On Inverse and Implicit Function Theorem for Sobolev Mappings

Abstract

We extend Clarke’s local inversion theorem for Sobolev mappings. We use this result to find a general implicit function theorem for continuous locally Lipschitz mapping in the first variable and satisfying just a topological condition in the second variable. An application to control systems is given.

1. Introduction

The local inversion theorem and the implicit function theorem are the basic theorems of classical mathematical analysis (see [1] for more information). Also, Sobolev spaces are important instruments in analyses, PDEs, and ordinary differential equations, and in our paper, we find a link between these two fields.

The classical local inversion theorem affirms that if E is a Banach space, $a\in U$, $U\subset E$ is open, $f\in C^1(U,E)$ and $f^{\prime }(a):E\to E$ is bijective, then there exists $U_a\in \mathcal{V}(a)$, $V_a\in \mathcal{V}(f(a))$ such that $f|U_a:U_a\to V_a$ is a homeomorphism. Here, if E is a Hausdorff space and $x\in E$, we set $\mathcal{V}(x)=\{U\subset E$ to open in $|x\in U\}$. If $U_a=U$, we say that f is a global homeomorphism.

In [2,3,4,5], it is shown that if $n\ge 2$, $D\subset {\mathbb{R}}^n$ is open and $f:D\to {\mathbb{R}}^n$ is Fréchet differentiable and $J_f(x)\ne 0$ for every $x\in D$, then it results that f is a local homeomorphism. We can have some “singular” sets $K\subset D$ such that f is only continuous on K and differentiable with $J_f(x)\ne 0$ on $D\setminus K$, and if $n\ge 3$, we also find that f is a local homeomorphism (K may be countable dense; see [3]). We see, in this way, that we can have a local inversion theorem without assuming the continuity of the derivative.

Let $D\subset {\mathbb{R}}^n$ be open, and $m\le n$ and $f:D\to {\mathbb{R}}^m$ be locally Lipschitz. Denote with $E_f=\{x\in D|f$ a function not differentiable at $x\}$, and with ${\mu }_n$ the Lebesgue measure in ${\mathbb{R}}^n$. Rademacher’s theorem affirms that ${\mu }_n(E_f)=0$, and we define the Clarke Jacobian of f at a point $x_0\in D$ using $\partial f(x_0)=co\{A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^m)|$. There exist $x_j\in D\setminus E_f$, $x_j\to x_0$ such that $f^{\prime }(x_j)\to A\}$. Here, $coM$ is the convex null of a subset M from ${\mathbb{R}}^n$.

The known Clarke’s local inversion theorem says that if $D\subset {\mathbb{R}}^n$ is open, $f:D\to {\mathbb{R}}^n$ is locally Lipschitz, $x_0\in D$, and $detA\ne 0$ for every $A\in \partial f(x_0)$, then there exist $U\in \mathcal{V}(x_0)$ and $V\in \mathcal{V}(f(x_0))$ such that $f|U:U\to V$ is a homeomorphism (see [6,7,8]). We see from Lemma 2 that if $D\subset {\mathbb{R}}^n$ is open, $x_0\in D$, and $f:D\to {\mathbb{R}}^n$ is locally Lispchitz, then

$$detA\ne 0\mathrm{for}\mathrm{every}A\in \partial f(x_0)$$

(1)

if and only if

$$\mathrm{there}\mathrm{exist}r,\delta >0\mathrm{such}\mathrm{that}\underset{|x|=1}{inf}|A(x)|>\delta \mathrm{for}\mathrm{every}A\in co\{f^{\prime }(D\setminus E_f)\cap B(x_0,r)\}.$$

(2)

Also, if $n\ge m$, $D\subset {\mathbb{R}}^n$ is open, $f:D\to {\mathbb{R}}^m$ is locally Lipschitz, and $\partial f(x)$ contains only surjective mappings for every $x\in D$, then f is surjective (see [9,10]).

A special field of complex analysis is dedicated to global univalence, and a large volume of papers are dedicated to this subject. We remind the reader of just the univalence on the border theorem, which affirms that if $D\subset \mathbb{C}$ is a Jordan domain, $\overline{D}\subset \Omega$, and $f:\Omega \to \mathbb{C}$ is analytic and injective on $\partial D$, then it results that f is injective on D. Some extensions of this theorem were given in [11,12] in ${\mathbb{R}}^n$, and in $n\ge 2$ for Sobolev mappings, and the paper of Ball [11] was a seminal one in the theory of Nonlinear Elasticity.

A known theorem of global univalence is the Banach–Mazur–Browder theorem, which says that if $E,F$ are Hausdorff spaces that are pathwise connected, F is simply connected, and $f:E\to F$ is a local homeomorphism which is proper or closed, then it results that $f:E\to F$ is a global homeomorphism (see [13]).

Another known theorem of global univalence is the Hadamard–Levy–John theorem, which says that if E is a Banach space, $\omega :(0,\infty )\to (0,\infty )$ is continuous and $\underset{0}{\overset{\infty }{\int }}{\displaystyle \frac{dt}{\omega (t)}}=\infty$, and $f:E\to E$ is a local homeomorphism such that $D^{-}f(x)\ge \omega (\parallel x\parallel )$ for every $x\in E$, then it results that $f:E\to E$ is a homeomorphism. Here,

$$D^{-}f(x)=\underset{y\to x}{lim\; inf}{\displaystyle \frac{\parallel f(y)-f(x)\parallel }{\parallel y-x\parallel }}.$$

The theorem was proven first on ${\mathbb{R}}^n$ in [14,15] and on Banach spaces in [16]. Some other connected results may be found in [17,18,19,20,21,22,23,24,25,26,27,28].

The classical implicit function theorem says that if $E,F$ are Banach spaces, $U\subset E$ and $V\subset F$ are open sets, $f:U\times V\to F$ is continuous, $(a,b)\in U\times V$, $f(a,b)=0$, ${\displaystyle \frac{\partial f}{\partial y}}:U\to \mathcal{L}(E,F)$ exists and is continuous, and ${\displaystyle \frac{\partial h}{\partial y}}(a,b)$ is bijective, then there exists $U_a\in \mathcal{V}(a)$, such that $U_a\subset U$, and an unique, continuous implicit function $\phi :U_a\to F$ such that $\phi (a)=b$ and $f(x,\phi (x))=0$ for every $x\in U_a$.

