The splitting lemmas for nonsmooth functionals on Hilbert spaces I

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,..., D^mu)dx$ as in (\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincar\'e-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai \cite{DHK} with some techniques from \cite{JM, Skr, Va1}. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.


Motivation
Morse theory is an important tool in critical point theory. Morse inequalities, which provide the appropriate relations between global topological notions and the critical groups of the critical points, had been generalized to very general frameworks, see [11,37] (for C 1 -functionals on manifolds of infinite dimension) and [17] (for continuous functionals on complete metric spaces) and the references therein. These inequalities and precise computations of critical groups are extremely useful in distinguishing different types of critical points and obtaining multiple critical points of a functional (cf. [4,11,37,40]). However, the calculation of critical groups in applications is a complex problem. Gromoll-Meyer's generalization of Morse lemma to an isolated degenerate critical point in [23], also called the splitting theorem, provides a basic tool for the effective computation of critical groups. Since then many authors made their effort to improve the splitting theorem, see [11,24,37,26,27,30,19,20,31] and related historical and bibliographical notes in [11,Remark 5.1] and [37, page 202]. Probably, the most convenient formulations in the present applications are ones given in [10,Th. 2.1] (see also [11,Th. 5.1]) and [37,Th.8.3] (see also [36]). It was only assumed therein that f is a C 2 -functional on a neighborhood U of the origin θ in a Hilbert space H and that θ is an isolated critical point of f such that 0 is either an isolated point of the spectrum σ(d 2 f (θ)) or not in σ(d 2 f (θ)). This can be used to deal with many elliptic boundary value problems of form △u = f (x, u) on bounded smooth domains in R n with Dirichlet boundary condition.
However, the action functionals in many important variational problems are at most C 2−0 on spaces where the functionals can satisfy the (PS) condition. Let Ω ⊂ R n be a bounded domain with smooth boundary ∂Ω, x = (x 1 , · · · , x n ) ∈ R n , and let α = (α 1 , · · · , α n ) be a multi-index of nonnegative integer components α i , and |α| = α 1 + · · · + α n be its length. Denote by M(m) the number of such α of length |α| ≤ m, and by ξ = {ξ α : |α| ≤ m} ∈ R M (m) . Consider the variational problem F (u) = Ω f (x, u, · · · , D m u)dx, (1.1) where the function f : Ω×R M (m) → R, (x, ξ) → f (x, ξ) is measurable in x for all values of ξ, and twice continuously differentiable in ξ for almost all x; and there are continuous, positive, nondecreasing function g 1 and nonincreasing function g 2 such that the functions Generally speaking, under the assumptions above, as stated on the pages 118-119 of [44] (see [43] for detailed arguments) the functional F in (1.1) is C 1 and satisfies the (PS) condition on W m,2

0
(Ω), and the mapping F ′ is only G-differentiable on W m,2

0
(Ω); moreover, on Banach spaces on W m,p 0 (Ω) with p > 2, it is C 2 , but does not satisfy the (PS) condition. Furthermore, Morse inequalities were also obtained in [43,Chapter 5] under the assumptions that the functional F have only nondegenerate critical points. A similar question appears in some optimal control problems (see Vakhrameev [46]).
Under these assumptions the functional L τ is only C 2−0 on the Hilbert manifold H τ (as showed [1] recently), but satisfies the (PS) condition on H τ . The usual regularity theory shows that all critical points of L τ on H τ sit in the Banach manifold X τ = C 1 (R/τ Z, M).
It is very unfortunate that the (PS) condition cannot be satisfied on X τ though L τ is C 2 on it. So far one do not find a suitable space on which the functional L τ is not only C 2 but also satisfies the (PS) condition. The common points of the two functionals above are: one hand on a Hilbert manifold they have smoothness lower than C 2 , but satisfy the (PS) condition; on the other hand their critical points are contained in a densely and continuously imbedded Banach manifold on which the functional possesses at least C 2 smoothness, but does not satisfy the (PS) condition. To my knowledge there is no a suitable splitting lemma, which can be used to deal with the above functionals. These motivate us to look for a new splitting theorem.
With the regularity theory and prior estimation techniques of differential equations our theory can also be applied to some variational problems not satisfying our theorems (such as general Tonelli Lagrangian systems and geodesics on Finsler manifolds, see [32,Remarks 5.9,6.1], [35] and the references cited therein) by modifying the original Euler-Lagrangian functions.

