,πœ“0).
Next, we consider πœβ‰ 0. Without loss of generality, we assume that 𝜏=1. Notice that πΉπœ†(πœ€0,πœ†0,𝑒0)βˆˆπ‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0)) from (𝐹2ξ…ž), we can define that 𝑣1βˆˆπ‘‹3⧡{0} is unique solution of (3.5). Substituting 𝜏=1, 𝑣=𝑣1 into (3.10), we have πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣1,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]=0.(3.12) From (3.4), there exists a unique πœ“1βˆˆπ‘‹3 satisfies (3.9). Then, 𝑁(𝐾)=spanξ€½ξ€·0,𝑀0,πœ“0ξ€Έ,ξ€·1,𝑣1,πœ“1ξ€Έξ€Ύ,(3.13) that is, dim𝑁(𝐾)=2.
(2) codim𝑅(𝐾)=1. We only claim that 𝑅𝐹(𝐾)=π‘…π‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€ΈΓ—π‘Œ.(3.14) Let (β„Ž,𝑔)βˆˆπ‘…(𝐾) and (𝜏,𝑣,πœ“)βˆˆπ‘Γ—π‘‹Γ—π‘‹3 satisfy πœπΉπœ†ξ€·πœ€0,πœ†0,𝑒0ξ€Έ+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[𝑣]=β„Ž,(3.15)πœπΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]=𝑔.(3.16) Using (3.15) and (𝐹2ξ…ž), then (β„Ž,𝑔)βˆˆπ‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘Œ and 𝑅(𝐾)βŠ‚π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘Œ.
Conversely, for any (β„Ž,𝑔)βˆˆπ‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘Œ, from (𝐹3), set 𝜏1=βŸ¨π‘™,π‘”βŸ©ξ«π‘™,πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0,(3.17) where π‘™βˆˆπ‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0))βŸ‚βŠ‚π‘Œβˆ—. From (𝐹2ξ…ž), we have β„Žβˆ’πœ1πΉπœ†ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€·πΉβˆˆπ‘…π‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€Έ.(3.18) Set 𝑣2=[𝐹𝑒|𝑋3]βˆ’1[β„Žβˆ’πœ1πΉπœ†(πœ€0,πœ†0,𝑒0)]βˆˆπ‘‹3, we obtain that 𝜏1πΉπœ†ξ€·πœ€0,πœ†0,𝑒0ξ€Έ+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0𝑣2ξ€»=β„Ž.(3.19) Substituting 𝜏=𝜏1, 𝑣=𝑣2 into (3.16), we have 𝜏1πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣2,𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έ[πœ“]=𝑔.(3.20) Using (𝐹1), (𝐹3), then there exists 𝑣3βˆˆπ‘‹3, 𝜏2βˆˆπ‘ satisfies πΉπ‘’π‘’ξ€·πœ€0,πœ†0,𝑒0𝑣2,𝑀0ξ€»=𝜏2πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0𝑣3ξ€».(3.21) Substituting (3.21) into (3.20), we have ξ€·πœ1+𝜏2ξ€ΈπΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»+πΉπ‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€Ίπœ“+𝑣3ξ€»=𝑔.(3.22) Applying 𝑙 to (3.22), we have 𝜏2=0 because of the definition of 𝜏1 and π‘”βˆ’πœ1πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»ξ€·πΉβˆˆπ‘…π‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€Έ.(3.23) Thus we can define πœ“2=ξ€ΊπΉπ‘’βˆ£π‘‹3ξ€»βˆ’1ξ€½π‘”βˆ’πœ1πΉπœ†π‘’ξ€·πœ€0,πœ†0,𝑒0𝑀0ξ€»ξ€Ύβˆ’π‘£3βˆˆπ‘‹3.(3.24) Therefore, 𝐾(𝜏1,𝑣2,πœ“2)=(β„Ž,𝑔), that is, 𝑅(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘ŒβŠ‚π‘…(𝐾). Hence, 𝑅(𝐾)=𝑅(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘Œ. That is, codim𝑅(𝐾)=1.
(3) π»πœ€(πœ€0,πœ†0,𝑒0,𝑀0)βˆ‰π‘…(𝐾). Since 𝑅(𝐾)=𝑅(𝐹𝑒(πœ€0,πœ†0,𝑒0))Γ—π‘Œ, we need only to show that πΉπœ€(πœ€0,πœ†0,𝑒0)βˆ‰π‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0)) but that is exactly assumed in (𝐹5). So, the statement of the theorem follows from Theorem 2.4.