If $U_a=U$, we say that $\phi$ is a global implicit function. It is known that the classical local inversion theorem can be used to prove the classical implicit function theorem, and hence, if we work on ${\mathbb{R}}^n$, we can easily prove implicit functions theorems without assuming the continuity of ${\displaystyle \frac{\partial f}{\partial y}}(·)$ on U (see [4]). Other results concerning extensions of the implicit function theorem can be found in [29,30,31,32,33,34,35,36,37,38,39,40], and most of them are established for locally Lipschitz mappings.

Global implicit functions are studied in [41,42,43,44,45,46,47,48,49,50,51,52,53,54]. In [24,25,26,47,53], the authors apply methods of ordinary differential equations to obtain the results of global implicit functions. For instance, in [47], we show the following:

Theorem 1. 

Let $n\ge 2$, let $D\subset {\mathbb{R}}^n$ be a starlike domain, let $f:D\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ be Fréchet differentiable such that $det{\displaystyle \frac{\partial f}{\partial y}}(x,y)\ne 0$ for every $(x,y)\in D\times {\mathbb{R}}^m$, and let $\omega :(0,\infty )\to (0,\infty )$ be continuous such that $\underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{dt}{\omega (t)}}=\infty$.

Suppose that $|{\displaystyle \frac{\partial f}{\partial x}}(x,y)|/l({\displaystyle \frac{\partial f}{\partial y}}(x,y))\le \omega (|y|)$ for every $(x,y)\in D\times {\mathbb{R}}^n$. Then, there exists differentiable $\phi :D\to {\mathbb{R}}^n$ such that $\phi (a)=b$ and $f(x,\phi (x))=f(a,b)$ for every $(a,b)\in D\times {\mathbb{R}}^n$.

Here, if $A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^m)$, we set $|A|=\underset{|x|=1}{sup}|A(x)|$, and if $A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$, we set $l(A)=\underset{|x|=1}{inf}|A(x)|$. We set $|x|={({\displaystyle \sum _{i=1}^n}{x}_i^2)}^{{\displaystyle \frac{1}{2}}}$ if $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$. We set $B(x,r)=\{z\in {\mathbb{R}}^n||z-x|<r\}$ and $S^n=\{z\in {\mathbb{R}}^n||z|=1\}$.

The previous extensions of the local inversion theorem or of the implicit functions theorem work with a.e. differentiable mappings (i.e., with mappings $f:D\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ such that ${\mu }_n(E_f)=0$, where $E_f=\{x\in D|f$ is not Fréchet differentiable at $x\}$). It is a very difficult task to apply these methods for Sobolev mappings where $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$, $D\subset {\mathbb{R}}^n$ is open, and $n\ge 2$. Indeed, there exist homeomorphisms $f\in W_{loc}^{1,n-1}(U,V)$ between open sets in ${\mathbb{R}}^n$ such that f and $f^{-1}$ are nowhere Fréchet differentiable (see [55]), and hence, we must work with weak derivatives. The Sobolev spaces play a basic role in PDE theory and also in function theory, namely in the theory of the well-known class of quasiregular mappings. Zhuravlev [56] proved an implicit function theorem using some basic facts from the theory of quasiregular mappings. He denoted for a matrix that $A\in {\mathcal{M}}_{m,n}$ with $|A|=\underset{|x|=1}{sup}|A(x)|$ and with $\parallel A\parallel ={(trA\circ A^t)}^{{\displaystyle \frac{1}{2}}}$, where $A^t$ is the transpose of A. He proved the following:

Theorem 2. 

Let $D\subset {\mathbb{R}}^{n+m}$ be open and $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^m)$ be continuous; let $D^{\ast }\subset D$ be such that ${\mu }_n(D\setminus D^{\ast })=0$ and f has first partial derivatives on $D^{\ast }$; and let $f^{\prime }(x,y)$ be the matrix of the first partial derivatives of f at a point $(x,y)\in D^{\ast }$. Suppose that $|F^{\prime }(x,y)|>0$ a.e., and if $a\in D$, we set $N(f,a)=co(\{B\in {\mathcal{M}}_{m,n+m}|$. There exists $z_p\to a,z_p\in D^{\ast }$ such that ${\displaystyle \frac{f^{\prime }(z_p)}{|f^{\prime }(z_p)|}}\to B\})$, and we set $N_{y}f(a)$ as the family of all matrices in ${\mathcal{M}}_{m,m}$ formed by the last m columns of some matrix $B\in N(f,a)$.

Let $a=(x_0,y_0)\in D$, and suppose that $\parallel F^{\prime }(x,y)\parallel >C>0$ a.e. and that $detA\ne 0$ for every $A\in N_{y}f(a)$. Then, there exists $\rho >0$ and an unique continuous mapping $\phi :B(x_0,\rho )\to {\mathbb{R}}^m$ such that $f(x,\phi (x))=f(x_0,y_0)$ for every $x\in B(x_0,\rho )$ and $\phi (x_0)=y_0$.

We shall prove the local and global inversion theorems and an implicit function theorem for Sobolev mappings, and our results may be used in PDE theory and control systems and in the theory of quasiregular mappings.

2. Notations and Definitions

Let $p\ge 1$ and $D\subset {\mathbb{R}}^n$ be open and let $U\in L_{loc}^1(D)$. We say that $v\in L_{loc}^1(D)$ is the ith weak partial derivative of u if

$$\underset{D}{\int }\phi (x)v(x)d{\mu }_n=-\underset{D}{\int }u(x){\displaystyle \frac{\partial \phi }{\partial x_i}}(x)d{\mu }_{n}forevery\phi \in C_0^{\infty }(D).$$

If v exists, it is also denoted by $D_{i}u$. The Sobolev space $W_{loc}^{1,p}(D,{\mathbb{R}}^n)$ consists of all functions $f:D\to {\mathbb{R}}^n$ which are locally in $L^p$ together with their first-order weak partial derivatives.

If $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$, then f has a.e. first partial derivatives (see page 6 in [57]), and $S_f=\{x\in D|f$ has no classical first partial derivatives at $x\}$.

