A KIND OF GENERALIZED TRANSVERSALITY THEOREM FOR C r MAPPING WITH PARAMETER

. The author considers a generalized transversality theorem of the mappings with parameter in inﬁnite dimensional Banach space. If the mapping is generalized transversal to a single point set, and in the sense of exterior parameters, the mapping is a Fredholm operator, then there exists a residual set of parameter, such that the Fredholm operator is generalized transversal to the single point set.


1.
Introduction. Transversality is an important and basic notion in geometry and analysis. The celebrated transversality theorem of Thom is always used to character the stability, and has many important applications in nonlinear differential equations. Zeidler has pointed out that transversality is certainly one of the most important concepts in modern mathematics, which provides an answer to the question that when the preimage of a manifold is still a manifold.
We often meet an equation with a parameter s : F (x, s) = 0.
The following phenomenon has been observed: a branch of solutions x(s) depending on s, is either disappeared or split into several branches, as s attains some critical values. This kind of phenomenon is called bifurcation. It is well known that all eigenvalues of second-order ordinary differential equations on bounded intervals with Dirichlet data are simple, but it is not true for partial differential equations, for example, the Laplacian on a ball. In order to study the simplicity of eigenvalues of the Laplacian on bounded domains with Dirichlet data, Chang established a transversality theorem with one-parameter (see Theorem 1.3.18 in [1]). Let F (u, s) : X × S → Z be a C r mapping, where X, Z are Banach spaces, S is a C r Banach manifold. Assume that F (u, s) ⋔ {θ}, f s (u) = F (u, s) is a Fredholm mapping. Then there exists a residual set Σ ⊂ S, such that f s ⋔ {θ}, for all s ∈ Σ. This theorem showed that for most domains, all eigenvalues of secondorder partial differential equations on bounded intervals with Dirichlet data are simple. If {θ} is a critical value of F (u, s), does the transversality theorem hold?
In this paper, the author mainly considers the above transversality theorem if {θ} is a regular value or some special critical value, and establishes two examples and a generalized transversality theorem. In the Example 3.1, the mapping f : R 2 → R 2 does not have any nontrivial regular value. The preimage of every critical value of f is a smooth regular submanifold of preimage space R 2 . This example shows that there are some critical values which have similar properties to those of regular values. This kind of critical values and regular values are called generalized regular values. On the other hand, the concept of transversality, f ⋔ P mod N, requires that Im(f ′ (x)) + T P f (x) = T N f (x) for any x ∈ f −1 (P ). However, it is often the case that this equation does not hold.
In [4], The structure of this paper is as follows. In Section 2, some preliminaries needed in the sequel are presented. In Section 3, two examples for generalized regular value and generalized transversality are established. In Section 4, the generalized transversality theorem of Fredholm operators with parameter between infinite dimensional Banach spaces is established.
In this paper, all the derivatives are Fréchet derivatives, all symbols could be found in [5,6].

2.
Preliminaries. This section presents an introduction to generalized inverse, generalized regular value and generalized transversality.
Let M, N be C k -Banach manifolds, k ≥ 1. Then f : M → N is called C r (r ≤ k), which means that for each point x ∈ M and admissible charts (U, ϕ), There is no denying the fact that the tangent map is linear and continuous. If f is the representative of f in the admissible charts (U, ϕ) and (V, where f ′ (x ϕ ) is the Fréchet derivative of the representative f . The basic ideas of the theory of generalized inverse are as follows. Let X, Y be Banach spaces, and B(X, Y ) be the space of all bounded linear operators from the where the symbols Im(T x ) denotes the image of T x , and Ker(T + x0 ) denotes the kernel of T + x0 . (See [2]) This concept induces the following two concepts of locally fine point. One is defined in Banach spaces, the other in Banach manifolds. Let X, Y be Banach spaces, f : According to the concept of locally fine point mentioned above, the regular point x 0 is a locally fine point of f, but the converse statement may not be true (see the Example 3.1 in this paper).
If y ∈ F is a regular value of f , which means that f −1 (y) is empty or consists of only regular points of f, then y must be the generalized regular value of f. We call the complement of generalized regular value as sharp−critical value. The following Sard-Smale Lemma holds in the sense of sharp critical value.