4. Calculations of Bifurcation Directions

In Theorem 3.1, we have πœ€1(0)=πœ€2(0)=πœ€0, πœ€ξ…ž1(0)=πœ€ξ…ž2(0)=0, πœ†1(0)=πœ†2(0)=πœ†0, 𝑒1(0)=𝑒2(0)=𝑒0, 𝑀1(0)=𝑀2(0)=𝑀0, πœ†ξ…ž1(0)=0, π‘’ξ…ž1(0)=𝑀0, π‘€ξ…ž1(0)=πœ“0, πœ†ξ…ž2(0)=1, π‘’ξ…ž2(0)=𝑣1, π‘€ξ…ž2(0)=πœ“1.

To completely determine the turning direction of curve of degenerate solutions, we need some calculations.

Let {𝑇𝑖(𝑠)=(πœ€π‘–(𝑠),πœ†π‘–(𝑠),𝑒𝑖(𝑠),𝑀𝑖(𝑠))βˆΆπ‘ βˆˆ(βˆ’π›Ώ,𝛿)} be a curve of degenerate solutions which we obtain in Theorem 3.1. Differentiating 𝐻(πœ€π‘–(𝑠),πœ†π‘–(𝑠),𝑒𝑖(𝑠),𝑀𝑖(𝑠))=0 with respect to 𝑠, we obtain πΉπœ€πœ€ξ…žπ‘–(𝑠)+πΉπœ†πœ†ξ…žπ‘–(𝑠)+πΉπ‘’ξ€Ίπ‘’ξ…žπ‘–ξ€»πΉ(𝑠)=0,πœ€π‘’ξ€Ίπ‘€π‘–(ξ€»πœ€π‘ )ξ…žπ‘–(𝑠)+πΉπœ†π‘’ξ€Ίπ‘€π‘–(ξ€»πœ†π‘ )ξ…žπ‘–(𝑠)+𝐹𝑒𝑒𝑀𝑖(𝑠),π‘’ξ…žπ‘–(𝑠)+πΉπ‘’ξ€Ίπ‘€ξ…žπ‘–(𝑠)=0.(4.1) Setting 𝑠=0 in (4.1), we get exactly 𝐹𝑒[𝑀0]=0, (3.5), (3.8), and (3.9). We differentiate (4.1) again, and we have (omit the subscript 𝑖 in the equation) πΉπœ€πœ€ξ€Ίπœ€ξ…žξ€»(𝑠)2+πΉπœ€πœ€ξ…žξ…ž(𝑠)+πΉπœ†πœ†ξ€Ίπœ†ξ…žξ€»(𝑠)2+πΉπœ†πœ†ξ…žξ…ž(𝑠)+πΉπ‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)+πΉπ‘’ξ€Ίπ‘’ξ…žξ…žξ€»(𝑠)+2πΉπœ€πœ†πœ€ξ…ž(𝑠)πœ†ξ…ž(𝑠)+2πΉπœ€π‘’ξ€Ίπ‘’ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+2πΉπœ†π‘’ξ€Ίπ‘’ξ…žξ€»πœ†(𝑠)ξ…žπΉ(𝑠)=0,(4.2)πœ€πœ€π‘’[]ξ€Ίπœ€π‘€(𝑠)ξ…žξ€»(𝑠)2+πΉπœ€πœ†π‘’[]πœ€π‘€(𝑠)ξ…ž(𝑠)πœ†ξ…ž(𝑠)+πΉπœ€π‘’π‘’ξ€Ίπ‘’ξ…žξ€»πœ€(𝑠),𝑀(𝑠)ξ…ž(𝑠)+πΉπœ€π‘’ξ€Ίπ‘€ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+πΉπœ€π‘’[]πœ€π‘€(𝑠)ξ…žξ…ž(𝑠)+πΉπœ€πœ†π‘’[]πœ€π‘€(𝑠)ξ…ž(𝑠)πœ†ξ…ž(𝑠)+πΉπœ†πœ†π‘’[]ξ€Ίπœ†π‘€(𝑠)ξ…žξ€»(𝑠)2+