Notion and terminology
Since there often exists some small differences in references we state some necessary notions and terminologies for reader's conveniences. Let E 1 and E 2 be two real normed linear spaces. Denote by L(E 1 , E 2 ) the space of the continuous linear operator from E 1 to E 2 , and by L(E 1 ) = L(E 1 , E 1 ). A map T from an open subset U of E 1 to E 2 is called directional differentiable at x ∈ U if for every u ∈ E 1 there exists an element of E 2 , denoted by DT (x, u), such that lim t→0 T (x+tu)−T (x) t − DT (x, u) = 0; DT (x, u) is called the directional derivative of T at x in the direction u. If the map U × E 1 → E 2 , (x, u) → DT (x, u) is continuous we say T to be continuously directional differentiable on U. (This implies that T is Gâteaux differentiable at every point of U in the following sense). If there exists a B ∈ L(E 1 , E 2 ) such that DT (x 0 , u) = Bu ∀u ∈ E 1 , T is called Gâteaux differentiable at x 0 ∈ U, and B is called the Gâteaux derivative of T at x 0 , denoted by DT (x 0 ) (or T ′ (x 0 )). By Definition 3.2.2 of [42], T is called strictly G (Gâteaux) differentiable at x 0 ∈ U if for any v ∈ E 1 , if this convergence uniformly holds for v in any compact subset we say T to be strictly H (Hadamard) differentiable 3 at x 0 ∈ U; moreover T is called strictly 4 (this implies that T has Fréchet derivative T ′ (x 0 ) at x 0 ). By [15,Prop.2.2.1] or [42,Prop.3.2.4(iii)], T is strictly H-differentiable at x 0 ∈ U if and only if T is locally Lipschitz continuous around x 0 and strictly G-differentiable at x 0 ∈ U. Specially, the strict F -differentiability of T at x 0 implies that T is Lipschitz continuous in some neighborhood of x 0 . By [42,Prop.3.4.2], the continuous F-differentiability of T at x 0 implies that T is strictly F-differentiable at x 0 . If T is F -differentiable in U, then dT = T ′ is continuous at x 0 ∈ U (i.e. T is continuously differentiable at x 0 ) if and only if T is strictly F -differentiable at x 0 , see Questions 3a) and 7a) at the end of [18,Chap.8,§6]. By Proposition B.1 the continuously directional differentiability of T in U implies the strict H-differentiability of T in U (and thus the locally Lipschitz continuality of T in U).

Method and overview
The main methods to the splitting lemma in past references are the implicit function theorem method such as [23] and dynamical system one as in [11,Th. 5.1] and [37,Th.8.3]. Our method is different from theirs completely. Recently, Duc-Hung-Khai [19] gave a new proof to the Morse-Palais lemma based on elementary differential calculus. It seems that the parameterized versions of the new Morse lemma cannot be applied to the above two typical functionals yet. After carefully analyzing the functionals we combine it with some techniques from [27,43,46] to successfully design a splitting lemma which is applicable to our above functionals. For completeness and reader's convenience we state the parameterized versions of Duc-Hung-Khai's Morse-Palais lemma in [19] and outline its proof in Appendix A. Some results on functional analysis are given in Appendix B.
In Section 2 we state our main results, which include a new splitting lemma, Theorem 2.1, and the corresponding shifting theorem, Corollary 2.6. We also obtain critical group characteristics for local minimum and critical points of mountain pass type under weaker conditions in Corollaries 2.7, 2.9, respectively. Corollary 2.5 and Theorem 2.10 study relations between critical groups of a functional and its restriction on a densely imbedded Banach space, which are very key for our work [35]. A theorem of Poincaré-Hopf type, Theorem 2.12, is proved in Section 5. We also study the functor properties of our splitting lemma in Section 6, and estimate behavior of the functional L of Theorem 2.1 near θ in Section 7. As concluding remarks it is shown in Section 8 that the most results in Theorem 2.1 still hold true under weaker conditions. These result have been used in [34] to generalize some previous results on computations of critical groups and some critical point theorems to weaker versions. This paper consists of the sections 1,2 and the appendix of [33], which is not to be published elsewhere. The fourth section of [33] has been rewritten and extended into a separate paper. The author would like to express his deep gratitude to the anonymous referee for many valuable revision suggestions and for pointing out many misprints.