From now on, we set $f^{\prime }(x)={({\displaystyle \frac{\partial f_i}{\partial x_j}}(x))}_{i,j=1,\dots ,n}$ if $x\in D\setminus S_f$ and if $x_0\in D$ and $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$, and we set

$\partial f(x_0)=co\{A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)|$. There exist $x_j\in D\setminus S_f$ and $x_j\to x_0$ such that $f^{\prime }(x_j)\to A\}$.

Let $D\subset {\mathbb{R}}^n$ be open, let Y be a topological space, and let $x_0\in D$, $y_0\in Y$. Let $f:D\times Y\to {\mathbb{R}}^n$ be a mapping and let $f_y:D\to {\mathbb{R}}^n$ be given by $f_y(x)=f(x,y)$ for every $x\in D$ and every $y\in Y$. We define

$${\partial }_{x}f(x_0,y_0)=\partial (f_{y_0})(x_0)$$

and we suppose that ${\mu }_n(S_{f_{y_0}})=0$. Let $y\in Y$ be such that ${\mu }_n(S_{f_y})=0$ and $z\in D\setminus S_{f_y}$. We set

$${\displaystyle \frac{\partial f_y}{\partial x}}(z)={\left({\displaystyle \frac{\partial {(f_y)}_i}{\partial x_j}}(z)\right)}_{i,j=1,\dots ,n}.$$

If $y\in Y$ is such that ${\mu }_n(S_{f_y})=0$ and $E\subset D\setminus S_{f_y}$, we set

$$co\{{\displaystyle \frac{\partial f_y}{\partial x}}(E)\}=\{A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)|thereexist{\lambda }_k\ge 0,z_k\in E,A_k\in {\displaystyle \frac{\partial f_y}{\partial x}}(z_k),k=1,\dots ,n$$

such that ${\displaystyle \sum _{k=1}^m}{\lambda }_k=1$ and ${\displaystyle \sum _{k=1}^m}{\lambda }_k{A}_k=A\}$.

Let $X,Y$ be Hausdorff spaces. We say that $f:X\to Y$ is a multivalued function if for every $x\in X$, $f(x)$ is a nonempty subset of Y. For $B\subset Y$, we state that $f^{+}(B)=\{x\in X|f(x)\subset B\}$ and we say that the multivalued function $f:X\to Y$ is upper semicontinuous if $f^{+}(B)$ is open in X for every open set $B\subset Y$.

Let $X,Y$ be Hausdorff spaces and let $f:X\to Y$ be a mapping. We say that f is proper if $f^{-1}(K)$ is compact in X for every compact set $K\subset Y$. We say that f is closed if $f(F)$ is closed in Y for every closed set $F\subset X$.

Let $W\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$ be convex and let $v\in S^n$. We set $W(v)=\{w\in {\mathbb{R}}^n|$. There exists $A\in W|$ such that $w=A(v)\}$, and we see that $W(v)$ is convex.

3. Preliminaries

Lemma 1. 

Let $D\subset {\mathbb{R}}^n$ be open, let Y be a topological space with a countable base, and let $x_0\in D$, $y_0\in Y$. Let $f:D\times Y\to {\mathbb{R}}^n$ be locally Lipschitz in the first variable such that $detA\ne 0$ for every $A\in {\partial }_{x}f(x_0,y_0)$. Suppose that the set-valued mapping $(a,b)\to {\partial }_{x}f(a,b)$ is upper semicontinuous at $(x_0,y_0)$, and let $f_y:D\to {\mathbb{R}}^n$ be given by $f_y(z)=f(z,y)$ for every $z\in D$ and every $y\in Y$. Then

$$\begin{array}{c}\mathit{There}\mathit{exist}r>0,U_0\in \mathcal{V}(y_0)\mathit{and}\delta >0\mathit{such}\mathit{that}l(A)>\delta \\ \mathit{for}\mathit{every}A\in co\{{\displaystyle \frac{\partial f_y}{\partial x}}((D\setminus S_{f_y})\cap B(x_0,r))\}\mathit{and}\mathit{every}y\in U_0.\end{array}$$

(3)

Proof of Lemma 1. 

Let us show that there exists $\delta >0$ such that $l(A)>\delta$ for every $A\in {\partial }_{x}f(x_0,y_0)$. If not, we find $A_p\in {\partial }_{x}f(x_0,y_0)$, $v_p\in S^n$, and $p\in \mathbb{N}$ such that $v_p\to v\in S^n$ and $A_p(v_p)\to 0$. Since ${\partial }_{x}f(x_0,y_0)$ is compact, we find $A\in {\partial }_{x}f(x_0,y_0)$ and ${(A_{p_k})}_{k\in \mathbb{N}}$ to be a subsequence of ${(A_p)}_{p\in \mathbb{N}}$ such that $A_{p_k}\to A$. Then, $A_{p_k}(v_{p_k})\to A(v)$, and hence, $A(v)=0$, and we reach a contradiction, since $detA\ne 0$. □

Suppose that we find $r_p\to 0$, $v_p\in S^n$, and ${(U_p)}_{p\in \mathbb{N}}$ to be a fundamental system of neighborhoods of $y_0$, $x_p\in (D\setminus S_{f_{y_p}})\cap B(x_0,r_p)$, $y_p\in U_p$, and $A_p\in co\{{\displaystyle \frac{\partial f_{y_p}}{\partial x}}(x_p)\}$ for every $p\in \mathbb{N}$ such that $A_p(v_p)\le \delta$ for every $p\in \mathbb{N}$, and we can suppose that $v_p\to v\in S^n$.

Let $ϵ>0$. Since the set-valued mapping $(a,b)\to {\partial }_{x}f(a,b)$ is upper semicontinuous at $(x_0,y_0)$, there exists $p_{ϵ}\in \mathbb{N}$ such that $A_p\in \overline{B}({\partial }_{x}f(x_0,y_0),ϵ)$ for every $p\ge p_{ϵ}$, and we can suppose that there exists $A\in {\partial }_{x}f(x_0,y_0)$ such that $A_p\to A$. Then, $A_p(v_p)\to A(v)$, and let $q_{ϵ}\ge p_{ϵ}$ be such that $|A(v)-A_p(v_p)|<ϵ$ for every $p\ge q_{ϵ}$. Then, $|A(v)|\le |A(v)-A_p(v_p)|+|A_p(v_p)|\le ϵ+|A_p(v_p)|$ for every $p\ge q_{ϵ}$.

