A Global Diffeomorphism Theorem for Fr\'{e}chet spaces

TL;DR

This paper establishes conditions under which a local diffeomorphism between Fréchet spaces is global, extending Clarke's gradient theory and proving a mountain pass theorem in this infinite-dimensional setting.

Contribution

It extends Clarke's generalized gradient theory to Fréchet spaces and proves a global diffeomorphism theorem using the Chang Palais-Smale condition.

Findings

Extended Clarke's theory to Fréchet spaces

Proved a mountain pass theorem for Lipschitz maps in Fréchet spaces

Established conditions for global diffeomorphisms in infinite-dimensional spaces

Abstract

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fr\'{e}chet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.

Equations

$$d_F(f,g)=\sum _{i=0}^{\infty }\frac{\parallel f-g{\parallel }_F^i}{1+\parallel f-g{\parallel }_F^i}.$$

$${\rho }_{{\varphi }^{\prime }}(x):-\underset{y\in {\varphi }^{\prime }}{\sup }\mid y(x)\mid ,$$

$${\rho }_{\varphi }(y):-\underset{x\in \varphi }{\sup }\mid y(x)\mid ,$$

$${\phi }^{\circ }(f,g):-\underset{h\to f,t\downarrow 0}{\mathrm{limsup}}\frac{\phi (h+tg)-\phi (h)}{t},t\in \mathbb{R},h\in F.$$

$$\partial\varphi(f)\coloneq\Big{\{}f^{\prime}\in F^{\prime}\mid(\forall g\in F)\langle f^{\prime},g\rangle\leqq\varphi^{\circ}(f,g)\Big{\}}.$$

$$\phi (g)-\phi (f)\in ⟨\partial \phi (g+(\theta (g-f))),g-f⟩.$$

$$\partial \tau (e)⫅\partial \psi (\phi (e))\circ D\phi (e).$$

$$\parallel L{\parallel }_{\mathcal{B}}^n:-\sup \{\parallel L(f){\parallel }_E^n:f\in B\&B\in \mathcal{B}\}.$$

$$D\phi (x)(h)=<D\phi (x),h{>}_{\mathcal{B}}.$$

$$<D\phi (x),h{>}_{\mathcal{B}}\geqq <D\phi (x),h>$$

$${\lambda }_{\phi ,\varphi }(f)=\underset{y\in \partial \phi (f)}{\min }{\rho }_{\varphi }(y).$$

$${\lambda }_{\phi ,\varphi }(f_0)\leqq {\rho }_{\varphi }(y_0)\leqq \underset{n_i\to \infty }{\mathrm{liminf}}{\rho }_{\varphi }(y_{n_i})$$

$$\underset{n\to \infty }{\lim }{\lambda }_{\phi ,\varphi }(f_n)=0,$$

$$\underset{n\to \infty }{\lim }{\lambda }_{\phi ,\varphi }(f_n)=\underset{n\to \infty }{\mathrm{liminf}}{\lambda }_{\phi ,\varphi }(f_n)\geqq {\lambda }_{\phi ,\varphi }(f_0).$$

$$\alpha :-\underset{\parallel f{\parallel }^1\to \infty }{\mathrm{liminf}}\phi (f)$$

$$m(r):-\underset{\parallel f{\parallel }^1\geqq r}{\inf }\phi (f).$$

$$\underset{r\to \infty }{\lim }m(r)=\alpha .$$

$$\alpha -{\epsilon }^2\leqq m(r).$$

$$r_2\geqq \max \{r_1,2\epsilon \}.$$

$$\phi (z_0)<\alpha +{\epsilon }^2$$

$$\phi (u)\geqq m(\parallel u{\parallel }_F^1)\geqq \alpha -{\epsilon }^2,\forall u\in F\mathrm{with}\parallel u{\parallel }_F^1\geqq r_2.$$

$$\phi (g)\leqq \phi (x)+\epsilon d_F(g,x),\forall x\in \mathbf{F}$$

$$\phi (g)\leqq \phi (z_0)-\epsilon d_F(g,z_0)$$

$$\alpha -{\epsilon }^2\leqq m(r_2)\leqq \phi (g)\leqq \phi (z_0)-\epsilon d_F(g,z_0)\leqq \alpha +{\epsilon }^2-\epsilon d_F(g,z_0)$$

$$d_F(g,z_0)\leqq 2\epsilon .$$

$$d_F(g,0)\ge d_F(z_0,0)-d_F(g,z_0)\geqq 2r_2-2\epsilon \geqq r_2.$$

$$\tilde{\phi }(h):-d_F(g,h)+\phi (h).$$

$$0\in \partial \tilde{\phi }(g)\subseteq \partial \phi (g)+\epsilon B_F^{\prime }$$