πΉπœ†π‘’π‘’ξ€Ίπ‘’ξ…žξ€»πœ†(𝑠),𝑀(𝑠)ξ…ž(𝑠)+πΉπœ†π‘’ξ€Ίπ‘€ξ…žξ€»πœ†(𝑠)ξ…ž(𝑠)+πΉπœ†π‘’[]πœ†π‘€(𝑠)ξ…žξ…ž(𝑠)+πΉπœ€π‘’π‘’ξ€Ίπ‘’ξ…žξ€»(𝑠),𝑀(𝑠)πœ€β€²(𝑠)+πΉπœ†π‘’π‘’ξ€Ίπ‘’ξ…žξ€»πœ†(𝑠),𝑀(𝑠)ξ…ž(𝑠)+πΉπ‘’π‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠),𝑀(𝑠)+πΉπ‘’π‘’ξ€Ίπ‘€ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)+𝐹𝑒𝑒𝑀(𝑠),π‘’ξ…žξ…žξ€»(𝑠)+πΉπœ€π‘’ξ€Ίπ‘€ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+πΉπœ†π‘’ξ€Ίπ‘€ξ…žξ€»πœ†(𝑠)ξ…ž(𝑠)+πΉπ‘’π‘’ξ€Ίπ‘€ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)+πΉπ‘’ξ€Ίπ‘€ξ…žξ…žξ€»πΉ(𝑠)=0,(4.3)πœ€πœ€π‘’[]ξ€Ίπœ€π‘€(𝑠)ξ…žξ€»(𝑠)2+πΉπœ€π‘’[]πœ€π‘€(𝑠)ξ…žξ…ž(𝑠)+πΉπœ†π‘’[]πœ†π‘€(𝑠)ξ…žξ…ž(𝑠)+πΉπœ†πœ†π‘’[]ξ€Ίπœ†π‘€(𝑠)ξ…žξ€»(𝑠)2+πΉπ‘’π‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠),𝑀(𝑠)+𝐹𝑒𝑒𝑀(𝑠),π‘’ξ…žξ…žξ€»(𝑠)+πΉπ‘’ξ€Ίπ‘€ξ…žξ…žξ€»(𝑠)+2πΉπœ€πœ†π‘’[]πœ€π‘€(𝑠)ξ…ž(𝑠)πœ†ξ…ž(𝑠)+2πΉπœ€π‘’π‘’ξ€Ίπ‘’ξ…žξ€»πœ€(𝑠),𝑀(𝑠)ξ…ž(𝑠)+2πΉπœ†π‘’π‘’ξ€Ίπ‘’ξ…žξ€»πœ†(𝑠),𝑀(𝑠)ξ…ž(𝑠)+2πΉπœ€π‘’ξ€Ίπ‘€ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+2πΉπœ†π‘’ξ€Ίπ‘€ξ…žξ€»πœ†(𝑠)ξ…ž(𝑠)+2πΉπ‘’π‘’ξ€Ίπ‘€ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)=0.(4.4)

Setting 𝑠=0 in (4.2), we obtain πΉπœ€πœ€1ξ…žξ…ž(0)+πΉπœ†πœ†1ξ…žξ…ž(0)+𝐹𝑒𝑒𝑀0,𝑀0ξ€»+𝐹𝑒𝑒1ξ…žξ…žξ€»(0)=0,(4.5)πΉπœ€πœ€2ξ…žξ…ž(0)+πΉπœ†πœ†+πΉπœ†πœ†2ξ…žξ…ž(0)+𝐹𝑒𝑒𝑣1,𝑣1ξ€»+𝐹𝑒𝑒2ξ…žξ…žξ€»(0)+2πΉπœ†π‘’ξ€Ίπ‘£1ξ€»=0.