Letting $ϵ\to 0$, we find that $|A(v)|\le \delta$, and we reach a contradiction, since $l(A)>\delta$ for every $A\in {\partial }_{x}f(x_0,y_0)$.

We also prove the following:

Lemma 2. 

Let $D\subset {\mathbb{R}}^n$ be open, and let $x_0\in D$ and $f:D\to {\mathbb{R}}^n$ be locally Lipschitz such that every $A\in \partial f(x)$ is surjective. Then, there exist $r>0$ and $\delta >0$ such that $l(A)>\delta$ for every $A\in co\{f^{\prime }(D\setminus E_f)\cap B(x_0,r)\}$.

Lemma 3. 

Let $n\ge 2$ and $D\subset {\mathbb{R}}^n$ be open, let $v=(v_1,\dots ,v_n)\in S^n$, and let $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ be continuous. Then, there exists a set $E_v\subset D$ such that ${\mu }_n(E_v)=0$ and ${\displaystyle \frac{\partial f}{\partial v}}(x)$ exist for every $x\in D\setminus E_v$, and ${\displaystyle \frac{\partial f}{\partial v}}(x)={\displaystyle \sum _{j=1}^n}{\displaystyle \frac{\partial f}{\partial x_j}}(x)v_j$ for every $x\in D\setminus E_v$. Let Q be a cube with a side of length l parallel to v such that $\overline{Q}\subset D$, and let S be a face of Q perpendicular to v. Then, it results that $f\circ {\gamma }_y$ is absolutely continuous for ${\mu }_{n-1}$-almost $y\in S$, where ${\gamma }_y:[0,l]\to Q$ is given by ${\gamma }_y(t)=y+tv$ for every $t\in [0,l]$ and every $y\in S$.

Proof. 

Let $A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$ be such that $A(e_k)=l{w}_k$, $k=1,\dots ,n$, $<w_k,w_m>={\delta }_{km}$ if $k,m=1,\dots ,n$, $w_n=v$, $A({[0,l]}^n)=Q$, let $V_0={[0,l]}^{n-1}$, and let $S=A(V_0)$. Here, $e_1,\dots ,e_n$ is the standard base in ${\mathbb{R}}^n$. We see from Proposition 1.11 in [57] that

$${\partial }_i(f\circ A)(x)=\sum _{j=1}^n{\partial }_{j}f(A(x))a_{ij}$$

(4)

for $i=1,\dots ,n$, where $A={(a_{ij})}_{i,j=1,\dots ,n}$ and the derivatives are weak derivatives, and hence, $f\circ A\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$.

Let $C=[a,c]\subset [0,l]$ and $V=B^{n-1}(y,\rho )\subset V_0$, where $y\in V_0$ and $B^{n-1}(y,\rho )$ is the ball in ${\mathbb{R}}^{n-1}$. We consider test functions $\phi \in C^1(V\times C)$ of the form $\phi (x)=g(y)h(t)$, where supp $g\subset V$ and supp $h\subset C$. We have

$$\underset{V\times C}{\int }{\partial }_n(f\circ A)(x)h(t)g(y)dx=-\underset{V\times C}{\int }(f\circ A)(x)h^{\prime }(t)g(y)dx.$$

Letting g run through a sequence $g_k$ such that $0\le g_k\le 1$ and $g_k\to 1$, we find that

$$\underset{V\times C}{\int }{\partial }_n(f\circ A)(x)h(t)dx=-\underset{V\times C}{\int }(f\circ A)(x)h^{\prime }(t)dx.$$

Using Fubimi’s theorem, this gives

$${\displaystyle \frac{1}{|V|}}\underset{V}{\int }(\underset{a}{\overset{c}{\int }}{\partial }_n(f\circ A)(y,t)h(t)dt)dy={\displaystyle \frac{-1}{|V|}}\underset{V}{\int }(\underset{a}{\overset{c}{\int }}(f\circ A)(y,t)h^{\prime }(t)dt)dy.$$

Letting $\rho \to 0$ and using Lebesgue’s theorem, we find that for ${\mu }_{n-1}$-almost $y\in V_0$, we have

$$\underset{a}{\overset{c}{\int }}{\partial }_n(f\circ A)(y,t)h(t)dt=-\underset{a}{\overset{c}{\int }}f(A(y)+tv)h^{\prime }(t)dt.$$

Let us fix $y\in V_0$ and let $h_k:[a,c]\to {\mathbb{R}}_{+}$ be such that $h_k=1$ on $[a+{\displaystyle \frac{1}{k}},c-{\displaystyle \frac{1}{k}}]$, $|h_k(t)|\le 1$ and $|h_k^{\prime }(t)|\le 2k$ for every $t\in [a,c]$ and supp $h_k\subset [a,c]$ for every $k\in \mathbb{N}$. We see that the following are true:

$$\underset{c-{\displaystyle \frac{1}{k}}}{\overset{c}{\int }}(f(A(y)+cv)-f(A(y)+tv)h_k^{\prime }(t)dt\to 0\mathrm{if}k\to \infty$$

$$\underset{a}{\overset{a+{\displaystyle \frac{1}{k}}}{\int }}(f(A(y)+tv)-f(A(y)+av))h_k^{\prime }(t)dt\to 0\mathrm{if}k\to \infty .$$

Then,

$$-\underset{a}{\overset{c}{\int }}f(A(y)+tv)h_k^{\prime }(t)dt=ϵ(k)-((f(A(y)+cv)(h_k(c)-$$

$$-h_k(c-{\displaystyle \frac{1}{k}}))+f(A(y)+av)(h_k(a+{\displaystyle \frac{1}{k}})-h_k(a))))$$

and $ϵ(k)\to 0$ if $k\to \infty$.

Letting $k\to \infty$, we find from (4) that

$$\underset{a}{\overset{c}{\int }}\sum _{j=1}^n{\partial }_{j}f(A(y)+tv)v_{j}dt=\underset{a}{\overset{c}{\int }}{\partial }_n(f\circ A)(y,t)dt=f(A(y)+cv)-f(A(y)+av).$$

Now, letting $c\to a$, we find that there exists a Gateaux derivative of f on direction v at the point $A(y)+av$ and

$${\displaystyle \frac{\partial f}{\partial v}}(A(y)+av)=\sum _{j=1}^n{\displaystyle \frac{\partial f}{\partial x_j}}(A(y)+av)v_j$$

for ${\mu }_{n-1}$-almost $y\in V_0$ and ${\mu }_1$-almost $a\in [0,l]$.