$${\lambda }_{\phi ,\varphi }(g)=\min \{{\rho }_{\varphi }(h)\mid h\in \partial \phi (h)\}\leqq \epsilon .$$

$$\max \{\phi (0),\phi (f)\}<m\leqq \underset{bd\mathcal{U}}{\inf }\phi .$$

Full text

The author would like to thank the reviewer for the valuable comments

The problem of finding sufficient conditions for a local diffeomorphism to be a global one has been investigated by many authors in the framework of Banach spaces, cf. [14] and references therein. But it has not been the subject of study for more general Fréchet spaces. In [6] we found sufficient conditions that indicate when smooth tame maps are global diffeomorphisms. The purpose of this paper is to find weakened conditions for $C^{1}_{c}$-maps. To do this, we will apply the Clarke’s theory of generalized gradients. By means of this theory the problem of global invertibility of non-differentiable maps has been studied in Banach spaces by many authors cf. [10, 5, 13], but nothing exists for Fréchet spaces.

The calculus of generalized gradients involves Lipschitz maps also on dual spaces weak*∗* topology suffices. Thus, we may expect to carry it over to the Fréchet setting without much difficulty. To this end, we start with the definition of the Clarke’s subdifferential of Lipschitz functions and present some of its basic properties. We then naturally formulate the Palais-Smale condition in the sense of Chang [5]. By means of Ekeland’s variational principle we prove that any lower bounded function that satisfies the Chang Palais-Smale condition at all levels is coercive.

As pointed out by Kartiel [13], mountain pass theorems can be used to obtain global homemorphism theorems. These theorems has many extensions and variations particularly, Shuzhong [16] generalizes this theorem to locally Lipschitz functions on Banach spaces. Following his ideas we prove the mountain pass theorem for Fréchet spaces, see Theorem (3.2). The desired advantage of this theorem is that an obtained convergent subsequent satisfies the Chang Palais-Smale condition.

Finally, we prove the main theorem which roughly states that if $\varphi$ is a $C^{1}_{c}$-locally diffeomorphism of Fréchet spaces and if for an appropriate coercive auxiliary function $\imath$, a function $x\mapsto\imath(\varphi(x)-y)$ for any $y$ satisfies the Chang Palais-Smale condition then $\varphi$ is a global diffeomorphism.

It might not always be easy even for Banach spaces to check if a map satisfies the Chang Palais-Smale condition, therefore, for the Banach case another approach which is based on the path lifiting property has been developed, see Plastock [15]. A potential line for further studies would be the generalization of this approach for Fréchet spaces as well.

Despite the fact that the theory of Fréchet spaces has a remarkable relation with both linear and non-linear problems but not many methods for solving different types of differential equations are known. Our motivation here has an eye on future applications to ordinary differential equations. It is known that each global existence theorem for an autonomous system in Banach spaces has a correspondence with a global inversion theorem. Analogously, we would expect that such theorems will play notable role in the theory of differential equations in Fréchet spaces.

1. Clarke’s subdifferential

In this section we extend some basic concepts of the generalized gradients calculus to the Fréchet setting. In most cases the proofs have elementary calculus nature and similar to their Banach analogues so we merely give references.

We denote by $F$ a Fréchet space and by $F^{\prime}$ its dual. We assume that the topology of $F$ is defined by an increasing sequence of seminorms $\parallel\cdot\parallel^{1}_{F}\leqq\parallel\cdot\parallel^{2}_{F}\leqq\cdots$. A translation-invariant complete metric inducing the same topology on $F$ can then be defined by

[TABLE]

A ball with center $x$ and radius $r$ in $F$ and $F^{\prime}$ is denoted by $B_{r}(x)$ and $B^{\prime}_{r}(x)$, respectively. The boundary of a set $U$ is denoted by $bd\,U$. We will use the Keller’s notion of $C^{k}_{c}$-maps, see [7, Definition 2.2].

The weak topology $\sigma(F,F^{\prime})$ on $F$ is given by the fundamental system of seminorms

[TABLE]

where $\phi^{\prime}$ runs through the set $\Phi^{\prime}$ of finite subsets of the dual space $F^{\prime}$. The weak∗ topology $\sigma(F^{\prime},F)$ on $F^{\prime}$ is given by the fundamental system of seminorms

[TABLE]

where $\phi$ runs through the set $\Phi$ of finite subsets of $F$. Let $\langle.,.\rangle$ be the dual pairing between $F$ and $F^{\prime}$.

