1$, as provided therein. The second improvement is the simplification of the proof in the spirit described above. A normed linear space $E$ is called strictly convex if the unit sphere contains no line segments on its surface, i.e., condition $\|x\| =1,\| y\| =1,x\neq y$ implies that \[ \| \frac{1}{2}(x+y) \| <1 \] or in other words, that the unit sphere is a strictly convex set. The space $E$ is called uniformly convex, if for each $\varepsilon \in (0,2]$ there exists $\delta (\varepsilon ) >0$ such that if $\| x\| =1,\| y\| =1$ and $\|x-y\| \geq \varepsilon $, then $\| x+y\| \leq 2(1-\delta (\varepsilon ) ) $. A uniformly convex space is necessarily strictly convex and reflexive. We recall from \cite{MahJeb} the notion of duality mapping from $E$ to $E^{\ast }$ relative to a normalization function. In the sequel we shall write duality mapping with the understanding that we mean a duality mapping relative to some normalization function. A continuous function $\varphi :\mathbb{R}_{+}\to \mathbb{R}_{+}$ is called a normalization function if it is strictly increasing, $\varphi (0) =0$ and $\varphi(r)\to \infty $ with $r\to \infty $. A duality mapping on $E$ corresponding to a normalization function $\varphi $ is an operator $A:E\to 2^{E^{\ast }}$ such that for all $u\in E$ and $u^{\ast }\in A(u) $, \[ \| A(u) \| _{\ast }=\varphi (\|u\| ) , \quad \langle u^{\ast },u\rangle =\| u^{\ast }\| _{\ast }\| u\| . \] Some remarks are in order, especially those concerning the assumptions on a duality mapping. We recall from \cite{MahJeb} that \begin{itemize} \item[(i)] for each $u\in E$, $A(u) $ is a bounded, closed and convex subset of $E^{\ast };$ \item[(ii)] $A$ is monotone \[ \langle u_1^{\ast }-u_2^{\ast },u_1-u_2\rangle \geq (\varphi (\| u_1\| ) -\varphi (\| u_2\| ) ) (\| u_1\| -\| u_2\| ) \] for each $u_1,u_2\in E$, $u_1^{\ast }\in A(u_1),u_2^{\ast }\in A(u_2)$; \item[(iii)] for each $u\in E$, $A(u) =\partial \psi (u) $, where $\partial \psi (\cdot ):E\to 2^{E^{\ast }}$ denotes the subdifferential in the sense of convex analysis of the functional $\psi (u) =\int_0^{\| u\| }\varphi (t) dt$, i.e. \[ \partial \psi (u) =\big\{ u^{\ast }\in E^{\ast }:\psi ( y) -\psi (u) \geq \langle u^{\ast },y-u\rangle _{E^{\ast },E}\text{ for all }y\in E\big\} ; \] \item[(iv)] if $E^{\ast }$ is strictly convex, then $card(A(u) )=1$, for all $x\in E$; \item[(v)] when $E$ is reflexive and strictly convex, then operator $A:E\to E^{\ast }$ is demicontinuous, which means that if $x_{n}\to x$ in $E$ then $A(x_{n}) \rightharpoonup A(x) $ in $E^{\ast }$. \end{itemize} We see by \cite[Proposition 2.8]{phelps} that since $A$ is demicontinuous, we obtain that functional $\psi $ is differentiable in the sense of G\^{a}teaux and operator $A$ being its derivative. \begin{lemma} \label{lem_duality_mapping} Assume that $E$ is a reflexive Banach space with a strictly convex dual $E^{\ast }$. Then the duality mapping $A:E\to E^{\ast }$\ corresponding to a normalization function $\varphi $ is single-valued and the functional $\psi $ is differentiable in the sense of G\^{a}teaux with $A$ being its G\^{a}teaux