We also see that $f\circ {\gamma }_y$ is absolutely continuous for ${\mu }_{n-1}$-almost $y\in S=A(V_0)$. It results that there exists ${\displaystyle \frac{\partial f}{\partial v}}(x)$ for ${\mu }_n$-almost $x\in Q$ and ${\displaystyle \frac{\partial f}{\partial v}}(x)=\sum _{j=1}^n{\displaystyle \frac{\partial f}{\partial x_j}}(x)v_j$ for ${\mu }_n$-almost $x\in Q$. □

4. The Main Results

In the following theorem, we extend Clarke’s local inversion theorem in the class of Sobolev mappings.

Theorem 3. 

Let $D\subset {\mathbb{R}}^n$ be open, let $x\in D$, and let $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ be continuous, and suppose that

$$thereexistr,\delta >0suchthatl(A)>\delta foreveryA\in co\{f^{\prime }(D\setminus S_f)\cap B(x,r)\}.$$

(5)

Then, there exists $V\in \mathcal{V}(f(x))$ such that $f|B(x,r):B(x,r)\to V$ is a homeomorphism and its inverse is ${\displaystyle \frac{1}{\delta }}$ Lipschitzian, and $D^{-}f(z)\ge \delta$ for every $z\in B(x,r)$ and $B(f(x),\delta r)\subset f(B(x,r)).$

Here, if $E\subset D\setminus S_f$, we set $f^{\prime }(E)=\{A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)|$. There exists $x\in E$ such that $A=f^{\prime }(x)\}$, and we set $D^{-}f(z)=\underset{y\to z}{lim\; inf}{\displaystyle \frac{|f(y)-f(z)|}{|y-z|}}$ if $z\in D$.

Proof of Theorem 3. 

Let $y,z\in B(x,r)$, $v_0=z-y$ and $v={\displaystyle \frac{v_0}{|v_0|}}$. Let H be a hyperplane perpendicular to v and $S=H\cap B(x,r)$. Using Lemma 3, we see that there exists a set $E_v\subset D$ such that ${\mu }_n(E_v)=0$, and there exist ${\displaystyle \frac{\partial f_i}{\partial x_j}}(w)$, $i,j=1,\dots ,n$, and ${\displaystyle \frac{\partial f}{\partial v}}(w)$ for every $w\in D\setminus E_v$ and ${\displaystyle \frac{\partial f}{\partial v}}(w)={\displaystyle \sum _{j=1}^n}{\displaystyle \frac{\partial f}{\partial x_j}}(w)v_j$ if $w\in D\setminus E_v$. We also see from Lemma 3 that $f\circ {\gamma }_w$ is absolutely continuous for ${\mu }_{n-1}$-almost $w\in S$, where ${\gamma }_w:[0,1]\to {\mathbb{R}}^n$ is given by ${\gamma }_w(t)=w+tv$ for every $t\in [0,1]$.

Let $ϵ>0$. We can find a direction d parallel to v and points $y_0,z_0\in d\setminus E_v$ such that $|z_0-y_0|\ge |v_0|$, f is absolutely continuous on $[y_0,z_0]$, ${\mu }_1([y_0,z_0]\setminus E_v)=0$,

$$|f(y)-f(y_0)|<{\displaystyle \frac{ϵ}{4}}|z-y|,|f(z)-f(z_0)|<{\displaystyle \frac{ϵ}{4}}|z-y|,$$

and there exists ${\displaystyle \frac{\partial f}{\partial v}}(w)={\displaystyle \sum _{j=1}^n}{\displaystyle \frac{\partial f}{\partial x_j}}(w)v_j$ for ${\mu }_1$-almost $w\in [y_0,z_0]$. We also see that ${\displaystyle \frac{\partial f}{\partial v}}(w)=f^{\prime }(w)(v)$ for ${\mu }_1$-almost $w\in [y_0,z_0]$.

Using the absolute continuity of the mapping f on $[y_0,z_0]$, we find $0<{\delta }_{ϵ}<ϵ$ such that if $[a_0,b_0],\dots ,[a_m,b_m]$ are disjoint intervals in $[0,|z_0-y_0|]$ and ${\displaystyle \sum _{k=1}^m}{\mu }_1([a_k,b_k])<{\delta }_{ϵ}|z_0-y_0|$, it results that

$$\sum _{k=1}^m|f(y_0+b_{k}v)-f(y_0+a_{k}v)|<{\displaystyle \frac{ϵ}{4}}|z-y|.$$

Let $W=co\{f^{\prime }(D\setminus S_f)\cap B(x,r)\}$ and $Q=W(v)$. Then, Q is convex. Let either $E=\{w\in [y_0,z_0]|$ or ${\displaystyle \frac{\partial f}{\partial v}}(w)$ not exist, or some partial first derivatives ${\displaystyle \frac{\partial f_i}{\partial x_j}}(w)$ not exist, or ${\displaystyle \frac{\partial f}{\partial v}}(w)\ne {\displaystyle \sum _{j=1}^m}{\displaystyle \frac{\partial f}{\partial v}}(w)v_j\}$, and we can suppose that ${\mu }_1([y_0,z_0]\setminus E)=0$.

Let $a\in [y_0,z_0]\setminus E$, $a=y_0+t_{a}v$. Then, there exists ${ϵ}_a>0$ such that $I_a=(t_a-{ϵ}_a,t_a+{ϵ}_a)\subset [0,|y_0-z_0|]$ and

$${\displaystyle \frac{f(y_0+tv)-f(y_0+t_{a}v)}{t-t_a}}\in B(Q,{\displaystyle \frac{ϵ}{4}})$$

for every $t\in I_a$.