Let $\operatorname{Lip_{loc}}(F,\mathbb{R})$ be the set of all locally Lipschitz functions on $F$ and $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$. As in [2] we define for each $f\in F$ the generalized directional derivative, denoted by $\varphi^{\circ}(f,g)$, in the direction $g\in F$ by

[TABLE]

It can be easily seen that the function $f\rightarrow\varphi^{\circ}(f,g)$ is locally Lipschitz, positively homogeneous and sub-additive. For any $f\in F$ we define the Clarke’s subdifferential of $\varphi$, denoted by $\partial\varphi$, as follows:

[TABLE]

Lemma 1.1**.**

• (a)

$\varphi^{\circ}(f;g)$ is upper semi-continuous as a function of $(f,g)$ and, as a function of $g$ alone, is Lipschitz on $F$.

• (b)

$\varphi^{\circ}(f,-g)=(-\varphi^{\circ})(f,g)$.

• (c)

For every $g\in F$, $\varphi^{\circ}(f;g)=\max\{\langle h,g\rangle:h\in\partial\varphi(f)\}$.

• (d)

$g\in\partial\varphi(f)$ if and only if $\varphi^{\circ}(f;h)\geq\langle g,h\rangle,\,\forall h\in F$.

• (e)

Suppose sequences $(f_{j})\subset F$ and $(g_{j})\subset F^{\prime}$ are such that $g_{j}\in\partial\varphi(f_{j})$. If $f_{j}\rightarrow f$ and $g$ is a cluster point of $(g_{j})$, then $g\in\partial(f)$.

• (f)

$\partial(t\varphi)(f)=t\partial\varphi(f),\,\forall t\in\mathbb{R}$.

• (g)

If $f$ is a local minimum of $\varphi$, then $0\in\partial\varphi(f)$.

• (h)

$\partial(\varphi+\psi)(f)\subset\partial\varphi(f)+\partial\psi(f)$.

Proof.

The proofs of $(a)-(h)$ are easy and similar to the Banach case cf. [3, Prop. 2.1.1(b)];[3, Prop. 2.1.1(c)]; [3, Prop. 2.1.2(b)]; [3, Prop. 2.1.5(a)];[3, Prop. 2.1.5(b)]; [3, Prop. 2.3.1]; [3, Prop. 2.3.2]; [3, Prop. 2.3.3], respectively. ∎

Lemma 1.2**.**

The subdifferential $\partial\varphi(f)$ is a nonempty, convex and weak∗ compact subset of $F^{\prime}$.

Proof.

The Hahan-Banach theorem and Bourbaki-Alaoglu theorem are available for Fréchet spaces, therefore, it is enough to apply the arguments of Clarke [3, Prop. 2.1.2(a)]. ∎

Lemma 1.3** (Mean Value Theorem).**

Let $f,g\in F$ and $\varphi$ be a Lipschitz function on an open set containing the line segment $[f,g]$. Then there exists $\theta\in(0,1)$ such that

[TABLE]

Proof.

The proof is very similar to that of [3, Theorem 2.3.7]. ∎

Lemma 1.4** (Chain rule).**

Let $E$ be a Fréchet space, $\varphi:E\rightarrow F$ a $C^{1}_{c}$-map in a neighborhood of $e\in E$, and $\psi:F\rightarrow\mathbb{R}$ a locally Lipschitz map. Then $\tau=\psi\circ\varphi$ is locally Lipschitz and

[TABLE]

If $\psi$ is regular at $\varphi(e)$ then the equality holds.

Proof.

The proof is also quite analogous to the Banach case [3, Theorem 2.3.10]. ∎

We recall that a family $\mathcal{B}$ of bounded subsets of $F$ that covers $F$ is called a bornology on $F$ if it is directed upwards by inclusion and if for every $B\in\mathcal{B}$ and $r\in\mathbb{R}$ there is a $C\in\mathcal{B}$ such that $r\cdot B\subset C$.

Let $E$ be a Fréchet space, $\mathcal{B}$ a bornology on $F$ and $L_{\mathcal{B}}(F,E)$ the space of all linear continuous maps from $F$ to $E$. The $\mathcal{B}$-topology on $L_{\mathcal{B}}(F,E)$ is a Hausdorff locally convex topology defined by all seminorms obtained as follows:

[TABLE]

Suppose that $\mathcal{B}$ consists of all compact sets, then the $\mathcal{B}$-topology on the space $L_{\mathcal{B}}(F,\mathbb{R})=F_{\mathcal{B}}^{\prime}$ of all continuous linear functional on $F$, the dual of $F$, is the topology of compact convergence. Let $U\subset F$ be open and $\varphi:F\rightarrow E$ a Keller $C^{1}_{c}$-map at $x\in U$. The derivative of $\varphi$ at $x$, $\operatorname{D}\varphi(x)$, is an element of $F_{\mathcal{B}}^{\prime}$. We denote by $<.,.>_{\mathcal{B}}$ the duality pairing between $F$ and $F^{\prime}_{\mathcal{B}}$

Lemma 1.5**.**

Let $\varphi:U\subset F\rightarrow\mathbb{R}$ be Lipschitz in open neighbourhood $U$ of $x$. If $\varphi$ is a $C^{1}_{c}$-map at $x$, then $\operatorname{D}\varphi(x)\in\partial\varphi(x)$.