derivative. \end{lemma} \begin{proof} The duality mapping $A$ is now single valued and its potential, i.e. $\psi $, has a single-valued subdifferential which is demicontinuous. Now the argument given prior to the formulation finishes the proof. \end{proof} In view of Lemma \ref{lem_duality_mapping}, it is apparent that assuming the continuous differentiability of a potential of a duality mapping is not a very restrictive condition. Indeed, in order to get some reasonable result we will have to assume that the duality mapping has the potential which is continuously differentiable. This is necessary in order to obtain that the functional $h:X\to \mathbb{R}$ given by the formula \[ h(x) =\psi (f(x) -y) \] is continuously differentiable. Moreover, functional $\psi $ can be considered as a potential of a duality mapping $A$ in case it is single-valued. In this case we write $A:Y\to Y^{\ast }$ and by writing $A:Y\to Y^{\ast }$ we implicitly assume that $A$ is single valued. We will follow this observation in the sequel. We formulate our second global diffeomorphism result. \begin{theorem} \label{GLOBIFnew_v1} Let $X$, $Y$ be a real Banach spaces. Let the potential $\psi $ of a duality mapping $A:Y\to Y^{\ast }$ corresponding to a normalization function $\varphi $ be continuously G\^{a}teaux differentiable. Assume that $f:X\to Y$ is a $C^1$-mapping such that: \begin{itemize} \item[(A5)] for any $y\in Y$ the functional $h:X\to \mathbb{R}$ given by the formula \[ h(x) =\psi (f(x) -y) \] satisfies the Palais-Smale condition, \item[(A6)] $f'(x)\in \operatorname{Isom}(X,Y) $ for any $x\in X$. \end{itemize} Then $f$ is a diffeomorphism. \end{theorem} \begin{proof} Let us fix a point $y\in Y$. Functional $h$ is a composition of two $C^1-$mappings, so it is $C^1$ itself. Moreover, $h$ is bounded from below and it satisfies the Palais-Smale condition by (A5). Thus it follows from Theorem \ref{CPT} that there exists an argument of a minimum which we denote by $\overline{x}$. We see by the chain rule and Fermat's Principle and by the assumptions on a duality mapping that \[ 0=h'(\overline{x})=A(f(\overline{x}) -y) \circ f'(\overline{x}) . \] Since by (A6), mapping $f'(\overline{x}) $ is invertible, we get that $A(f(\overline{x}) -y) =0$. Now, by the property that $\| A(u) \| _{\ast }=\varphi (\| u\| ) $ we note that \[ \| A(f(\overline{x}) -y) \| _{\ast }=\varphi (\| f(\overline{x}) -y\| ) . \] So it follows, since $\varphi (0) =0$ and since $\varphi $ is strictly increasing, that \[ f(\overline{x}) -y=0, \] which proves the existence of $x\in X$ for every $y\in Y$, such that $ f(\overline{x}) =y$. The uniqueness can be shown by contradiction exactly as before. \end{proof} Now we provide a simple example of a space and a duality mapping for which the assumptions of the above result hold. Let us define a single valued operator $A:W_0^{1,p}\to ( W_0^{1,p}) ^{\ast }$ such that \[ \langle Au,v\rangle =\int_0^1| u'(t)| ^{p-2}u'(t)v'(t)dt. \] It follows from Remark \ref{rem_diff_wp} that if $2\leq p<+\infty $, then $A$ is a potential operator and its $C^1-$potential is $h$ defined by \eqref{def_fi}. Therefore by