Suppose that $I_a\cap I_b\ne \varphi$ and let $t\in I_a\cap I_b\cap (t_a,t_b)$. We can find $Z_k\in B(Q,{\displaystyle \frac{ϵ}{4}}$) $k=1,2,3,4$ such that

$${\displaystyle \frac{f(y_0+(t_b+{ϵ}_b)v)-f(y_0+(t_a-{ϵ}_a)v)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}={\displaystyle \frac{f(y_0+(t_b+{ϵ}_b)v)-f(y_0+t_{b}v)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}+$$

$$+{\displaystyle \frac{f(y_0+t_{b}v)-f(y_0+tv)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}+{\displaystyle \frac{f(y_0+tv)-f(y_0+t_{a}v)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}+{\displaystyle \frac{f(y_0+t_{a}v)-f(y_0+(t_a-{ϵ}_a)v)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}=$$

$$+{\displaystyle \frac{{ϵ}_b{Z}_1+(t_b-t)Z_2+(t-t_a)Z_3+{ϵ}_a{Z}_4}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}.$$

and hence,

$${\displaystyle \frac{f(y_0+(t_b+{ϵ}_b)v)-f(y_0+(t_a-{ϵ}_a)v)}{t_b+{ϵ}_b-(t_a-{ϵ}_a)}}\in B(Q,{\displaystyle \frac{ϵ}{4}}).$$

(6)

Now, let intervals $I_q=(t_q-{ϵ}_q,t_q+{ϵ}_q)\subset [0,|z_0-y_0|]$ and points $w_q=y_0+t_{q}v\in [y_0,z_0]\setminus E$, $q=1,\dots ,m$, such that

$${\displaystyle \frac{f(w_q+tv)-f(w_q)}{t-t_q}}\in B(Q,{\displaystyle \frac{ϵ}{4}})ift\in I_q,q=1,\dots ,m$$

(7)

and

$${\mu }_1(\bigcup _{q=1}^m{I}_q)>(1-{\delta }_{ϵ})|z_0-y_0|.$$

(8)

Let ${\Delta }_k=(a_k,b_k)$, $k=0,1,\dots ,p$ be the components of ${\displaystyle \bigcup _{q=1}^m}{I}_q$. Using (6), we find that $Z_k\in B(Q,{\displaystyle \frac{ϵ}{4}})$ such that

$$f(y_0+b_{k}v)-f(y_0+a_{k}v)=Z_k(b_k-a_k),k=0,1,\dots ,p.$$

(9)

Since $(a_0+{\displaystyle \sum _{k=1}^p}(a_k-b_{k-1})+|z_0-(y_0+b_{p}v)|<{\delta }_{ϵ}|z_0-y_0|$, we find that

$$|f(y_0)-f(y_0+a_{0}v)|+|f(z_0)-f(y_0+b_{p}v)|+\sum _{k=1}^p|f(y_0+a_{k}v)-f(y_0+b_{k-1}v)|<{\displaystyle \frac{ϵ}{4}}|z-y|.$$

(10)

Since ${\displaystyle \sum _{k=0}^p}(b_k-a_k)\ge |z-y|(1-{\delta }_{ϵ})>|z-y|(1-ϵ)$, we can find $0<\mu <{\displaystyle \frac{1}{1-ϵ}}$ such that ${\displaystyle \sum _{k=0}^p}{\displaystyle \frac{\mu (b_k-a_k)}{|z-y|}}=1$. Let $w\in Q$ and $A\in W$ be such that $A(v)=w$ and $|{\displaystyle \sum _{k=0}^p}{\displaystyle \frac{\mu (b_k-a_k)}{|z-y|}}{Z}_k-w|<{\displaystyle \frac{ϵ}{4}}$. We have

$|z-y|(\delta -{\displaystyle \frac{ϵ}{4}})\le |z-y|(l(A)-{\displaystyle \frac{ϵ}{4}})\le (|w|-|w-{\displaystyle \sum _{k=0}^p}{\displaystyle \frac{\mu (b_k-a_k)}{|z-y|}}{Z}_k|\le |{\displaystyle \sum _{k=0}^p}\mu (b_k-a_k)Z_k|$.

Then,

$$|f(z_0)-f(y_0)|=|\sum _{k=0}^p(f(y_0+b_{k}v)-f(y_0+a_{k}v))+\sum _{k=1}^p(f(y_0+a_{k}v)-f(y_0+b_{k-1}v))|+$$

$$+f(z_0)-f(y_0+b_{p}v)+f(y_0+a_{0}v)-f(y_0)|\ge$$

$$\ge {\displaystyle \frac{1}{\mu }}|\sum _{k=0}^p\mu (b_k-a_k)Z_k|-|\sum _{k=1}^p(f(y_0+a_{k}v)-f(y_0+b_{k-1}v)$$

$$+f(z_0)-f(y_0+b_{p}v)+f(y_0+a_{0}v)-f(y_0)|\ge$$

$$\ge {\displaystyle \frac{|z-y|}{\mu }}(\delta -{\displaystyle \frac{ϵ}{4}})-{\displaystyle \frac{ϵ}{4}}|z-y|)\ge |z-y|((1-ϵ)(\delta -{\displaystyle \frac{ϵ}{4}})-{\displaystyle \frac{ϵ}{4}}).$$

We have

$$|f(z)-f(y)|\ge |f(z_0)-f(y_0)|-|f(z)-f(z_0)|-|f(y)-f(y_0)|\ge |z-y|((1-ϵ)(\delta -{\displaystyle \frac{ϵ}{4}})-{\displaystyle \frac{3ϵ}{4}}).$$

Letting $ϵ\to 0$, we find that

$$|f(z)-f(y)|\ge \delta |z-y|$$

(11)

for every $y,z\in B(x,r)$.

This shows that $D^{-}f(z)\ge \delta$ for every $z\in B(x,r)$ and that $B(f(x),\delta r)\subset f(B(x,r))$, and also that f is injective on $B(x,r)$. Using Brouwer’s theorem, we see that there exists $V\in \mathcal{V}(f(x))$ such that $f|B(x,r):B(x,r)\to V$ is a homeomorphism and its inverse is ${\displaystyle \frac{1}{\delta }}$ Lipschitzian. □

The following example shows that our extension of Clarke’s local inversion theorem is effective.

Example 1. 

Let $\rho :\mathbb{R}\to \mathbb{R}$, $\rho (t)=1$ if $t\in (-\infty ,0]\cup [1,\infty )$, $\rho (t)=t^{-{\displaystyle \frac{1}{2}}}$ if $t\in [{\displaystyle \frac{1}{2n-1}},{\displaystyle \frac{1}{2n}})$, and $\rho (t)=1+a_n$ if $t\in [{\displaystyle \frac{1}{2n}},{\displaystyle \frac{1}{2n+1}})$ for $n\in \mathbb{N}$, where $Q\cap [0,1]={(a_n)}_{n\in \mathbb{N}}$.