Proof.

By definition for all $h\in F$ we have

[TABLE]

By our definition of differentiability we get $\operatorname{D}\varphi(x)(h)\leqq\varphi^{\circ}(x,h)$, but

[TABLE]

therefore $<\operatorname{D}\varphi(x),h>\leqq\varphi^{\circ}(x,h)$ thereby by Lemma 1.1(d) we obtain $\operatorname{D}\varphi(x)\in\partial\varphi(x)$. ∎

2. Chang Palais-Smale condition

A point $f\in F$ is called a critical point of $\varphi$ if $0\in\partial\varphi(f)$, that is $\varphi^{\circ}(f;g)\geqq 0,\forall g\in F$. The value of a critical point is called a critical value.

We define for each $\phi\in\Phi$ the function $\lambda_{\varphi,\phi}$ on $F$ as follows

[TABLE]

The seminorms $\rho_{\phi}(\cdot)$ are bounded below and weak∗ lower semi-continuous because they arise as the pointwise supremum of the continuous absolute value function. Also, $\partial\varphi(x)$ is weak∗-compact therefore the minimum is obtained.

Lemma 2.1**.**

The set-valued mapping $f\mapsto\partial\varphi{(f)}$ is locally bounded and weak*∗* upper semi-continuous.

Proof.

The proof is a slight modification of the Banach case, cf. [11, Theorem 1.1.2] ∎

Lemma 2.2**.**

For each $\phi\in\Phi$ the function $\lambda_{\varphi,\phi}(f)$ is sequentially lower semi-continuous

Proof.

If $\lambda_{\varphi,\phi}(f)$ is not sequentially lower-continuous there exist a sequence $f_{n}\rightarrow f_{0}$ such that $\lim_{n\rightarrow\infty}\lambda_{\varphi,\phi}(f_{n})<\lambda_{\varphi,\phi}(f_{0})$. Let a sequence $y_{n}\in\partial\varphi(f_{n})$ be such that $\rho_{\phi}(y_{n})=\lambda_{\varphi,\phi}(f_{n})$. By Lemma 2.1 there exist a weak*∗* open set $U$ in $F^{\prime}$ such that $\partial\varphi(f_{0})\subseteq U$ and a neighborhood $V$ of $f_{0}$ on which the mapping is bounded such that there exists a subsequence $(f_{n_{i}})$ of $(f_{n})$ in $V$ so $y_{n_{i}}\in\partial\varphi(f_{n_{i}})$ and $y_{n_{i}}\in U$. Since $\{y_{n_{i}}\}$ is bounded it has a weak cluster $y_{0}$ and hence by Lemma 1.1(e) we have $y_{0}\in\partial\varphi(f_{0})$ but

[TABLE]

which is a contradiction. ∎

Definition 2.1** (Chang PS-condition).**

Let $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$. We say that $\varphi$ satisfies the Palais-Smale condition in the Chang’s sense, Chang PS condition for short, if any sequence $(f_{n})$ in $F$ such that $\varphi(f_{n})$ is bounded and for all $\phi\in\Phi$

[TABLE]

possesses a convergent subsequence. Also, if any sequence $(f_{n})\subset F$ such that $\varphi(f_{n})\rightarrow c\in\mathbb{R}$ and satisfies (2.2) possesses a convergent subsequence we say that $\varphi$ satisfies the Chang PS condition at level $c$.

Suppose that $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ satisfies the Chang PS-condition. Let $(f_{n})$ be any sequence in $F$ that converges to $f_{0}$ and satisfies (2.2). Since by Lemma 2.2 the functions $\lambda_{\varphi,\phi}(f_{n})$ are sequentially lower semi-continuous it follows that $\forall\phi\in\Phi$

[TABLE]

Whence $\lambda_{\varphi,\phi}(f_{0})=0$, that is the $zero$ function in $F^{\prime}$ belongs to $\partial\varphi(f_{0})$, hence $f_{0}$ is a critical point.

Now we prove that a functional $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ that satisfies the Chang PS condition at all levels is coercive. The idea of proof is inspired by the work of Brezis and Nirenberg [1].

A functional $\varphi:F\rightarrow\mathbb{R}$ is said to be coercive if $\varphi(f)\rightarrow+\infty$ as $\parallel f\parallel_{F}^{1}\rightarrow\infty$.

We will need the following version of Ekeland’s variational principle.