the cited remarks from \cite{MahJeb} we get that a duality mapping on $W_0^{1,p}$ corresponding to a normalization function $ t\to t^{p-1}$ is defined by $A$. \section{Applications} \subsection{Applications to algebraic equations} We conclude this paper with some applications to the unique solvability of nonlinear equations of the form $Ax=F(x)$\ where $A$\ is a nonsingular matrix and $F$\ is a $C^1$ nonlinear operator. Some motivation to extend existence tools for such equations can be found in \cite{add3,add4,add2,add1}. We consider the problem \begin{equation} Ax=F(x) , \label{equAPP} \end{equation} where $A$ is an $n\times n$ matrix (possibly singular) and $F:\mathbb{R}^n\to\mathbb{R}^n$ is a $C^1-$mapping. We consider $\mathbb{R}^n$ with the Euclidean norm in both, theoretical results and in the example which follows. To apply Theorem \ref{MainTheo copy(2)} to the solvability of \eqref{equAPP} we need some assumptions. Let us recall that if $A^{\ast }$ denotes the transpose of matrix $A$, then $A^{\ast }A$ is symmetric and positive semidefinite. Let $\delta _{\max }(A) $ denote the greatest of the singular values of $A$, i.e. the eigenvalues of $A^{T}A$, with the obvious meaning of $\delta _{\min }(A) $. We assume that $A$ is different from the $zero$ matrix, so that both mentioned values are positive. \begin{theorem}\label{firstAlgTheo} Assume that $F:\mathbb{R}^n\to\mathbb{R}^n$ is a $C^1$-mapping and the following conditions hold: \begin{itemize} \item[(i)] either there exists a constant $0\delta _{\max }(A) $ such that \[ \| F(x) \| \geq b\| x\| \] for all $x\in \mathbb{R}^n$ $x\in\mathbb{R}^n$ with sufficiently large norm; \item[(ii)] $\det (A-F{'}(x) ) \neq 0$ for every every $x\in \mathbb{R}^n$. \end{itemize} Then Problem \eqref{equAPP} has exactly one solution for any $\xi \in\mathbb{R}^n$. \end{theorem} \begin{proof} We put $\varphi (x) =Ax-F(x) $. From the first possibility of assumption (ii) it follows for $x\in\mathbb{R}^n$ with sufficiently large norm, \begin{align*} \| \varphi (x) \| &=\| Ax-F( x) \| \geq \| Ax\| -\| F(x) \| \\ &\geq \sqrt{\langle A^{\ast }Ax,x\rangle }-a\|x\| \geq (\delta _{\min }(A) -a) \| x\| . \end{align*} Hence the function $\varphi $ is coercive. Since $\det \varphi '(x) \neq 0$ for every $x\in\mathbb{R}^n$, it follows by Theorem \ref{MainTheo copy(2)} that $\varphi $ is a global homeomorphism and thus equation \eqref{equAPP} has exactly one solution for any $\xi \in\mathbb{R}^n$. The second case of assumption (ii) follows likewise. \end{proof} \begin{remark} \rm We note that to obtain coercivity of the function $\varphi $ in the above theorem we can employ the following assumption instead of (ii), \begin{itemize} \item[(iia)] either there exist constants $\alpha >0$, $0<\gamma <1$ such that \[ \| F(x) \| \leq \alpha \| x\| ^{\gamma } \] for all $x\in\mathbb{R}^n$ with sufficiently large norm, or \item[(iib)] there exist constants $\beta >0$, $\theta >1$ such that \[ \| F(x) \| \geq \beta \| x\|^{\theta } \] for all $x\in\mathbb{R}^n$ with sufficiently large norm. \end{itemize} \end{remark} Now we provide some examples of problems which