Let $g:\mathbb{R}\to \mathbb{R}$, $g(x)=\underset{0}{\overset{x}{\int }}\rho (t)dt$ if $x\in \mathbb{R}$. Then, g is absolutely continuous, and let $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$, $f(x,y)=(g(x),y)$ if $x,y\in \mathbb{R}$. Then,

$$f^{\prime }(x,y)=\left(\begin{array}{cc}\rho (x)& 0\\ 0& 1\end{array}\right)$$

a.e. in ${\mathbb{R}}^2$.

Since $\underset{-1}{\overset{1}{\int }}\rho (t)dt<\infty$, we see that $f\in W_{loc}^{1,1}({\mathbb{R}}^2,{\mathbb{R}}^2)$.

Let $W=\{A\in \mathcal{L}({\mathbb{R}}^2,{\mathbb{R}}^2)|A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, $a\in [1,2]$, $b=c=0$, $d=1\}$∪$\{\infty \}$. Then, $\partial f((0,0))=W$ and $l(A)\ge 1$ for every $A\in W$, and we see that $f^{\prime }$ is not bounded near $(0,0)$, and hence, f is not locally Lipschitz. We can apply our Theorem 1 to see that f is injective, and we cannot apply Clarke’s local inversion theorem to prove that f is a local homeomorphism around $(0,0)$.

In the next theorem, we prove a Hadamard–Levy–John-type theorem for Sobolev mappings, extending a result of Pourciau [10] given for locally Lipschitz mappings.

Theorem 4. 

Let $f\in W_{loc}^{1,1}({\mathbb{R}}^n,{\mathbb{R}}^n)$ be continuous such that for every $x\in {\mathbb{R}}^n$, there exist $r_x>0$ and $\omega :(0,\infty )\to (0,\infty )$, which are continuous such that $\underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{dt}{\omega (t)}}=\infty$ and such that

$$l(A)\ge \omega (|x|)foreveryA\in co\{f^{\prime }({\mathbb{R}}^n\setminus S_f)\cap B(x,r_x)\}.$$

(12)

Then, $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a homeomorphism whose inverse is locally Lipschitz.

Proof of Theorem 4. 

We see from Theorem 3 that $D^{-}f(x)\ge \omega (|x|)$ for every $x\in {\mathbb{R}}^n$ and that f is a local homeomorphism, and since $\underset{1}{\overset{\infty }{\int }}{\displaystyle \frac{dt}{\omega (t)}}=\infty$, we use John’s result from [17] to see that $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a homeomorphism. □

The next theorem is a Banach–Mazur–Browder-type theorem for Sobolev mappings:

Theorem 5. 

Let $E,F\subset {\mathbb{R}}^n$ be domains such that F is simply connected, and let $f:E\to F$ be proper or closed, and suppose that for every $x\in E$, there exists $r_x>0$ and ${\delta }_x>0$ such that $l(A)>{\delta }_x$ for every $A\in co\{f^{\prime }(E\setminus S_f)\cap B(x,r_x)\}$. Then, $f:E\to F$ is a global homeomorphism.

Proof. 

We see from Theorem 3 that f is a local homeomorphism, and we apply the Banach–Mazur–Browder theorem.

Recently, a very general implicit function theorem was proved in [39] in terms of Clarke’s Jacobian in the class of locally Lipschitzian functions. □

Theorem 6 

([39]). Let $D\subset {\mathbb{R}}^n$ be open, let Y be a topological space, and let $x_0\in D$, $y_0\in Y$. Let $f:D\times Y\to {\mathbb{R}}^n$ be continuous such that $f(x_0,y_0)=0$, the mapping $f_y:D\to {\mathbb{R}}^n$ be Lipschitz continuous on D for every $y\in Y$, and the set-valued mapping $(x,y)\to {\partial }_{x}f(x,y)$ be upper semicontinuous at $(x_0,y_0)$. Here, $f_y(z)=f(z,y)$ for every $z\in D$ and every $y\in Y$.

Suppose that

$$\mathrm{There}\mathrm{exists}\delta >0\mathrm{such}\mathrm{that}l(A)\ge \delta \mathrm{for}\mathrm{every}A\in {\partial }_{x}f(x_0,y_0).$$

(13)

Then, there exist $U\in \mathcal{V}(y_0)$, $r>0$ and a continuous implicit function $g:U\to D$ such that the following are true:

$$f(g(y),y)=0\mathrm{for}\mathrm{every}y\in U.$$

(14)

$$|g(y)-x_0|\le {\displaystyle \frac{|f(x_0,y)|}{\delta }}\mathrm{for}\mathrm{every}y\in U\mathrm{and}g(y_0)=x_0.$$

(15)

$$\{g(y)\}=\{x\in B(x_0,r)|f(x,y)=0\}\mathrm{for}\mathrm{every}y\in U.$$

(16)

$$|g(y_1)-g(y_2)|\le {\displaystyle \frac{|f(g(y_1),y_2)|}{\delta }}\mathrm{for}\mathrm{every}{y}_1,y_2\in U.$$

(17)

We now extend Theorem 6 in the class of Sobolev mappings in the first variable, using condition (3) instead of condition (13). Condition (3) may be valid for Sobolev mappings in the first variable, for which ${\partial }_{x}f(x_0,y_0)$ may not be compact.

Theorem 7. 

Let $D\subset {\mathbb{R}}^n$ be open, let Y be a topological space, let $x_0\in D$, let $y_0\in Y$, let $f:D\times Y\to {\mathbb{R}}^n$ be continuous such that $f(x_0,y_0)=0$, and let $f_y:D\to {\mathbb{R}}^n$ be given by $f_y(z)=f(z,y)$ for every $z\in D$ and every $y\in Y$. Suppose that $f_y\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ for every $y\in Y$ and suppose that condition (3) holds. Then, there exist $U\in \mathcal{V}(y_0)$, $U\subset U_0$, $r>0$ and a continuous implicit function $g:U\to B(x_0,r)$ such that conditions (14), (15), (16), and (17) hold.

