ON THE EXISTENCE OF A GLOBAL DIFFEOMORPHISM BETWEEN FRÉCHET SPACES

We provide sufficient conditions for existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the mountain pass theorem which plays a key ingredient in the proof of the main theorem.


Introduction
In this paper we consider the problem of finding sufficient conditions under which a tame map between tame Fréchet spaces becomes a global diffeomorphism. Tame maps are important because they appear not only as differential equations but also as their solutions (see [4] for examples). Although, the theory of differential equations in Fréchet spaces has a significant relation with problems in both linear and nonlinear functional analysis but not many methods for solving different type of differential equations are known. Our result would provide an approach to solve an initial value nonlinear integrodifferential equation x 1 ptq`ż t 0 φpt, s, xpsqqds " yptq, t P r0, 1s.
We follow the ideas in [5] and [6] where the analogue problem for Banach and Hilbert spaces was studied. There are two approaches to calculus on Fréchet spaces. The Gâteaux-approach (see [7]) and the so called convenient analysis (see [9]). We will apply the first one because to define the Palais-Smale condition, which plays a significant role in the calculus of variation, we need an appropriate topology on dual spaces that compatible with our notion of differentiability; only in the first approach there exists such a topology.
In [3], the author defined the Palais-Smale condition for C k -maps between Fréchet spaces and obtained some existence results for locating critical points. In this paper by means of this condition we generalize the mountain pass theorem of Ambrosetti and Rabinowitz to Fréchet spaces. Our proof of the mountain pass theorem relies on the Ekeland's variational principle. Since, in general, we can not acquire deformation results for Fréchet spaces because of the lack of a general solvability theory for differential equations. It is worth mentioning that for every Fréchet space the projective limit techniques gives a way to solve a wide class of differential equations (see [1]). This technique would be a way of obtaining many results (such as deformation lemmas) for Fréchet spaces.
Roughly speaking, the main theorem states that if ϕ is a smooth tame map that satisfies the assumptions of the Nash-Moser inverse function theorem and if for an appropriate auxiliary functional ι, a functional e Þ Ñ ιpϕpeq´f q satisfies the Palais-Smale condition then ϕ is a global diffeomorphism.

Mountain Pass Theorem
We denote by F a Fréchet space whose topology is defined by a sequence p ¨ n F q nPN of seminorms, which we can always assume to be increasing (by considering max kőn ¨ k F , if necessary). Moreover, the complete translation-invariant metric We recall that (cf. [11]) a family B of bounded subsets of F that covers F is called a bornology on F if ‚ @A, B P B there exists C P B such that A Y B Ă C, ‚ @B P B and @r P R there is a C P B such that r¨B Ă C. We use the Keller's definition of C k c -maps which is equivalent to the notion of C k -maps in the sense of Michal and Bastiani.
Let E, F be Fréchet spaces, U an open subset of E, and ϕ : U Ñ F a continuous map. If the directional (Gâteaux) derivatives d ϕpxqh " lim tÑ0 φpx`thq´φpxq t exist for all x P U and all h P E, and the induced map d ϕ : UˆE Ñ F, pu, hq Þ Ñ d ϕpuqh is continuous in the product topology, then we say that ϕ is a C 1 -map in the sense of Michal and Bastiani. Higher directional derivatives and C k -maps, k ŕ 2, are defined in the obvious inductive fashion.
Let E be a Fréchet space, B a bornology on E and L B pE, F q the space of all linear continuous maps from E to F . The B-topology on L B pE, F q is a Hausdorff locally convex topology defined by all seminorms obtained as follows: (2) L n B -supt Lpeq n F : e P Bu, where B P B. One similarly may define the space L k B pE, F q of k-linear jointly continuous maps from E k to F . If B is generated by all compact sets, in the sense that every B P B is contained in some compact set, then the B-topology on the space L B pE, Rq " E 1 B of all continuous linear functional on E, the dual of E, is the topology of compact convergence. If B contains all compact sets of E, then L k B pE, L l B pE, F qq is canonically isomorphic to L l`k B pE, F q as a topological vector space, see [7, Theorem 0.1.3]. In particular, Under the above condition on B, we define the differentiability of class C k c : Let U Ă E be open, a map ϕ : U Ñ F is called C 1 c if its directional derivatives exist and the induced map d ϕ : B , and the directional derivative of φ at x toward h P E is given by d ϕpxqh " xφ 1 pxq, hy, where x¨,¨y is duality pairing.