we can consider by the above method. \begin{example} \rm Consider the indefinite matrix \[ A=\begin{bmatrix} -2 & 1 \\ 6 & -3 \end{bmatrix} \] and the function $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by \[ F(x,y) =(x^{3}+y+1,6x+y+y^{3}+1)\,. \] On $\mathbb{R}^{2}$ consider the Euclidean norm, $\| (x,y) \| = \sqrt{x^{2}+y^{2}}$. We recall that $\| (x,y)\| \leq 2^{\frac{1}{3}}\sqrt[6]{x^{6}+y^{6}}$. Note that $F(x,y) =(x^{3},y^{3}) +(0,6x) +(y,y)+(1,1)$. Let \[ \varphi (x,y) =F(x,y)-A(x,y),\quad (x,y)\in \mathbb{R}^{2}. \] Hence \begin{align*} \| \varphi (x,y) \| &\geq \frac{1}{2}\| (x,y)\| ^{3}-6\sqrt{2}\| (x,y)\| -\sqrt{2}-\| A\| \| (x,y)\| \\ &=\| (x,y)\| \Big(\frac{1}{2}\| (x,y)\| ^{2}-(6\sqrt{2}\mathbf{+}\| A\| ) -\frac{\sqrt{2}}{\| (x,y)\| }\Big) . \end{align*} From the above sequence of inequalities it follows that $\varphi $ is coercive. One can easily see that for any $(x,y) \in\mathbb{R}^{2}$ \[ F'(x,y) -A=\begin{bmatrix} 3x^{2}+3 & 0 \\ 0 & 3y^{2}+4 \end{bmatrix}. \] Since $\det (F'(x,y) -A) >0$, we see that the problem $Ax=F(x) $ has exactly one (nontrivial) solution. \end{example} \subsection{Application to an integro-differential system} In this section we propose some improvement of results from \cite{galewskikoniorczyk} as far as the growth assumptions and methods of the proof are concerned. Namely, we use the Bielecki type norm on the underlying space instead of the regular one. Since the proofs do not differ that much apart from some estimation, we provide only the main differences referring to \cite{galewskikoniorczyk} for the more detailed reasoning. We were inspired by \cite{majewski} to come up with these results. Prior to formulating the problem under consideration we introduce some required function space setting. We introduce \[ W^{1,p}([0,1],\mathbb{R}^n) =\big\{x:[0,1]\to \mathbb{R}^n\text{ is absolutely continuous, }x'\in L^p([0,1],\mathbb{R}^n)\big\}. \] Here $x'$ denotes the a.e. derivative of $x$. Further, we denote $L^p([0,1],\mathbb{R}^n)$ by $L^p$ and $W^{1,p}([0,1],\mathbb{R}^n)$ by $W^{1,p}$. The $W^{1,p}$ space is equipped with the usual norm $\|x\|_{W^{1,p}}^p=\|x\|_{L^p}^p+\|x'\|_{L^p}^p$. The Sobolev space is defined as \[ \tilde{W}_0^{1,p}([0,1],\mathbb{R}^n)=\{x\in W^{1,p},x(0)=0\} \] and it is equipped with the norm \begin{equation} \|x\|_{\tilde{W}_0^{1,p}} =\Big(\int_0^1| x'(t)| ^pdt\Big)^{1/p},\quad x\in \tilde{W}_0^{1,p} \label{W01p_norm} \end{equation} equivalent to $\|x\|_{W^{1,p}}$. By definition, for any $p>1$, we have the following chain of embeddings \begin{equation} \tilde{W}_0^{1,p}\hookrightarrow W^{1,p}\hookrightarrow L^p. \label{space_imbedding} \end{equation} There exists a constant $C$ such that for any $u\in \tilde{W}_0^{1,p}$ \[ \|u\|_{W^{1,p}[0,1]}\leq C\|u'\|_{L^p[0,1]}. \] The space $\tilde{W}_0^{1,p}$ is uniformly convex. In the literature, the existence of the solution to integro-differential equation is obtained by the Banach fixed point theorem or another type of fixed point theorem, see \cite{integro1,integro2}. Let us formulate a nonlinear integro-differential equation with variable integration