Theorem 2.1**.**

[9] Let $(X,\sigma)$ be a complete metric space. Let a functional $f:X\rightarrow(-\infty,\infty]$ be semi-continuous, bounded from below and not identical to $+\infty$. Then, for any $\epsilon>0$ and every point $x_{0}\in X$ there exists $u\in X$ such that

1. (1)

$f(u)\leqq f(x_{0})-\varepsilon\sigma(u,x_{0})$ 2. (2)

$f(u)\leqq f(x)+{\epsilon}\sigma(x,u),\quad\forall x\in X$.

Theorem 2.2**.**

Let $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ and let

[TABLE]

be finite. Then there exists a sequence $(f_{n})\subset F$ such that $\parallel f_{n}\parallel^{i}\rightarrow\infty(\forall i\in\mathbb{N})$, $\varphi(f_{n})\rightarrow\alpha,$ and $\lambda_{\varphi,\phi}(f_{n})\rightarrow 0$ for all $\phi\in\Phi$.

Proof.

Define

[TABLE]

The function $m(r)$ is a non-decreasing and

[TABLE]

By (2.4) for each $\varepsilon>0$ there exists $r_{1}$ such that for all $r\geqq r_{1}$

[TABLE]

For a fixed $\varepsilon>0$ choose a number

[TABLE]

By our assumption we can fix some $z_{0}$ with $\parallel z_{0}\parallel^{1}\geqq 2r_{2}$ such that

[TABLE]

Let $\mathbf{F}=\{f\in F:\parallel f\parallel^{1}\geqq r_{2}\}$. It is closed in $F$, so it is a complete metric space by the induced metric (1.1). Moreover, $\varphi$ is lower semi-continuous on $F$ and so on $\mathbf{F}$. Also, by (2.3),(2.5) and (2.6)

[TABLE]

So $\varphi$ is lower bounded, and therefore, all assumptions of Theorem 2.1 are fulfilled for $\mathbf{F}$. Thus, there is $g\in\mathbf{F}$ such that

[TABLE]

[TABLE]

It follows that (2.3),(2.5), (2.6),(2.9) and (2.7)

[TABLE]

Hence

[TABLE]

Thereby, by (2.6)

[TABLE]

Whence $g$ is an interior point of $\mathbf{F}$. Define on $\mathbf{F}$ the function

[TABLE]

The function $\widetilde{\varphi}(h)$ attains its minimum in $g\in\mathrm{Int}\,\mathbf{F}$ by virtue of (2.8). Therefore

[TABLE]

where $B^{\prime}_{F}$ is the closed unit ball in $F^{\prime}$. Thus,

[TABLE]

Letting $\varepsilon=\varepsilon_{n}\downarrow 0$ completes the proof. ∎

Corollary 2.1**.**

If $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ is bounded below and satisfies the Chang PS condition at $c$ for all $c\in\mathbb{R}$, then it is coercive.

Proof.

If it is not coercive then $\alpha=\liminf_{\parallel f\parallel^{1}\rightarrow\infty}\varphi(f)$ is finite. Then by Theorem 2.2 there exists a sequence $(f_{n})\subset F$ such that $\parallel f_{n}\parallel^{i}\rightarrow\infty(\forall i\in\mathbb{N}),\,\varphi(f_{n})\rightarrow\alpha$ and $\lambda_{\varphi,\phi}(f_{n})\rightarrow 0$ for all $\phi\in\Phi$. Then the Chang PS-condition at $\alpha$ yields that $(f_{n})$ has a convergent subsequent which is a contradiction. ∎

3. The mountain pass theorem

Following the lines of the mountain pass theorem for Banach spaces due to Shuzhong [16] we prove a version of the mountain pass theorem for locally Lipschitz functions between Fréchet spaces. This is the most suitable version for our goals as it involves the Chang PS condition.

Let $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ be a function. Let $\mathcal{U}$ be an open neighborhood of $zero$ and $f\notin\overline{\mathcal{U}}$ be given such that for a real number $m$

[TABLE]

Let

[TABLE]

be the space of continuous paths joining [math] and $f$. Consider the Fréchet space $C([0,1];F)$ with the family of seminorms

[TABLE]

Let

[TABLE]

be the metric that defines the same topology. We can easily varify that $\Gamma$ is a closed subset of $C([0,1];F)$ so it is a complete metric space with the induced metric $d_{\Gamma}$.

In the sequel we will apply the following weak form of Ekeland’s variational principle.

Theorem 3.1**.**

[8, Theorem 1 bis.] Let $(X,\sigma)$ be a complete metric space. Let a functional $f:X\rightarrow(-\infty,\infty]$ be semi-continuous, bounded from below and not identical to $+\infty$. Then, for any $\epsilon>0$ there exists $x\in X$ such that

1. (1)

$f(x)<\inf_{X}f+\epsilon$ 2. (2)

$f(x)\leqq f(y)+{\epsilon}\sigma(x,y),\quad\forall y\neq x\in X$.