Because of the equivalency of the notions of differentiability we shall omit the index c in denoting differentiable maps of order k. We always assume that a bornology B on a Fréchet space contains all its compact sets.
(i) We say that φ satisfies the Palais-Smale condition, PS-condition in short, if each sequence px i q Ă F such that φpx i q is bounded and Let φ be a C k -functional pk ŕ 1q on a Fréchet space F . As usual, a point p in the domain of φ is said to be a critical point of if φ 1 ppq " 0, the corresponding value c " φppq will be called a critical value.
The next result is essential when we want to prove the existence of a critical point.
Theorem 2.2. Let pX, σq be a complete metric space. Let a functional Ψ : X Ñ p´8, 8s be semi-continuous, bounded from below and not identical to`8. Then, for any ǫ ą 0 there exists x P X such that Let pF, ¨ n F q be a Fréchet space and let φ P C 1 pF, Rq be a functional. Let be the set of continuous paths joining 0 and f . Consider the Fréchet space Cpr0, 1s; F q with the family of seminorms The metric is complete translation-invariant and induces the same topology on Cpr0, 1s; F q. We can easily show that Γ f is closed in Cpr0, 1s; F q and so it is a complete metric space with the metric, d Γ f , which is the restriction of d C to Γ f . The proof of the following mountain pass theorem is the refinement of the proof for the Banach spaces case (see [10,Theorem 4.10]). The idea of the proof is straightforward: for a given φ P C 1 pF, Rq that satisfies the PS-condition and a point f P F if a particular condition hold (the condition (5)), we define a functional Ψ on Γ f so that it satisfies the assumptions of the Ekeland's variational principle (Theorem 2.2). Then this theorem yields that Ψ has almost minimizers points satisfying some certain conditions. We use a sequence of these points on Γ f and associate this sequence of almost minimizers with a sequence on F , which satisfies the requirement of the PS-condition for φ. The limit of a subsequence of this sequence on F is a critical point of φ. The difficult step is to find a connection between the sequence of almost minimizers of Ψ and a sequence on F , which satisfies the requirements of the PS-condition. φpγptqq.
Proof. Let Suppose the metric d Γ f which is the restriction of the metric d C (4) to Γ f , defines the topology of Γ f . With this metric Γ f is a complete metric space.
Since Ψ is the least upper bound of a family of lower semi-continuous functions it follows that it is lower semi-continuous too. Since U separates 0 and f for all γ P Γ f , we have Therefore, by (6) and then it follows from (5) that c ŕ c 1 ą a. Thus, Ψ is bounded from below.
Let p γ P Γ f , we show that Ψ is continuous at p γ. Given ε ą 0, choose ̺ ą 0 (here we use the continuity of φ) such that @y P p γpr0, 1sq and @x P F such that d F px, yq ă ̺ we have | φpxq´φpyq |ă ε, where |¨| is the usual absolute modulus. Now for each γ P Γ f such that d Γ f pp γ, γq ă ̺, we have Ψpγq´Ψpp γq " φpγpt m qq´max tPr0,1s φpp γptqq ő φpγpt m qq´φpp γpt m qq, where t m P r0, 1s is the point where the maximum of φpγptqq is attained. Since it follows that Ψpγq´Ψpp γq ă ε. Reverting the roles of p γ and γ yields that Thus Ψ satisfies all conditions of Theorem 2.2, and hence, for every ǫ ą 0 there exists γ ǫ P Γ f such that Without loss of generality we may assume (10) 0 ă ǫ ă c´a.
Now we show that there is s P r0, 1s such that for all seminorms we have (11) φ 1 pγ ǫ psqq n B ő ǫ, We prove the inequality (11) by contradiction. Notice that φ 1 pγ ǫ psqq n B " sup g n n F "1, g n PB xφ 1 pγ ǫ psqq, g n y.