Statements of main results
Let H be a Hilbert space with inner product (·, ·) H and the induced norm · , and let X be a Banach space with norm · X , such that (S) X ⊂ H is dense in H and the inclusion X ֒→ H is continuous, i.e. we may assume For an open neighborhood V of the origin θ ∈ H, V ∩ X is also an open neighborhood of θ in X, denoted by V X for clearness without special statements. Suppose that a functional L : V → R satisfies the following conditions: (F1) L is continuously directional differentiable (and thus C 1−0 ) on V .
(F2) There exists a continuously directional differentiable (and thus C 1−0 ) map A : V X → X, which is strictly Fréchet differentiable at θ, such that (This actually implies that L| V X ∈ C 1 (V X , R).) (F3) There exists a map B from V X to the space L s (H) of bounded self-adjoint linear operators of H such that (This and (F1)-(F2) imply: (a) A is Gâtuax differentiable and The origin θ ∈ X is a critical point of L| V X (and thus L), 0 is either not in the spectrum σ(B(θ)) or is an isolated point of σ(B(θ)). 5 where P (x) : H → H is a positive definitive linear operator and Q(x) : H → H is a compact linear operator with the following properties: (D1) All eigenfunctions of the operator B(θ) that correspond to negative eigenvalues belong to X; (D2) For any sequence {x k } ⊂ V ∩ X with x k → 0 it holds that P (x k )u − P (θ)u → 0 for any u ∈ H; (D3) The map Q : V ∩ X → L(H) is continuous at θ with respect to the topology induced from H on V ∩ X; (D4) For any sequence {x n } ⊂ V ∩ X with x n → 0 (as n → ∞), there exist constants C 0 > 0 and n 0 ∈ N such that Sometimes we need to replace the condition (D4) by the following slightly stronger (D4*) There exist positive constants η 0 > 0 and C ′ 0 > 0 such that 5 The claim in the latter sentence is actually implied in the following condition (D) by Proposition B.2. In order to state some results without the condition (D) we still list it. 6 Actually, this and (D4) imply the claim in the second sentence in (C1) by Proposition B.2.
Here is a way looking for the map B. Suppose that L| V X is twice Gâteaux differentiable at every point x ∈ V X , i.e. for any u 1 , u 2 ∈ X the limit exists and is linear continuous with respect to u i , i = 1, 2, where .
is self-adjoint with respect to the inner (·, ·) H . By Question 17) at the end of [18,Chap.11, §5], A ′ (x) can be extended into an elementB(x) ∈ L s (H) with the following properties: By the assumption (D) each B(x) is Fredholm. In particular, H 0 := Ker(B(θ)) is finitely dimensional. Let H ± := (H 0 ) ⊥ be the range of B(θ). There exists an orthogonal decomposition H = H 0 ⊕ H ± = H 0 ⊕ H − ⊕ H + , where H − and H + are subspaces invariant under B(θ) such that B(θ)| H + is positive definite and B(θ)| H + is negative definite. Clearly, we have also The conditions (C2) and (D) imply that both H 0 and H − are finitely dimensional subspaces contained in X by Proposition B.2. Denote by P * the orthogonal projections onto H * , * = +, −, 0, and by X * = X ∩ H * = P * (X), * = +, −. Then X + is dense in H + , and (I − P 0 )| X = (P + + P − )| X : (X, · X ) → (X ± , · ) is also continuous because all norms are equivalent on a linear space of finite dimension, where X ± := X ∩ (I − P 0 )(H) = X ∩ H ± = X − + P + (X) = X − + H + ∩ X. These give the following topological direct sum decomposition: Let ν = dim H 0 and µ = dim H − . We call them the nullity and the Morse index of critical point θ of L, respectively. In particular, the critical point θ is said to be nondegenerate if ν = 0. Since the norms · and · X are equivalent on the finite dimension space H 0 we shall not point out the norm used without occurring of confusions. In this paper, for a normed vector space (E, · ) and Moreover, we always use θ to denote the origins of all linear spaces without occurring of confusions.
an open neighborhood W of θ in H and an origin-preserving homeomorphism Moreover, the homeomorphism Φ has also properties: ) ⊂ X even if the topologies on these two sets are chosen as the induced one by X.
The map h and the function B H 0 (θ, ǫ) ∋ z → L • (z) := L(z + h(z)) 7 also satisfy: and dL • is strictly F-differentiable at θ ∈ H 0 and d 2 L • (θ) = 0; 7 If A is C 1 then maps h and L • have higher smoothness too, see Remark 3.2.
(iii) If θ is an isolated critical point of L| V X , then θ is also an isolated critical point of L • .
If the strictly Fréchet differentiability at θ of the map A : V X → X in (F2) is replaced by weaker conditions we shall show in Section 8 that the most results in Theorem 2.1 still hold true.
Under the conditions (L1)-(L3) it was proved in [32] that the functional L τ in (1.2) satisfies the assumptions of Theorem 2.1 near a critical point of it. In fact, a special version of Theorem 2.1 was used there. As stated in [43, §5.2] the arguments of [43,Chap.3] showed that the functional F in (1.1) satisfies the assumptions of Theorem 2.1 near a critical point of it too. Our frame conditions in Theorem 2.1 seem strange and complex. But they come from abstract and analysis for the studies in [43]. Of course, the theory of this paper can be used to improve one of [43]. This work is in progress.