Lemma 2.2 (Sard-Smale Lemma).
Suppose that X is a separable Banach space and Y is a Banach space. Let f ∈ C r (U, Y ) be a Fredholm mapping, where U ⊂ X is an open set. If r > max{0,ind(f )}, then the set of critical values is of the first category.
Transversality is a promotion of regular value. Assume that M, N are C r (r ≥ 1) Banach manifolds, P is a submanifold of N, f : M → N is a C r mapping. Then f is called transversal to P in N if and only if for any y ∈ f (M ) ∩ P, the preimage (f ′ (x)) −1 (T P f (x) ) splits T M x for all x with f (x) = y. We write as f ⋔ P mod N.
Thom's famous result, the transversality theorem, which provides an answer to the question that when the preimage of a manifold is still a manifold. Let M, N be C r (r ≥ 1) Banach manifolds, P be a submanifold of N . Suppose that f : M → N is a C r mapping. If f ⋔ P mod N, then the preimage S = f −1 (P ) is a submanifold of M with the tangent space T S x = (f ′ (x)) −1 (T P f (x) ), for any x ∈ S.
However, in our daily studying, we often need transversality theorem in case that Im(f ′ (x)) + T P f (x) = T N f (x) . So, the concept of generalized transversality and generalized transversality theorem are necessary. Definition 2.3. Let f : M → N be a C r mapping and P be a submanifold of N . Then f is generalized transversal to P , and write as f ⋔ G P mod N, if for each x 0 ∈ f −1 (P ), the following two conditions are satisfied.
(ii) For any x 0 ∈ f −1 (P ), there exists a neighborhood U 0 at x 0 and a subbundle Lemma 2.4. Let M, N be C k Banach manifolds, f : M → N be a C r mapping, and P be a submanifold of N. If f ⋔ G P mod N, then the preimage S = f −1 (P ) is a submanifold of M with the tangent space for any x ∈ S. If P only consists of a single point y ∈ N, then f is generalized transversal to P if and only if y is a generalized regular value of f. (See [4]) 3. Examples. There are two examples in this section. The difference between regular value and generalized value is showed in the first example. The preimage of every critical value of f in this example is a smooth regular submanifold of the preimage space. The second one is an example of generalized transverlality. By the way, the mapping in the first example is generalized transversal to the origin {θ}, the proof of this generalized transversality is presented in the beginning of the next section.
Notice that f ′ (p) is not surjective, for all p ∈ U, f has not regular points and nontrivial regular values. However, for any p ∈ U, p is a locally fine point for f, then f has nontrivial generalized regular values. All q ∈ f (U ) are critical values. In fact, for z > −1, f −1 (e z+1 − e, z) = {p = (x, y) | x 2 + y 2 − 1 = z} consists of only locally fine points of f. That is to say, (e z+1 − e, z) is a generalized regular value of f. In particular, we obtain that the critical value θ = (0, 0) of f is a generalized regular value of f.
2. An example of generalized transversality. Let f : R 2 → R 3 be defined by (x, y, z) = f (s, t) = (s, s 3 , t), and let P be the z-axis in R 3 . Then, f −1 (P ) is the t-axis in R 2 , and t-axis is a submanifold of R 2 . f is not transversal to P mod R 3 . Indeed, f ⋔ G P mod R 3 (see Figure 1).
So for each (s, t) ∈ f −1 (P ), we have Im(T f (0,t) ) + T P (0,0,t) R 3 , that is, f is not transversal to P mod R 3 . However, f ⋔ G P mod R 3 . In fact, (T f (s,t) ) −1 (T P (s,s 3 ,t) ) is the t−axis in R 2 , let E 0 be the s−axis in R 2 , then for any (s, t) ∈ R 2 , This proves that f ⋔ G P mod R 3 .
Then there exists a residual set Σ ⊂ S, which is the countable intersection of open dense sets, such that for all s ∈ Σ, f s ⋔ G {θ}.
Proof. Let V = F −1 (θ), by Lemma 2.4, we can draw the conclusion that V is a submanifold of X since F ⋔ G θ.
Define the injection: i : V → X × S, and the projection π : X × S → S, and p : X × S → X. In the following, all of them are restricted on V. We claim that π • i is a Fredholm mapping with ind(π • i) =ind(f s ).