It follows from (10) that a ă c´ǫ. Since γ ǫ p0q " 0 and φp0q ő a, we obtain that 0 R Spǫq. Furthermore, the functional φ is continuous on F and the set Spǫq is closed so Spǫq is compact. Suppose for all s P r0, 1s the inequality (11) does not hold. Then, for all s P Spǫq φ 1 pγ ǫ psqq n B ą ǫ, @n P N, @B P B. Thus, for each s P Spǫq there exist points g n s P F with g n s n F " 1p@n P Nq such that (13) xφ 1 pγ ǫ psq, g n s y ă´ǫ. Since φ 1 is continuous on F , it follows from (13) that for each s P Spǫq there exist α s ą 0 and an open interval B s Ă r0, 1s such that (14) xφ 1 pγ ǫ ptq`hq, g n s y ă´ǫ, for all t P B s and all h P F with h n F ă α s p@n P Nq. The family tB s u sPSpǫq covers the compact set Spǫq so there exists a finite subcovering B s1 , . . . , B s k of Spǫq. Since 0 R Spǫq, we may assume 0 R B si . Thus, r0, 1szB si is closed and not empty for all i " 1, . . . , k. Therefore, if t P Ť k i"1 B si , then Now define the function χ j ptq : r0, 1s Ñ r0, 1s by It is easily seen that χ j is continuous and (15) k ÿ j"1 χ j ptq ő 1 @t P r0, 1s and χ j ptq " 0 if t R B j . Fix a continuous function χ : r0, 1s Ñ r0, 1s such that χptq " Let α " mintα s1 , . . . , α s k u. For an arbitrary fixed n P N, define the continuous function µ n : r0, 1s Ñ F by µ n ptq " γ ǫ ptq`αχptq k ÿ j"1 χ j ptqg n sj .
Now it suffices to consider the sequence f n " γ 1{n pt 1{n q and use the pP Sq c -condition. By (18) we have φpf n q Ñ c, and by (17) we have φ 1 pf n q Ñ 0, therefore, f n has a convergent subsequence, denoted again by f n . If the limit of f n is h, in view of Theorem 2.1 it follows that h is a critical point and φphq " c.
Remark 2.1. Note that we could have used the weaker pP Sq c -condition instead of P Scondition, with c being the c in the proof. But in application c is not known explicitly. This constrains us to verify pP Sqc-condition for all possible values c.

The existence of a global diffeomorphism
In this section we prove a global diffeomorphism theorem in the category of tame Fréchet spaces. With respect to the metric (1), d F , we define an open ball B r pxq centered at x with radius r. Its closure and boundary is denoted by B r pxq and BB r pxq, respectively. Theorem 3.1. Let E and F be tame Fréchet spaces and τ : E Ñ F a smooth tame map. Let ι : F Ñ r0, 8s be a C 1 -functional such that ιpxq " 0 if and only if x " 0 and ι 1 pxq " 0 if and only if x " 0. If the following conditions hold C1: the derivative τ 1 peqp " k has a unique solution p " νpeqk for all e P E and all k, and the family of inverses ν : EˆF Ñ E is a smooth tame map; C2: for any f P F the functional φ f defined on E by φ f peq " ιpτ peq´f q satisfies the Palais-Smale condition at all levels. Then τ is a global diffeomorphism.
Proof. The map τ satisfies the assumptions of the Nash-Moser inverse function theorem, the condition C1, which implies that τ is a local diffeomorphism. Thus, it suffices to show that τ is surjective and bijective.
To prove that τ is surjective suppose that f P F is any arbitrary point. Then, by the Palais-Smale condition the functional φ f has a critical point p P E, and the assumptions on ι yields that τ ppq " f . The functional φ f is bounded from below and is of class C 1 as it is the composition of two C 1 maps. Since φ f satisfies the Palais-Smale condition it follows by Theorem 2.1 that it attains its critical point at some p P E so φ 1 f ppq " 0. From the chain rule [7, Corollary 1.3.2] it follows that (19) φ 1 f ppq " ι 1 pτ ppq´f q˝τ 1 ppq " 0. By the assumption the map τ 1 is invertible and hence (19) implies that ι 1 pτ ppq´f q " 0 therefore τ ppq " f . Thus, τ is surjective.
To prove that τ is injective we argue by contradiction, assume e 1 ‰ e 2 P E and τ pe 1 q " τ pe 2 q " l. Then we will construct the functional φ l on E that satisfies the assumptions of Theorem 2.3 and hence it has a critical point h which its existence violates the assumptions on ι.
Since τ is a local diffeomorphism it is an open map. Therefore, there exists α r ą 0 such that (20) B αr plq Ă τ pB r pe 1 qq, for r ą 0. Let ρ ą 0 be small enough such that (21) e 2 R B ρ pe 1 q.
Thus, all assumptions of Theorem 2.3 hold for the functional φ l and the points 0 and e 2 . Therefore, there exists a critical point h P E with φ l phq " c for some c ą 0. But c " φ l phq " ιpτ phq´lq ą 0, therefore, the assumption on ι implies (22) τ phq ‰ l.