Remark 2.2.
(i) Note that our proof only use the Banach fixed point theorem or the implicit function theorem in the case H 0 = {0}. If H 0 = {0}, we do not require the completeness of (X, · X ), that is, the condition (S) can be replaced by the following And the conclusions of Theorem 2.1 become: There exist a positive ǫ ∈ R, an open neighborhood W of θ in H and an origin-preserving homeomorphism, φ : ) ⊂ X are chosen as the induced ones by X.
(ii) Suppose that L is only defined on V ∩ X and that the condition (F1) can be replaced by the following (F1') L is continuously directional differentiable (and so C 1−0 ) on V ∩ X with respect to the topology of H.
Then the origin-preserving homeomorphism in (2.4) should be changed into (with respect to the topology of H), which satisfies (2.5) for all (z,  Clearly, this holds if L ∈ C 2 (V, R). In fact, the condition (C1) for B(θ) = d 2 L(θ) also imply the condition (D) in the case dim H 0 ⊕ H − < ∞. In order to see this we can write B(x) = P (x) + Q(x), where P (x) = P + B(x) − P − B(x) + P 0 and Q(x) = 2P − B(x) + P 0 + P 0 B(x). The latter is finite rank and therefore compact. The continuity of the map B : V → L s (H) implies that both maps P and Q are continuous, and that there exists a δ > 0 such that Note that (P (θ)u, u) H ≥ min{a 0 , 1} u 2 ∀u ∈ H and that We get Hereafter H q (A, B; K) denotes the qth relative singular homology group of a pair (A, B) of topological spaces with coefficients in K.
One of important applications of the splitting lemma is to compute critical groups of critical points. Recall that for q ∈ N ∪ {0} the qth critical group (with coefficients in K) of a real continuous functional f on a metric space M at a point x ∈ M is defined by where c = f (x) and U is a neighborhood of x in M. The definition of the critical groups are independent of the special choice of U because of the excision property of the singular homology. If M is a Banach space and f is C 1 then the qth critical group of an isolated critical point x may equivalently be defined as where c = f (x),f c = {f < c} and U is as above. (See [17,Prop.3.7]).
Actually our proof shows that (iii) implies θ to be a strict minimum.
Step 3 in the proof of Lemma 3.5 we easily derive a similar conclusion of Tromba's main result Theorem 1.3 in [45] without requirement for completeness of (X, · X ). Corollary 2.8. Under the assumptions of Theorem 2.1, but no requirement for completeness of (X, · X ), i.e., the condition (S) is replaced by (S ′ ), suppose also that d 2 (L| V X )(θ)(u, u) > 0 for any u ∈ X \ {θ}. Then θ is a strict minimum for L and thus L| V X .
According to Hofer [24] the critical point θ is called mountain pass type if for any small neighborhood O of θ in H the set {x ∈ O | L(x) < 0} is nonempty and not pathconnected.
(ii) If ν = dim Ker(B(θ)) = 1 in the case µ = dim H − = 0, then θ is mountain pass type if and only if C q (L, θ; K) ∼ = δ q1 K ∀q ∈ Z; The proofs of (i) and (ii) are the same as those of [ Gâteaux differentiable, if V = X and DL : X → X * is continuous from the norm topology of X to the weak*-topology of X * one may use a generalized version of mountain pass lemma in [22] to yield a critical point of mountain pass type provided that L also satisfies the condition (C) (weaker than (PS)).
If the critical point θ of L is isolated, Corollary 2.5 yields surjective homomorphisms from critical groups C * (L| V X , θ; K) to C * (L, θ; K), which are also isomorphisms provided that K is a field and both groups are finite dimension vector spaces over K of same dimension. When L ∈ C 2 (V, R) and A ∈ C 1 (V X , X) it follows from [27, Cor.2.8] that C * (L| V X , θ; K) ∼ = C * (L, θ; K) for any Abel group K. The following theorem generalizes and refines this result.
as an open subset of X (resp. Y ) as before. Assume also that Then for any open neighborhood W of θ in V and a field F the inclusions .
The first isomorphism in the final claims is due to Jiang [27], see Corollary 4.4. Taking Y = X we get Corollary 2.11. Under the assumptions of Theorem 2.1, also assume: (i) θ is an isolated critical point of L, (ii) L| V X ∈ C 2 (V X , R), (iii) the map A in (F2) belongs to C 1 (V X , X), (iv) the map B in (F3) is continuous, Then for any open neighborhood W of θ in V and a field F the inclusion induces isomorphisms between their relative homology groups with coefficients in F. Specially, C * (L| V X , θ; F) ∼ = C * (L, θ; F).
If Ω ⊂ R n is a bounded open domain with smooth boundary ∂Ω, and f ∈ C 1 (Ω × R, R) satisfies the condition: |f ′ t (x, t)| ≤ C(1 + |t| α ) for some constants C > 0 and α ≤ n+2 n−2 (if n > 2), then for an isolated critical point u 0 of the functional is also an isolated critical point of J| X . This result was obtained by Chang [13] under the assumption that J satisfies the (PS) c condition. Brézis and Nirenberg [8] firstly proved it as u 0 is a minimizer. Theorem 2.1 and Corollary 2.6 cannot be applied to the geodesic problems on Finsler geometry directly. But as outlined in Remark 5.9 of [32] we may develop an method of infinite dimensional Morse theory for geodesics on Finsler manifolds based on them in [35], that is, giving the shifting theorem of critical groups of the energy functional of a Finsler manifold at a nonconstant critical orbit and relations of critical groups under iterations. In particular, Corollary 2.5 is a key for us to realize the second goal.
Finally we give a theorem of Poincaré-Hopf type. By the condition (F1) the functional L : V → R is Gâteaux differentiable. Its gradient ∇L is equal to A on V ∩X by the condition (F2). Furthermore, under the assumptions (F3) and (D) we can prove that for a small ǫ > 0 the restriction of ∇L to B H (θ, 2ǫ) has a unique zero θ and is a demicontinuous map of class (S) + . According to [9] and [44] we have a degree deg BS (∇L, B H (θ, ǫ), θ). Under the conditions (C1) and (C2), A ′ (θ) : X → X is a bounded linear Fredholm operator of index zero, see the first paragraph in Step 1 of proof of Lemma 3.1. If the map A in (F2) is C 1 , then A is a Fredholm map of index zero near θ ∈ X and thus for sufficiently small ǫ > 0 there exists a degree deg FPR (A, B X (θ, ǫ), θ) or deg BF (A, B X (θ, ǫ), θ) according to [21,39] or [5,6].
provided a suitable orientation for A.
(ii) If θ is also an isolated critical point of L, and the condition (D4 * ) holds true, then for a small ǫ > 0, Here deg is the classical Brouwer degree.
The first equality in (ii) of Theorem 2.12 is a direct consequence of [14, Th.1.2] once we prove that the map ∇L is a demicontinuous map of class (S) + near θ ∈ H.
Using Theorem 2.1 we also gave a handle body theorem under the our weaker framework in Theorem 2.8 of [34].