The idea of the proof of the following mountain pass theorem is to define a function $\Psi(\gamma)=\max_{[0,1]}\varphi(\gamma(t))$ on $C([0,1];F)$ and show that it is locally Lipschitz. Then we find almost minimizers with some certain conditions by using Ekeland’s variational principle. We pick a sequence of these points on $\Gamma$ and associate it with a sequence on $F$ which satisfies the requirement of the Chang PS-condition for $\varphi$. The limit of a subsequence of this sequence on $F$ is a critical point of $\varphi$.

Theorem 3.2**.**

Suppose $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ satisfies (3.1) for a real number $m$. Let

[TABLE]

Then there exists a sequence $(f_{n})\subset F$ such that $\varphi(f_{n})\rightarrow c$ and satisfies (2.2). Moreover, if $\varphi$ satisfies the Chang PS condition then $c$ is a critical value of $\varphi$.

Proof.

Define the function $\Psi:C([0,1];F)\rightarrow\mathbb{R}$ by

[TABLE]

Let $\gamma\in C([0,1],F)$, for any $t\in[0,1]$ there are positive numbers $r_{t},c_{t}$ such that

[TABLE]

The family $\{B_{r_{t}}(\gamma(t))\}_{t\in[0,1]}$ is an open covering of the compact set $\gamma([0,1])$, therefore, there is a finite sub-covering $\{B_{r_{t_{j}}}(\gamma(t_{j}))\}_{j=1,\cdots,k}$ of $\gamma([0,1])$. Hence by the Lebesgue’s number lemma there exists a positive number $r$ such that for any $f\in\gamma([0,1])$ there exists some $1\leqslant j\leqslant k$ such that $B_{r}(f)\subset B_{r_{t_{j}}}(\gamma(t_{j}))$.

Set $c_{\gamma}\coloneq\max_{1\leqslant j\leqslant k}c_{t_{j}}$. Therefore

[TABLE]

If $\gamma_{1},\gamma_{2}\in C([0,1],F)$ satisfy

[TABLE]

Then

[TABLE]

[TABLE]

[TABLE]

Therefore $\Psi$ is locally Lipschitz.

Let $\varepsilon_{j}$ be a sequence of positive numbers converging to $zero$ and $(\eta_{j})\subset C([0,1];F)$ a sequence such that $\parallel\eta_{j}-\gamma\parallel_{\Gamma}^{i^{\prime}}(\forall i^{\prime}\in\mathbb{N})\rightarrow 0$ as $j\rightarrow\infty$ and for $\eta\in C([0,1];F)$

[TABLE]

Set

[TABLE]

For any $s_{j}\in M(\eta_{j}+\varepsilon_{j}\eta),j=1,2,\cdots$ it follows that

[TABLE]

By the mean value theorem, there exist $\epsilon_{j}\in(0,1)$ and $x^{*}_{j}\in\partial\varphi(\eta_{j}(s_{j})+\epsilon_{j}\varepsilon_{j}\eta(s_{j}))$ such that

[TABLE]

The sequence $(s_{j})$ has a convergent sequence, denoted again by $(s_{j})$, suppose that $s_{j}\rightarrow s$. Then $\eta_{j}(s_{j})+\epsilon_{j}\varepsilon_{j}\eta(s_{j})\rightarrow\gamma(s)$. By Lemma 1.1(e) the sequence $(x^{*}_{j})$ has a $w^{*}$-cluster point $x^{*}\in\partial\varphi(\gamma(s))$. So we have $\langle x_{j}^{*},\eta(s)\rangle\rightarrow\langle x^{*},\eta(s)\rangle$ and then

[TABLE]

Since $s_{j}\in M(\eta_{j}+\varepsilon_{j}\eta)$, we have

[TABLE]

Letting $t\rightarrow\infty$ yields

[TABLE]

and hence $s\in M(\gamma)$, therefore

[TABLE]

Set

[TABLE]

Suppose for some $\gamma\in C([0,1],F)$ we have $M(\gamma)\subset(0,1)$ and there exists $\varepsilon>0$ such that for $\eta\in C_{0}([0,1],F)$

[TABLE]

We prove that there exists $s\in M(\gamma)$ such that $\forall h\in F$

[TABLE]