Proof of Theorem 7. 

We see from Theorem 1 that $f_y|B(x_0,r):B(x_0,r)\to V_y$ is a homeomorphism for every $y\in U_0$, where $V_y\in \mathcal{V}(f(x_0,y))$ is such that $B(f(x_0,y),\delta r)\subset V_y$, and let $g_y:V_y\to B(x_0,r)$ be its inverse. We have the following:

• $f(g_y(z),y)=f_y(g_y(z))=z$ for every $z\in B(f(x_0,y),\delta r)$ and every $y\in U_0$;
• $g_y(f_y(x_0))=x_0$ for every $y\in U_0$;
• $|g_y(z)-x_0|=|g_y(z)-g_y(f(x_0,y))|\le {\displaystyle \frac{|z-f(x_0,y)|}{\delta }}$ for every $z\in B(f(x_0,y),\delta r)$ and every $y\in U_0$.

Let $U\in \mathcal{V}(y_0)$ be such that $U\subset U_0$ and $|f(x_0,y)|<\delta r$ for every $y\in U$. Then, $0\in B(f(x_0,y),\delta r)$ for every $y\in U$, and we define $g:U\to B(x_0,r)$ as $g(y)=g_y(0)$ for every $y\in U$.

We have the following:

• $f(g(y),y)=f(g_y(0),y)=0$ for every $y\in U$ and
• $|g(y)-x_0|=|g_y(0)-x_0|\le {\displaystyle \frac{|f(x_0,y)|}{\delta }}$ for every $y\in U$ and, hence, $g(y_0)=x_0$.

We see from Theorem 1 that

• $|f_y(x_1)-f_y(x_2)|\ge \delta |x_1-x_2|$ for every $x_1,x_2\in B(x_0,r)$ and every $y\in U_0$.

Then,

• $|g(y_1)-g(y_2)|\le {\displaystyle \frac{|f(g(y_1),y_2)-f(g(y_2),y_2)|}{\delta }}={\displaystyle \frac{|f(g(y_1),y_2)|}{\delta }}$ for every $y_1,y_2\in U$, and using the continuity of f, we see that the implicit function $g:U\to B(x_0,r)$ is also continuous.

Since $f(g(y),y)=0$ for every $y\in U$, we see that $\{g(y)\}\subset \{x\in B(x_0,r)|f(x,y)=0\}$ for every $y\in U$. Now, let $y\in U$ and $x\in B(x_0,r)$ be such that $f(x,y)=0$. Then, $f_y(x)=0$ and $f_y(g(y))=0$, and since $g(y)\in B(x_0,r)$ and $f_y$ is injective on $B(x_0,r)$, we have $x=g(y)$, and it results that $\{x\in B(x_0,r)|f(x,y)=0\}\subset \{g(y)\}$ for every $y\in U$.

Let us consider the following application of the implicit function theorem from Theorem 4. □

Let $D\subset {\mathbb{R}}^n$ be open, let $F:D\times {\mathbb{R}}^m\to {\mathbb{R}}^m$ be continuous, let $u_0\in D$, $x_0\in {\mathbb{R}}^m$, and let $f:D\times {\mathbb{R}}^m\to {\mathbb{R}}^n$ be continuous such that $f(u_0,x_0)=0$. Let $f_x:D\to {\mathbb{R}}^n$ be given by $f_x(u)=f(u,x)$ for every $u\in D$ and every $x\in {\mathbb{R}}^m$. Suppose that $f_x\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ for every $x\in {\mathbb{R}}^m$ and that there exist $r>0$, $U_0\in \mathcal{V}(x_0)$, and $\delta >0$ such that $l(A)>\delta$ for every $A\in co\{{\displaystyle \frac{\partial f_x}{\partial u}}(D\setminus S_{f_x})\cap B(u_0,r)\}$ and every $x\in U_0$.

Consider the control system

$$x^{\prime }=F(u,x),x(t_0)=x_0,f(u,x)=0.$$

(18)

We say that the system (18) is solvable at the point $(u_0,x_0)$ if there exist $\rho >0$, $U\in \mathcal{V}(x_0)$, and a continuous mapping $u:U\to {\mathbb{R}}^n$ such that $u(x_0)=u_0$, $f(u(x(t)),x(t))=0$ for every $t\in [t_0,t_0+\rho ]$, and $x:[t_0,t_0+\rho ]\to U$ is the solution to the Cauchy equation

$$x^{\prime }=F(u,x),x(t_0)=x_0.$$

(19)

Indeed, by using Theorem 4, we find $U\in \mathcal{V}(x_0)$ and $g:U\to B(u_0,r)$ to be continuous such that $f(g(x),x)=0$ for every $x\in U$ and $g(x_0)=u_0$.

Using Peano’s theorem and the continuity of the mapping F, we find $\rho >0$ and $x:[t_0,t_0+\rho ]\to U$ to be a continuously differentiable solution to the Cauchy problem

$$x^{\prime }=F(g(x),x),x(t_0)=x_0.$$

(20)

Then, $x^{\prime }(t)=F(g(x(t)),x(t))$ and $f(g(x(t)),x(t))=0$ for every $t\in [t_0,t_0+\rho ]$ and $x(t_0)=x_0$, and hence, the control system (18) is solvable at $(u_0,x_0)$.

5. Conclusions

We proved a local inversion theorem for Sobolev mappings $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$. The machinery of integration by parts and the use of weak derivatives is the new method for solving the problem of the local univalence of some Sobolev mappings. We prove some global inversion theorems of the Banach–Mazur–Browder type and of the Hadamarad–Levy–John type. These results are generally similar to those given for locally Lipschitz mappings or for differentiable mappings, which may not be continuously differentiable.

We study the equation

$$f(x,y)=0,f(x_0,y_0)=0,$$

where $f:D\times Y\to {\mathbb{R}}^n$ is continuous, $D\subset {\mathbb{R}}^n$ is open, x is the unknown variable, and y is the parameter. The mapping f is a Sobolev one in the unknown variable, and the parameter belongs to a topological space Y and satisfies some topological conditions. We use our new local inversion theorem and we prove some general implicit function theorems for Sobolev mappings. The results may be useful both in theoretical and applied mathematics (for instance, in proving the local or global existence of the solutions of some PDEs or ordinary differential equations).