Proof of Theorem 2.1
We shall complete the proof of Theorem 2.1 by a series of lemmas.

Lemma 3.1. Under the above assumption (S), for an open neighborhood
Then there exist a positive r 0 ∈ R, a unique map h : Proof. The proof method seems to be standard. For completeness and the reader's conveniences we give its detailed proof in two steps. Step In particular, this implies that A is continuous in B X (θ, r). Let Here the first two inequalities come from (3.2), and the third one is due to (3.1). In particular, for any z ∈ B H 0 (θ, r 1 ) and By (3.1), for any ε > 0 there exists a number δ > 0 such that Hence h is strictly F-differentiable at θ ∈ H 0 and h ′ (θ) = 0.
Step 2. Let us prove the remainder "Moreover" part. Since L| V ∩X is continuous and continuously directional differentiable (with respect to the induced topology on , by the mean value theorem we have s ∈ (0, 1) such that Let t → 0, we have because of (3.5) and the continuity of A in B X (θ, r). From this and (3.9) it follows that From this it easily follows that because of (3.1) and (3.6). Hence z 0 → DL • (z 0 ) is continuous and Since H and X induce equivalent norms on H 0 and thus on L(H 0 , R), the alternative cannot lead to any troubles for the arguments.) By [7, Th.2.1.13], this implies that L • is Fréchet differentiable at z 0 and its Fréchet differen- Now for any ε > 0 let δ > 0 such that (3.8) holds. For δ 0 ∈ (0, δ) below (3.8), by (3.10) and (3.6) we obtain Since · and · X are equivalent norms on H 0 we may choose δ > 0 so small that (3.11) and get that the maps h and L • are C 1 and C 2 , respectively. Precisely, Then for each z ∈B H 0 (θ, δ) the map F (z, ·) is continuously directional differentiable in B H ± (θ, δ), and the directional derivative of it at u ∈ B H ± (θ, δ) in any direction v ∈ H ± is given by It follows from this and (3.5) that (3.14) Now we wish to apply Theorem A.1 to the function F . In order to check that F satisfies the conditions in Theorem A.1 we need two lemmas.
Proof. Note that the condition (D2) can be equivalently expressed as: as x ∈ V ∩ X and x → 0 (because of the conditions (D2) and (D3)).
When H 0 = {θ} under the stronger assumptions the following lemma was proved in [43,46]. We also give proof of it for clearness.

Lemma 3.4.
There exists a small neighborhood U ⊂ V of θ in H and a number a 1 ∈ (0, 2a 0 ] such that for any x ∈ U ∩ X, Assume by contradiction that (i) does not hold. Then there exist sequences {x n } ⊂ V ∩X with x n → 0, and {u n } ∈ H + with u n = 1 ∀n, such that Passing a subsequence, we may assume that and that u n ⇀ u 0 in H. We claim: u 0 = θ. In fact, by the condition (D4) we have constants C 0 > 0 and n 0 ∈ N such that (P (x n )u, u) ≥ C 0 u 2 for any u ∈ H and n ≥ n 0 . Hence Moreover, a direct computation gives by the compactness of Q(θ), and by the condition (D3). Hence (3.18)- (3.20) give Then this and (3.16)-(3.17) yield This implies u 0 = θ. Note that u 0 also sits in H + . As above, using (3.20) we derive Note that It follows from this and (3.21)-(3.22) that because of the condition (D2) and (3.22). Similarly, we have From these, (3.16) and (3.23) it follows that By shrinking U (if necessary) we can require that ω(x) < a 0 for any x ∈ U ∩ X. Then the desired conclusion is proved.
Since h(θ) = θ, for the neighborhood U in Lemma 3.4 we may take ε ∈ (0, δ) so small that Lemma 3.5. For the above ε > 0 the restriction of the function F in (3.12) Proof. By (3.14) we only need to prove that F satisfies conditions (ii)-(iv) in Theorem A.1. Step Moreover, A is continuously directional differentiable so is the function By the mean value theorem we have t ∈ (0, 1) such that This implies the condition (ii).
By Lemma 3.5 we can apply Theorem A.1 to F to get a positive number ǫ, an open neighborhood W ofB H 0 (θ, ε) × {θ} inB H 0 (θ, ε) × H ± , and an origin-preserving homeomorphism even if the last two sets are equipped with the induced topology from X, or, equivalently, for (3.28) Consider the continuous map . Since H 0 and H − are finitely dimensional subspaces contained in X, from Steps 1,4 in the proof of Theorem A.1 it is easily seen that Then (2.6) follows from this and the fact that Im(h) ⊂ X ± ⊂ X. In particular, it holds that Φ(B H 0 (θ, ε) × B H − (θ, ǫ)) ⊂ X. Now we can complete the proof of Theorem 2.1 by the following lemma.  ǫ)). Then To prove the second claim, it suffices to prove that for (3.30) Note that h ∈ C(B H 0 (θ, δ), X ± ) and that X and H induce equivalent topologies on it follows from (3.28) that in (3.30) the left side implies the right side. Conversely, if the right of (3.30) holds, then . ǫ)). It is contained in X by (2.6). We write W 0− as W X 0− when it is considered a topological subspace of X. Clearly, It gives a deformation retract from L 0 ∩ W onto L 0 ∩ W 0− . Hence the inclusion induces isomorphisms between their relative singular homology groups with inverse (η 1 ) * , where η 1 (·) = η(1, ·). That means that each has a relative singular cycle representative, c = j g j σ j , such that By the conclusion (b) in Theorem 2.1 the identity map is a homeomorphism. So c is also a relative singular cycle in Denote by the inclusion and by the inclusion Since This completes the proof of Corollary 2.5.