If there there is no such $s$ then for any $t\in M(\gamma)$ there exits $h_{t}\in F$ with $\parallel h_{t}\parallel^{i}_{F}=1(i\in\mathbb{N})$ such that $\varphi^{\circ}(\gamma(t);h_{t})<-\varepsilon$. The continuity of $\gamma$ and the upper semi-continuity of $\varphi^{\circ}$ implies that for any $t\in M(\gamma)$ there exits $h_{t}\in F$ with $\parallel h_{t}\parallel^{i}_{F}=1(\forall i\in\mathbb{N})$ and $\varepsilon_{t}^{\prime}>0$ such that

[TABLE]

The family $\{B_{\varepsilon^{\prime}_{t}}(t)\}_{t\in M(\gamma)}$ is an open covering of $M(\gamma)$. Since $M(\gamma)$ is compact there exist $t_{1},\cdots,t_{k}\in M(\gamma)$ such that $M(\gamma)\subset\bigcup_{j=1}^{j=k}B_{\varepsilon_{t_{j}}^{\prime}}(t_{j})$. Since $M(\gamma)$ is a subset of $(0,1)$ it follows that $B_{\varepsilon_{t}^{\prime}}(t)$ does not contain $\{0,1\}$ for all $t\in M(\gamma)$. Thereby

[TABLE]

Define

[TABLE]

Where

[TABLE]

and

[TABLE]

By (3.14) it follows that $\Sigma_{j=0}^{j=k}d_{j}(t)>0$.

By the above arguments we obtain $\eta_{0}\in C_{0}([0,1];F)$ with $\parallel\eta_{0}\parallel_{\Gamma}^{i^{\prime}}\leqq 1$. Since $g\mapsto\varphi^{\circ}(f,g)$ is sublinear in $g$,

[TABLE]

Then for any $t\in M(\gamma)$ we get

[TABLE]

Therefore by Equation (3.10) we have

[TABLE]

which is a contradiction.

Since $\mathcal{U}$ separates 0 and $f$, for any $\gamma\in\Gamma$,

[TABLE]

Then by the assumptions of the theorem

[TABLE]

Therefore for every $\gamma\in\Gamma$,

[TABLE]

The restriction of $\Psi$ to $\Gamma$ is again locally Lipschitz and by (3.21) and (3.3) is bounded from below. Let $(\alpha_{n})$ be a sequence of positive numbers converging to $zero$. By Ekeland’s variational principle 3.1 there exists a sequence $(\gamma_{n})\subset\Gamma$ such that

[TABLE]

and

[TABLE]

Therefore for any $\eta\in C_{0}([0,1];F)$ we obtain

[TABLE]

Then by Equation (3.12), there exists $s_{n}\in M(\gamma_{n})$ such that $\varphi(\gamma_{n}(s_{n}))=\Psi(\gamma_{n})$ and

[TABLE]

Let $f_{n}=\gamma_{n}(s_{n})$ for $n=1,2,\cdots$, then $(f_{n})$ is the desired sequence and we have $\varphi(f_{n})\rightarrow c$, moreover

[TABLE]

where $U$ is a neighborhood of the origin in $F$. By the Chang PS condition $(f_{n})$ has a convergent subsequent, denoted again by $(f_{n})$, with the limit $z$. Then,

[TABLE]

and $\forall w\in F\&\,i\in\mathbb{N}$

[TABLE]

That is $0\in\partial\varphi(z)$. ∎

4. A global diffeomorphism theorem

In this section we apply the mountain pass theorem of the previous section to obtain a global diffeomorphism theorem.

Lemma 4.1**.**

Let $\varphi\in\operatorname{Lip_{loc}}(F,\mathbb{R})$ and bounded from bellow. Then there exists a sequence $(f_{n})$ such that $\lim_{n\rightarrow\infty}\varphi(f_{n})=\inf_{F}\varphi$ and for all $\phi\in\Phi$

[TABLE]

Proof.

Consider a sequence of positive nuumbers $(\epsilon_{n})$ converging to $zero$. The function $\varphi$ satisfies all assumptions of Ekeland variational principle 3.1, so we can find a sequence $(f_{n})$ such that

[TABLE]

and

[TABLE]

Assume $f=f_{n}+t(g-f_{n})$ for some $g\in F$ and a positive number $t$, then we obtain

[TABLE]

Thus, for all $g\in F$ if we let $t\rightarrow 0$ then

[TABLE]

For a fixed $f_{n}$ define the sets

[TABLE]

and

[TABLE]