(4.6) And applying 𝑙 to it, we have πœ€1ξ…žξ…ž(0)=0,(4.7)πœ€2ξ…žξ…žξ«(0)=βˆ’π‘™,πΉπœ†πœ†+𝐹𝑒𝑒𝑣1,𝑣1ξ€»+2πΉπœ†π‘’ξ€Ίπ‘£1ξ€»ξ¬βŸ¨π‘™,πΉπœ€βŸ©,(4.8) Using (𝐹2ξ…ž), (𝐹4ξ…ž), (𝐹5). From (4.7), (4.5) implies 𝑒1ξ…žξ…ž(0)=πœ†1ξ…žξ…ž(0)𝑣1+πœ“0+π‘˜π‘€0. Setting 𝑠=0 in (4.4), πΉπœ†π‘’ξ€Ίπ‘€0ξ€»πœ†1ξ…žξ…ž(0)+𝐹𝑒𝑒𝑒𝑀0,𝑀0,𝑀0ξ€»+𝐹𝑒𝑒𝑀0,𝑒1ξ…žξ…žξ€»(0)+𝐹𝑒𝑀1ξ…žξ…žξ€»(0)+2πΉπ‘’π‘’ξ€Ίπœ“0,𝑀0𝐹=0,(4.9)πœ€π‘’ξ€Ίπ‘€0ξ€»πœ€2ξ…žξ…ž(0)+πΉπœ†π‘’ξ€Ίπ‘€0ξ€»πœ†2ξ…žξ…ž(0)+πΉπœ†πœ†π‘’ξ€Ίπ‘€0ξ€»+𝐹𝑒𝑒𝑒𝑣1,𝑣1,𝑀0ξ€»+𝐹𝑒𝑒𝑀0,𝑒2ξ…žξ…ž(ξ€»0)+𝐹𝑒𝑀2ξ…žξ…žξ€»(0)+2πΉπœ†π‘’π‘’ξ€Ίπ‘£1,𝑀0ξ€»+2πΉπœ†π‘’ξ€Ίπœ“1ξ€»+2πΉπ‘’π‘’ξ€Ίπœ“1,𝑣1ξ€»=0.(4.10) Substituting the expression of 𝑒1ξ…žξ…ž(0) into (4.9), we have πœ†1ξ…žξ…žξ€·πΉ(0)πœ†π‘’ξ€Ίπ‘€0ξ€»+𝐹𝑒𝑒𝑣1,𝑀0ξ€»ξ€Έ+3πΉπ‘’π‘’ξ€Ίπœ“0,𝑀0ξ€»+𝐹𝑒𝑒𝑒𝑀0,𝑀0,𝑀0ξ€»+π‘˜πΉπ‘’π‘’ξ€Ίπ‘€0,𝑀0ξ€»+𝐹𝑒𝑀1ξ…žξ…ž(ξ€»0)=0.(4.11) And applying 𝑙 to it, we obtain βŸ¨π‘™,𝐹𝑒𝑒𝑒[𝑀0,𝑀0,𝑀0]+3𝐹𝑒𝑒[πœ“0,𝑀0]⟩=0, that is, 𝐹𝑒𝑒𝑒𝑀0,𝑀0,𝑀0ξ€»+3πΉπ‘’π‘’ξ€Ίπœ“0,𝑀0ξ€»ξ€·πΉβˆˆπ‘…π‘’ξ€·πœ€0,πœ†0,𝑒0ξ€Έξ€Έ.(4.12) Assume πΉπœ€π‘’[𝑀0]βˆˆπ‘…(𝐹𝑒(πœ€0,πœ†0,𝑒0)) and applying 𝑙 to (4.10), πœ†2ξ…žξ…žξ«(0)=βˆ’π‘™,πΉπœ†πœ†π‘’ξ€Ίπ‘€0ξ€»+𝐹𝑒𝑒𝑒𝑣1,𝑣1,𝑀0ξ€»+𝐹𝑒𝑒𝑀0,𝑒2ξ…žξ…žξ€»(0)+2πΉπœ†π‘’π‘’ξ€Ίπ‘£1,𝑀0ξ€»+2πΉπœ†π‘’ξ€Ίπœ“1ξ€»+2πΉπ‘’π‘’ξ€Ίπœ“1,𝑣1𝑙,πΉπœ†π‘’ξ€Ίπ‘€0.(4.13) We differentiate (4.2) again: πΉπœ€πœ€πœ€ξ€Ίπœ€ξ…žξ€»(𝑠)3+πΉπœ€πœ€ξ…žξ…žξ…ž(𝑠)+πΉπœ†πœ†πœ†ξ€Ίπœ†ξ…žξ€»(𝑠)3+πΉπœ†πœ†ξ…žξ…žξ…ž(𝑠)+πΉπ‘’π‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)+πΉπ‘’ξ€Ίπ‘’ξ…žξ…žξ…žξ€»(𝑠)+3πΉπœ€πœ€πœ€ξ…ž(𝑠)πœ€ξ…žξ…ž(𝑠)+3πΉπœ†πœ†πœ†ξ…ž(𝑠)πœ†ξ…žξ…ž(𝑠)+3πΉπ‘’π‘’ξ€Ίπ‘’ξ…žξ…ž(𝑠),π‘’ξ…žξ€»(𝑠)+3πΉπœ€πœ†πœ€ξ…žξ…ž(𝑠)πœ†ξ…ž(𝑠)+3πΉπœ€πœ†πœ€ξ…ž(𝑠)πœ†ξ…žξ…ž(𝑠)+3πΉπœ€π‘’ξ€Ίπ‘’ξ…žξ€»πœ€(𝑠)ξ…žξ…ž(𝑠)+3πΉπœ€π‘’ξ€Ίπ‘’ξ…žξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+3πΉπœ†π‘’ξ€Ίπ‘’ξ…žξ…žξ€»πœ†(𝑠)ξ…ž(𝑠)+3πΉπœ†π‘’ξ€Ίπ‘’ξ…žξ€»πœ†(𝑠)ξ…žξ…ž(𝑠)+3πΉπœ€πœ†πœ†πœ€ξ…žξ€·πœ†(𝑠)ξ…žξ€Έ(𝑠)2+3πΉπœ€πœ€πœ†ξ€·πœ€ξ…žξ€Έ(𝑠)2πœ†ξ…ž(𝑠)+3πΉπœ€πœ€π‘’ξ€Ίπ‘’ξ…žπœ€(𝑠)ξ€»ξ€·ξ…žξ€Έ(𝑠)2+3πΉπœ€π‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)+3πΉπœ†πœ†π‘’ξ€Ίπ‘’ξ…žπœ†(𝑠)ξ€»ξ€·ξ…žξ€Έ(𝑠)2+3πΉπœ†π‘’π‘’ξ€Ίπ‘’ξ…ž(𝑠),π‘’ξ…žξ€»πœ†(𝑠)ξ…ž(𝑠)+6πΉπœ€πœ†π‘’ξ€Ίπ‘’ξ…žξ€»πœ€(𝑠)ξ…ž(𝑠)πœ†ξ…ž(𝑠)=0.(4.14) Setting 𝑠=0 in (4.14), we obtain πΉπœ€πœ€1ξ…žξ…žξ…ž(0)+πΉπœ†πœ†1ξ…žξ…žξ…ž(0)+𝐹𝑒𝑒𝑒𝑀0,𝑀0,𝑀0𝑒+𝐹𝑒1ξ…žξ…žξ…žξ€»(0)+3𝐹𝑒𝑒𝑒1ξ…žξ…ž(0),𝑀0ξ€»+3πΉπœ†π‘’ξ€Ίπ‘€0ξ€»πœ†1ξ…žξ…ž(𝐹0)=0,(4.15)πœ†πœ†2ξ…žξ…žξ…ž(0)+πΉπœ€πœ€2ξ…žξ…žξ…ž(0)+3πœ†2ξ…žξ…žξ€·πΉ(0)πœ†πœ†+πΉπœ†π‘’ξ€Ίπ‘£1ξ€»ξ€Έ+3πœ€2ξ…žξ…žξ€·πΉ(0)πœ€πœ†+πΉπœ€π‘’ξ€Ίπ‘£1ξ€»ξ€Έ+πΉπœ†πœ†πœ†+𝐹𝑒𝑒𝑒𝑣1,𝑣1,𝑣1ξ€»+𝐹𝑒𝑒2ξ…žξ…žξ…ž(ξ€»0)+3𝐹𝑒𝑒𝑒2ξ…žξ…ž(0),𝑣1ξ€»+3πΉπœ†π‘’ξ€Ίπ‘’2ξ…žξ…ž(ξ€»0)+3πΉπœ†πœ†π‘’ξ€Ίπ‘£1ξ€»+3πΉπœ†π‘’π‘’ξ€Ίπ‘£1,𝑣1ξ€»=0.(4.16) Substituting the expression of 𝑒1ξ…žξ…ž(0) into (4.15) and applying 𝑙 to it, we have πœ€1ξ…žξ…žξ…ž(0)=0 using (3.4), (4.12), and (𝐹5).

Acknowledgments

The authors the referee for very careful reading and helpful suggestions on the paper. The paper was partially supported by NSFC (Grant no. 11071051), Youth Science Foundation of Heilongjiang Province (Grant no. QC2009C73), Harbin Normal University academic backbone of youth project, and NCET of Heilongjiang Province of China (1251–NCET–002).

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Copyright Β© 2011 Ping Liu and Yuwen Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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