Proofs of Theorem 2.10
Recall that H 0 = Ker(B(θ)) and We need the following theorem by Ming Jiang.

Remark 4.2.
(i) From the arguments of Lemma 3.1 and the proof of [27] it is easily seen that near θ ∈ N the map ρ is equal to h in Lemma 3.1.
(ii) It was proved in [27, Prop.2.1] that the condition (iii) in Theorem 2.10 can be derived from others of this proposition and the following two conditions: In this case, for y ∈ Y we can write y ⊥ = (I − P 0 )y = y + + y − = P + y + P − y and hence Then θ ∈ H 0 is its critical point, and also isolated if θ is an isolated critical point of L| V X . By Remark 3.2, ρ is C 1 , and Lemma 3.1 and Remark 4.2(i) show that near θ ∈ H 0 , If θ is an isolated critical point of L| V X (and hence L| V Y ), then by Theorem 4.1 we can use the same proof method as in [37,Th.8.4] Actually, from the proof of [27, Cor.2.8] one can get the following stronger conclusion: Proof. By the excision property of the singular homology theory we only need to prove it for some open neighborhood U Y of θ in V Y . By [27, Claim 1]) y D = (P 0 + P − )y Y + P + y Y gives a norm on Y equivalent to · Y . Let κ 0 ∈ (0, κ) be so small that is a neighborhood of θ in Y (resp. X) which only contains θ as a unique critical point of L| V Y (resp. L| V X ). (This can be assured by the second claim in Theorem 4.1). For conveniences let and isomorphisms Clearly, ℜ(0, ·) = id, ℜ(t, ·)| Ψ −1 (Y) 0− = id and ℜ(1, Ψ −1 (Y)) ⊂ Ψ −1 (Y) 0− . It was proved in [27] that ℜ is also a continuous map from [0, 1] × (B Y κ 0 ∩ X) to X (with respect to the induced topology from X) and that These show that ℜ gives not only a deformation retract from (with respect to the induced topology from X). Hence inclusions

It is obvious that
Hence the norms · X and · Y are equivalent on H 0 + H − . It follows from this and (4.2) that i xy 0 is a homeomorphism. This shows that (i xy 0 ) * and hence i xy * is an isomorphism. Note that Before proving Theorem 2.10 we also need the following observation, which is contained in the proof of [11, Th.3.2, page 100] and seems to be obvious. But the author cannot find where it is explicitly pointed out.

Remark 4.6.
Let H be a real Hilbert space, and let f ∈ C 2 (H, R) satisfy the (PS) condition. Assume that df (x) = x − T x, where T is a compact mapping, and that p 0 is an isolated critical point of f . Then for any field F and each q ∈ N ∪ {0}, C q (f, p 0 ; F) is a finite dimension vector space over F. In particular, if f ∈ C 2 (R n , R) has an isolated critical point p 0 ∈ R n then C q (f, p 0 ; F), q = 0, 1, · · · , are vector spaces over F of finite dimensions. In fact, by [11, for any Abel group K. Note that we may assume that W is given by Theorem 2.1 because of the excision property of the singular homology groups. By Proposition 4.5 the inclusion By (4.3) and Remark 4.6, for a field F and each q ∈ N ∪ {0}, are isomorphic vector spaces over F of finite dimension. Then any surjective (or injective) homomorphism among them must be an isomorphism. By Corollary 2.5 I xw * is a surjection and hence an isomorphism. Since I xw * = I yw * • I xy * , I yw * is also an isomorphism.

Proof of Theorem 2.12
We use the ideas of [24] to prove (i) in Step 1, and then derive (ii) in Step 2 from [14, Th.1.2] by checking that ∇L is a demicontinuous map of class (S) + .
By the first equality, (3.5) and the uniqueness we have u = h(z). So the second equality becomes This and (3.5) give A(z + h(z)) = θ. By (i) we get z + h(z) = θ. That is, z = θ and z + u = θ.
By Lemma 3.1(i), h ′ (θ) = θ. Using this it is easily proved that dΓ t (θ) = A ′ (θ) for any t ∈ [0, 3]. Since the C 1 Fredholm map is locally proper, we can shrink ǫ > 0 such that the restriction of each Γ t toB X (θ, ǫ) is Fredholm and that the restriction of Γ to [0, 3] ×B X (θ, ǫ) is proper. Hence Γ : [0, 3] × B X (θ, ǫ) → X satisfies the homotopy definition in the Benevieri-Furi degree theory [5,6], and we arrive at Moreover dim H 0 < ∞ implies that the map is compact. Hence the Leray-Schauder degree deg LS (I − K, B X (θ, ǫ), θ) exists, and for a suitable orientation of the map I − K. By Remark 3.2 and Lemma 3.1 L • is C 2 and Hence the gradient of L • with respect to the induced inner on H 0 (from H), denoted by ∇L • , is given by ∇L • (z) = P 0 A(z + h(z)) ∀z ∈ B H 0 (θ, r 0 ). By the definition and properties of the Leray-Schauder degree it is easily proved that ) is open, connected and simply connected. After a suitable orientation is chosen it follows from (5.3)-(5.5) that where the final equality comes from [37,Th.8.5]. Combing this with Corollary 2.6 the expected first conclusion is obtained.
Let us prove the remainder cases. Firstly, consider the case ν(L, θ) = ν( L, θ) = 0. We only need to remove z andẑ in the arguments below Claim 6.2 and then replace F and F by L and L, respectively.