They are open convex sets with empty intersection so by Hahn-Banach separation theorem there exists a separating hyperplane determined by a functional $\upsilon_{n}(h,t)=w_{n}(h)+\alpha t$ for some $\alpha\neq 0$, where $w_{n}$ is a linear functional on $F$ such that $w_{n}(0)=0$. Let $w^{*}(h)=\dfrac{-1}{\alpha}w_{n}(h)$, then $\upsilon_{n}(h,w^{*}(h))=0,\,\forall h\in F$. Thereby $w^{*}(h)\leqq\varphi^{\circ}(f_{n};h)$ hence by ((d))(g) we have $w_{n}^{*}\in\partial\varphi(f_{n})$. On the other hand $\mid w^{*}_{n}(h)\mid\leqq\epsilon_{n}\parallel h\parallel^{i}(\forall i\in\mathbb{N})$ so for all $\phi$ we have $\lambda_{\varphi,\phi}(f_{n})\leqq\rho_{\phi}(w^{*})\leqq\epsilon_{n}$. Letting $\varepsilon_{n}\downarrow 0$ completes the proof. ∎

Theorem 4.1**.**

Let $\imath:F\rightarrow[0,\infty)$ be a coercive locally Lipschitz function having the following two properties:

• •

$\imath(x)=0$ if and only if $x=0$,

• •

$0\in\partial\imath(y)$ if and only if $y=0$.

Further let $\tau:E\rightarrow F$ be a local $C^{1}_{c}$-diffeomorphism. Suppose that for each $f\in F$ the function $\jmath:F\rightarrow[0,\infty)$ given by

[TABLE]

satisfies the Chang Palais-Smale condition. Then $\tau$ is a global diffeomorphism.

Proof.

We need to show that $\tau$ is surjective and bijective. Let $e_{1}\neq e_{2}\in E$ if $\tau(e_{1})\neq\tau(e_{2})$ we have nothing to prove. Assume $\tau(e_{1})=\tau(e_{2})=l$. Since $\tau$ is a local diffeomorphism it follows that it is an open map, therefore, there exist $\sigma,\alpha>0$ such that

[TABLE]

Let $\mathbf{r}\in(0,\sigma)$ be the smallest number such that $e_{2}\notin\overline{B_{\mathbf{r}}(e_{1})}$. Consider the function $\jmath(e)=\imath(\tau(e)-l)$, therefore $\jmath(e_{1})=\jmath(e_{2})=0$.

Without the loose of generality we can suppose $e_{1}=0$. By (4.5) for $e\in bd\,B_{\mathbf{r}}(0)$ we have $0<m\leqq\jmath(e)$. Thus, all conditions of Theorem 3.2 hold so there exists $(e_{n})\subset E$ such that $\lim_{n\rightarrow\infty}\jmath(e_{n})=c$ for some $c\geqq m$ characterized by (3.3). Since $\jmath(e_{n})$ satisfies the Chang PS condition it has a convergent subsequent, denoted again by $(e_{n})$, with the limit $h$. Therefore, $h$ is a critical point so $0\in\partial\jmath(h)$ and $\tau(h)\neq l$ since $\lim_{n\rightarrow\infty}\jmath(e_{n})=\jmath(h)=c\geqq m>0$.

By the chain rule (1.4) we have $\partial\jmath(h)\subset\partial\imath(\tau(h)-l)\circ\operatorname{D}\tau(h)$. Therefore, there exists $v\in\partial\imath(\tau(h)-l)$ such that $0=v\circ\operatorname{D}\tau(h)$. Since $\tau$ is a local diffeomorphism it follows that $v=0$. Therefore, by our assumption on $\imath,\,\tau(h)-l$ must be $zero$ which is a contradiction.

Let $g\in F$ be given and consider the function $\jmath(e)=\imath(\tau(e)-g)$. By Lemma 4.1 there exists a sequence $(f_{n})$ such that $\lim_{n\rightarrow\infty}\jmath(f_{n})=\inf_{E}\jmath$ and for all $\phi\in\Phi$ we have

[TABLE]

Since $\jmath$ satisfies the Chang PS condition the sequence $(f_{n})$ has a convergent subsequent, denoted again by $(f_{n})$, with the limit $p$ which is a critical point of $\jmath$ so $0\in\partial\jmath(p)$. By the chain rule (1.4) we have $\partial\jmath(p)\subset\partial\imath(\tau(p)-g)\circ\operatorname{D}\tau(p)$. Thus, there exists $\xi\in\partial\imath(\tau(p)-g)$ such that $0=\xi\circ\operatorname{D}\tau(p)$. Since $\operatorname{D}\tau$ is invertible at $p$ we have $\xi=0$. Therefore by our assumption on $\imath,\,\tau(p)=g$. ∎

Remark 4.1**.**

In [12] the analogue of the theorem for Banach spaces is obtained, where the applied auxiliary function is $\dfrac{1}{2}\parallel\cdot\parallel^{2}$ and it satisfies the weighted Chang PS condition. The results of [12] may also work with the type auxiliary function that we use. Nevertheless, we may attempt to extend Theorem (4.1) for auxiliary functions that satisfy the weighted Chang PS condition which of course requires an appropriate mountain pass theorem.