An estimation for behavior of L
In this section we shall estimate behavior of L near θ. Such a result will be used in the proof of Theorem 5.1 of [34].
We shall replace the condition (D4) in Section 2 by the following stronger (D4**) There exist positive constants η ′ 0 and C ′ 2 > C ′ 1 such that For the neighborhood U in Lemma 3.4 we fix a small ρ ∈ (0, η ′ 0 ) so that We may assume that a 1 is no more than a 0 in Lemma 3.4. Set Since h(θ) = θ we can choose ρ 0 ∈ (0, ρ] so small that ω in Lemma 3.3 and Q in (D3) satisfy for all z ∈ B (0) ρ 0 and u ∈ B ± (ρ 0 ,ρ 0 ) ∩ X. As before we write B H ± (θ, δ) ∩ X as B H ± (θ, δ) X when it is considered as an open subset of X ± , and F X as the restriction of the functional F in (3.12) toB H 0 (θ, δ) × B H ± (θ, δ) X .
Then for positive constants ε = a ′ 1 s 2 and = a 1 8 s 2 (7.6) the following conclusions hold.

Concluding remarks
In this section we shall show that some conclusions of Theorem 2.1 can still be obtained if the strictly Fréchet differentiability at θ of the map A : V X → X is replaced by a weaker condition similar to (E ∞ ) or (E ′ ∞ ) in Theorems 4.1 and 4.3 of [33]. That is, the condition (F2) can be replaced by the following weaker (F2 ′ ) or (F2 ′′ ).
(F2 ′ ) There exists a continuously directional differentiable (and thus C 1−0 ) map A : V X → X such that DL(x)(u) = (A(x), u) H for all x ∈ V X and u ∈ X (which actually implies that L| V X ∈ C 1 (V X , R)), and that for some positive numbers κ > 1, r 1 > 0 and all Here C 1 is given by (3.2).
Next we consider the case (F2 ′ ) holding. By the proof of (3.4) we easily see and thus S(z, for any z ∈B H 0 (θ, r 0 ). Hence for each z ∈B H 0 (θ, r 0 ) we may apply the Banach fixed point theorem to the mapB to get a unique map h :B H 0 (θ, r 0 ) →B X ± (θ, r 0 ) such that S(z, h(z)) = h(z). From the latter and (8.5) it easily follows that for any z i ∈B X ± (θ, r 0 ), i = 1, 2. That is, h is Lipschitz continuous. Using this we may prove as in Step 2 of the proof of Lemma 3.1 that L • has a linear bounded Gâteaux derivative at each z 0 ∈B H 0 (θ, r 0 ) and Moreover, checking the proof of (3.10) we have still (3.10), i.e., for all z 0 ∈B H 0 (θ, r 0 ) and z ∈ H 0 . Note that A is continuously directional differentiable and hence C 1−0 . It follows from (8.6) that the mapB H 0 (θ, r 0 ) ∋ z 0 → DL • (z 0 ) ∈ L(H 0 , R) is C 1−0 . As before we derive from [7,Th.2.1.13] that L • is Fréchet differentiable at z 0 and its Fréchet differential dL • (z 0 ) = DL • (z 0 ) is C 1−0 in z 0 ∈ B H 0 (θ, r 0 ). Summarizing the above arguments we obtain  (2.5) and (2.6) Consequently, θ is an isolated critical point of L • provided that θ is an isolated critical point of L| V X .
Carefully checking the arguments in Section 2 and the proofs in Section 4 it is not hard to derive: By Claim 6.1, In order to assure that Theorem 6.1 also holds when Theorem 2.1 with (F2) is replaced by Theorem 8.1 with (F2 ′′ ) we should require not only that J| X : X → X is a Banach isometry but also that C 1 in (8.2) for (A, B) is replaced by C 1 . For Theorem 6.3 being true after Theorem 2.1 is replaced by Theorem 8.1 it is suffice to assume that J| X : X → X is a Banach isometry. Theorem 6.4 also holds if we replace "Theorem 2.1" by "Theorem 8.1" there.
Finally, we have also a corresponding result with Proposition 7.1 provided that the sentence "Under the assumptions of Theorem 2.1 with (D4) replaced by (D4**), suppose that the map A : V X → X in the condition (F2) is Fréchet differentiable." in Proposition 7.1 is replaced by "Under the assumptions of Theorem 8.1 with (D4) replaced by (D4**), suppose that the map A : V X → X in the condition (F2 ′′ ) is Fréchet differentiable."

A Parameterized version of Morse-Palais lemma due to Duc-Hung-Khai
Almost repeating the proof of Theorem 1.1 in [19] one easily gets the following parameterized version of it ( [33]). Actually we give more conclusions, which are key for proofs of some results in this paper.    with φ(λ, x) = (λ, φ λ (x)), such that The claim in "Moreover" part was not stated in [19], and can be seen from the proof therein. It precisely means: for any two norms · 1 and . This leads to the proof of Theorem 2.1(b), which is a key for the proofs of Corollary 2.5 and Theorem 2.10. So it is helpful for readers to outline the proof of Theorem A.1.

Sketches of proof of Theorem
This is actually contained in the proof of [19].
We here give a detailed proof of it because the compactness of Λ is very key in the following proof. They are helpful for understanding the proof of the noncompact case in Section 4 of [33].
In order to give the corresponding version at critical submanifolds we need a more general result than Theorem A.1. For future conveniences we here present it because many arguments and notations can be saved. Let Λ and E be two topological spaces. Imitating [29, §1 of Chap.III] one can naturally define a topological normed vector bundle over Λ to be a triple (E, Λ, p), where p : E → Λ is a continuous surjection (projection). In particular we have the notions of a topological Banach (resp. Hilbert) vector bundle. Corresponding to Definition 3.1 in Chapter 2 of [25], a bundle morphism from the normed vector bundles p 1 : E (1) → Λ 1 to p 2 : E (2) → Λ 2 is a pair of continuous maps (f , f ), wheref : E (1) → E (2) and f : Λ 1 → Λ 2 such that p 2 •f = f • p 1 . As on the pages 43-44 of [29] we may define the notion of a normed vector bundle morphism. If Λ 1 = Λ 2 = Λ and f = id Λ we get the notions of a Λ-bundle morphism and a Λ-normed vector bundle morphism. When f andf are homeomorphisms onto Λ 2 and E (2) the corresponding bundle morphism and normed vector bundle morphism (f , f ) are called bundle isomorphism and normed vector bundle isomorphism from E (1) onto E (2) . See [29] for more notions such as subbundles and so on. As in [11, Def.2.2, page 15] we can define a Finsler structure on the bundle p : E → Λ, and show the existence of such a structure on the vector bundle if Λ is paracompact.
Let G be a topological group. For a normed vector bundle p : E → Λ, let both E and Λ be also G-spaces and let p be a G-map (or G-equivariant map), we call it a G-normed vector bundle if for all g ∈ G the action of g : E λ → E gλ is a vector space isomorphism.
This leads to a contradiction. The uniqueness of ϕ λ (x) can also be proved by contradiction.
Then the desired conclusion follows from this and (A.12)-(A.13).

B Several results on functional analysis
Perhaps the results in this appendix can be founded in some references. For the readers's convenience we shall give proofs of them. Let E 1 and E 2 be two real normed linear spaces and let T be a map from an open subset U of E 1 to E 2 . For a positive integer n we call T finite n-continuous at x ∈ U if for any h 1 , · · · , h n ∈ E 1 the map R n ⊇ B n (0, ǫ) ∋ t = (t 1 , · · · , t n ) → T (x + t 1 h 1 + · · · + t n h n ) is continuous at the origin 0 ∈ R n .

Proposition B.1. (i)
If for any u ∈ E 1 the map x → DT (x, u) is finite 2-continuous at x 0 ∈ U then u → DT (x 0 , u) is additive.
(ii) If T is continuously directional differentiable on U then it is strictly H-differentiable at every x ∈ U, and restricts to a C 1 -map on any finitely dimensional subspace.
(So the continuously directional differentiability is a notion between the strict Hdifferentiability and C 1 .) (iii) If T : U → E 2 is G-differentiable near x 0 ∈ U and also strictly G-differentiable at x 0 , then T ′ is strongly continuous at x 0 , i.e. for any v ∈ E 1 it holds that T ′ (x)v − T ′ (x 0 )v → 0 as x − x 0 → 0. In particular, if E 2 = R this means that T ′ is continuous with respect to the weak* topology on E * 1 .
Since the map x → DT (x, u) is finite 2-continuous at x 0 ∈ U it follows that lim t→0 y * 1 t △ 2 tu,tv T (x 0 ) = 0.

Proposition B.2. Suppose that a bounded linear self-adjoint operator B on a Hilbert
space H has a decomposition B = P + Q, where Q ∈ L s (H) is compact and P ∈ L s (H) is positive, i.e., ∃ C 0 > 0 such that (P u, u) H ≥ C 0 u 2 ∀u ∈ H. Then every λ ∈ (−∞, C 0 ) is either a regular value of B or an isolated point of σ(B), which is also an eigenvalue of finite multiplicity.
Actually, this result may also follow from Proposition B.3 below. By Proposition 4.5 of [16], if A is a continuous linear normal operator (i.e. A * A = AA * ) on a Hilbert space H, then for λ ∈ σ(A) the range R(A − λI) is closed if and only if λ is not a limit point of σ(A). As a consequence we deduce that (i) and (ii) of the following proposition are equivalent. By the Banach inverse operator theorem we arrive at (ii)⇒ (iii). Conversely, R(A) = A(W ) = W is closed.