limit with an initial condition, which reads as follows \begin{gather} x'(t)+\int_0^{t}\Phi (t,\tau ,x(\tau ))d\tau =y(t),\quad \text{for a.e. }t\in [ 0,1], \label{I_D_eq} \\ x(0)=0, \label{I_D_iv} \end{gather} where $y\in L^p$ is fixed for the time being. Now we impose assumptions on the nonlinear term. These ensure that the problem is well posed in the sense that the solution to \eqref{I_D_eq}-\eqref{I_D_iv} exists, it is unique and the solution operator depends in a differentiable manner on a parameter $y$ provided we allow it to vary. This implies that problem \eqref{I_D_eq}-\eqref{I_D_iv} is well posed in the sense of Hadamard. Let $P_{\Delta }=\{(t,\tau )\in [ 0,1]\times [ 0,1];\tau \leq t\} $. We assume that function $\Phi :P_{\Delta }\times \mathbb{R} ^n\to \mathbb{R}^n$ satisfies the following conditions: \begin{itemize} \item[(A7)] %\label{App_A1} $\Phi (\cdot ,\cdot ,x)$ is measurable on $ P_{\Delta }$ for any $x\in \mathbb{R}^n$ and $\Phi (t,\tau ,\cdot )$ is continuously differentiable on $\mathbb{R}^n$ for a.e. $(t,\tau )\in P_{\Delta }$; \item[(A8)] %\label{App_A2} there exist functions $a$, $b\in L^p(P_{\Delta},R_0^{+})$ such that \[ |\Phi (t,\tau ,x)|\leq a(t,\tau )|x|+b(t,\tau ) \] for a.e. $(t,\tau )\in P_{\Delta }$, all $x\in \mathbb{R}^n$ and there exists a constant $\overline{a}>0$ such that \[ \int_0^{t}a^p(t,\tau )d\tau \leq \overline{a}^p \] for a.e. $t\in [ 0,1]$. \item[(A9)] %\label{App_A3} there exists functions $c\in L^p(P_{\Delta },\mathbb{R}_0^{+})$, $\alpha \in C(\mathbb{R}_0^{+},\mathbb{R}_0^{+})$ and a constant $C>0$ such that \[ |\Phi _{x}(t,\tau ,x)|\leq c(t,\tau )\alpha (|x|) \] for a.e. $(t,\tau )\in P_{\Delta }$ and all $x\in \mathbb{R}^n$; moreover \[ \int_0^{t}c^{q}(t,\tau )d\tau \leq C,\text{ for }a.e.\text{ }t\in [ 0,1]. \] \end{itemize} \begin{remark} \rm In \cite{galewskikoniorczyk} it was assumed that \[ \|a\|_{L^p(P_{\Delta },\mathbb{R})}<2^{-\frac{(p-1)}{p}} \] which considerably restricts the growth. \end{remark} For any $k>0$ let us define another form of the Bielecki type norm \begin{equation} \|x\|_{\tilde{W}_0^{1,p},k} =\Big(\int_0^1e^{-kt}|x'dt\Big) ^{1/p}. \label{Bielecki_1_def} \end{equation} For $k=0$ the above function defines a norm introduced by \eqref{W01p_norm} and therefore hereafter we will skip index $0$. It is easy to notice that \begin{equation} e^{-k/p}\|x\|_{\tilde{W}_0^{1,p}}\leq \|x\|_{\tilde{W} _0^{1,p},k}\leq \|x\|_{\tilde{W}_0^{1,p}} \label{rel_equ_biel} \end{equation} For any $k>0$ and $x\in \tilde{W}_0^{1,p}$ we assert the following relations: \begin{gather} \|x\|_{k}\leq \frac{\|x\|_{\tilde{W}_0^{1,p},k}}{k^{1/p}}, \label{Bielecki_1_1}\\ \| \int_0^{\cdot }|x(\tau )|d\tau \| _{k} =\Big(\int_0^1e^{-kt}\Big(\int_0^{t}|x(\tau )|d\tau \big) ^pdt\Big) ^{1/p}\leq \frac{\|x\|_{\tilde{W}_0^{1,p},k}}{k^{2/p}} \label{Bielecki_1_2} \end{gather} where the symbol $\int_0^{\cdot }u(\tau )d\tau $ denotes the function $[0,1]\ni t\to \int_0^{t}u(\tau )d\tau $. Now let us prove the stated relations, starting with \eqref{Bielecki_1_1}. Fix $k>0$ and $x\in\tilde{W}_0^{1,p}$: \begin{align*} \|x\|_{k}^p &=\int_0^1e^{-kt}|x(t)|^pdt =\int_0^1e^{-kt}|\int_0^{t}x'(\tau )d\tau | ^pdt \\ &\leq \int_0^1e^{-kt}\int_0^{t}| x'(\tau )|^pd\tau dt = \int_0^1| x'(\tau )| ^p(\int_{\tau }^1e^{-kt}dt) d\tau \\ &=\frac{1}{k}\int_0^1e^{-kt}|x'^pdt-\frac{e^{-k}}{k}\int_0^1|x'^pdt \\ &\leq \frac{1-e^{-k}}{k}\int_0^1e^{-kt}|x'^pdt \leq \frac{ \|x\|_{\tilde{W}_0^{1,p},k}^p}{k}. \end{align*} Now let us turn to the relation \eqref{Bielecki_1_2}: \begin{align*} \|\int_0^{\cdot }|x(\tau )|d\tau \|_{k}^p &=\int_0^1e^{-kt}(\int_0^{t}|x(\tau )|d\tau )^pdt \leq \int_0^1e^{-kt}(\int_0^{t}|x(\tau )|^pd\tau )dt \\ &= \int_0^1|x(\tau )|^p(\int_{\tau }^1e^{-kt}dt)d\tau \\ &=\frac{1}{k}\int_0^1e^{-kt}|x(t)|^pdt-\frac{e^{-k}}{k} \int_0^1|x(t)|^pdt \\ &\leq \frac{\|x\|_{k}^p}{k}\leq \frac{\|x\|_{\tilde{W}_0^{1,p},k}^p }{k^{2}}. \end{align*} To apply Theorem \ref{MainTheo} we can define functional $\varphi :\tilde{W} _0^{1,p}\to \mathbb{R}$ as follows \begin{align*} \varphi (x) &=(1/p)\| f(x)-y\| _{k}^p \\ &=(1/p)\int_0^1e^{-kt}| x'(t)-y(t)+\int_0^{t}\Phi (t,\tau ,x(\tau ))d\tau | ^pdt. \end{align*} We can define functional $\varphi :\tilde{W}_0^{1,p}\to \mathbb{R}$ in the form \begin{align*} \varphi (x) &=(1/p)\| f(x)-y\| _{k}^p \\ &=(1/p)\int_0^1e^{-kt}| x'(t)-y(t)+\int_0^{t}\Phi (t,\tau ,x(\tau ))d\tau | ^pdt. \end{align*} Having in mind the relation \eqref{rel_equ_biel}, which states that $L^p$ norm $\|\cdot \|_{L^p}$ and the Bielecki norm $\|\cdot \|_{k}$ are equivalent, the following inequality can be deduced for any $x\in \tilde{W}_0^{1,p}$: \begin{align*} (p\varphi (x)) ^{1/p} &=\| x'(\cdot) -y(\cdot ) +\int_0^{\cdot }\Phi (\cdot ,\tau ,x(\tau ))d\tau \| _{k} \\ &\geq \|x'\|_{k}-\|y\|_{k}-\|\int_0^{\cdot }\Phi (\cdot ,\tau ,x(\tau ))d\tau \|_{k} \\ &\geq \|x'\|_{k}-\|y\|_{k}-\overline{a}\| \int_0^{\cdot }x(\tau )d\tau \| _{k}-\| \int_0^{\cdot }b(\cdot ,\tau )d\tau \| _{k} \\ &\geq \|x\|_{\tilde{W}_0^{1,p},k}-\frac{\overline{a}}{k^{2/p}} \|x\|_{\tilde{W}_0^{1,p},k}+d, \end{align*} where $d=\|y\|_{k}-\| \int_0^{\cdot }b(\cdot ,\tau )d\tau \| _{k}$. For sufficiently large $k>0$, that is $k>\max \{1,\overline{a}^{\frac{p}{2}}\}$, we have the coercivity of functional $\varphi$. Using the above estimates and exactly the same arguments as in \cite{galewskikoniorczyk}, we can prove the following result. \begin{theorem}\label{appTheo_1} Under the above assumptions, for any fixed $y$ $\in L^p$, problem \eqref{I_D_eq}-\eqref{I_D_iv} has a unique solution $x_{y}\in \tilde{W}_0^{1,p}$. Moreover, the operator \[ L^p\ni y\to x_{y}\in \tilde{W}_0^{1,p} \] which assigns to each $y\in L^p$ a solution to \eqref{I_D_eq}-\eqref{I_D_iv}, is continuously differentiable. \end{theorem} We complete this section with an example of a nonlinear term satisfying our assumptions (A7)--(A9). Let us consider the function $\Phi :P_{\Delta }\times \mathbb{R}\to \mathbb{R}$ defined by \[ \Phi (t,\tau ,x)=\alpha (t-\tau )^{5/2}\ln (1+(t-\tau )^{2}x^{2}) \] for $t,\tau \in [0,1]$, $t>\tau $, $x\in \mathbb{R}$, where $\alpha >0$ is fixed. Since $\ln (1+s^{2}z^{2})\leq |s|+|z|$ for $s,z\in \mathbb{R}$, we see that \[ |\Phi (t,\tau ,x)|\leq \alpha (t-\tau )^{5/2}|x|+\alpha (t-\tau )^{5/2}. \] Let us put \[ a(t,\tau )=\alpha (t-\tau )^{5/2} \] for $t,\tau $ $\in [ 0,1]$, $t>\tau $. Then by a direct calculation we obtain \[ \|a\|_{L^p(P_{\Delta },\mathbb{R})}^p\leq \alpha ^p\frac{4}{(5p+2)(5p+4)} =:\overline{a}. \] Moreover, \begin{gather*} |\Phi _{x}(t,\tau ,x)|\leq \alpha (t-\tau )^{5/2}|x|, \\ \int_0^{t}c(t,\tau )^{q}d\tau =2^{-p}\int_0^{t}(t-\tau )^{5q/2}d\tau = \frac{2^{1-p}}{5q+2}t^{(5q/2)+1}\leq \frac{2^{1-p}}{5q+2}, \quad t\in [ 0,1]. \end{gather*} Hence, $\Phi $ satisfies assumptions (A7)--(A9). Theorem \ref{MainTheo} shows that the initial-value problem \[ x'(t)+\int_0^{t}2^{1-p}(t-\tau )^{1/2}\ln (1+(t-\tau )^{2}x^{2})d\tau =y(t),\quad \text{a.e. } t\in [ 0,1] \] has a unique solution $x_{y}\in \tilde{W}_0^{1,p}$ for any fixed $y\in L^p$. Moreover, the mapping \[ L^p\ni y\to x_{y}\in \tilde{W}_0^{1,p} \] is continuously differentiable. \subsection*{Acknowledgements} D. Repov\v{s} was supported by Slovenian Research Agency grants P1-0292, N1-0064, J1-8131, and J1-7025. \begin{thebibliography}{99} \bibitem{MMR} M. Be\l dzi\'{n}ski, M. Galewski, R. Stegli\'{n}ski; \emph{Solvability of abstract semilinear equations by a global diffeomorphism theorem}, submitted, ArXiv: 1712.03493 \bibitem{add3} G. Bonanno, G. Molica Bisci; \emph{Infinitely many solutions for a boundary value problem with discontinuous nonlinearities}, Bound. Value Probl., \textbf{2009 (2009)}, 1-20. \bibitem{BrezisBook} H. Br\'{e}zis; \emph{Functional Analysis, Sobolev Spaces and Partial Differential Equations}, Springer, 2010. \bibitem{MahJeb} G. Dinca, P. Jebelean, J. Mawhin; \emph{Variational and topological methods for Dirichlet problems with p-Laplacian}, Port. Math. (N.S.) \textbf{58} (2001), no. 3, 339--378. \bibitem{ekelend} I. Ekeland; \emph{An inverse function theorem in Fr\'{e} chet spaces}, Ann. Inst. Henri Poincar\textit{\'{e}}, Anal. Non Lin{e}aire \textbf{28} (2011), no. 1, 91-105. \bibitem{fig1} D.G. Figueredo; \emph{Lectures on the Ekeland Variational Principle with Applications and Detours}, Preliminary Lecture Notes, SISSA, 1988. \bibitem{fijalkowski2} P. Fija\l kowski; \emph{A global inversion theorem for functions with singular points}, Discrete Contin. Dyn. Syst., Ser. B \textbf{23} (2018), no. 1, 173-180. \bibitem{GGS} E. Galewska, M. Galewski, E. Schmeidel; \emph{Conditions for having a diffeomorphism between two Banach spaces}, Electron. J. Differ. Equ. \textbf{2014} (2014), no. 99, 6 p. \bibitem{galewskikoniorczyk} M. Galewski, M. Koniorczyk; \emph{On a global diffeomorphism between two Banach spaces and some application}, Stud. Sci. Math. Hung. \textbf{52} (2015), no. 1, 65-86. \bibitem{G_K_AMH} M. Galewski, M. Koniorczyk; \emph{On a global implicit function theorem and some applications to integro-differential initial value problem}, Acta Math. Hungar. \textbf{148} (2016), no. 2, 257-278. \bibitem{galewskiRadulescu} M. Galewski, M. R\u{a}dulescu; \emph{On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory}, accepted to Quaestiones Mathematicae 2017: 1--14, DOI: 10.2989/16073606.2017.1391353. \bibitem{Hadamard} J. Hadamard; \emph{Sur les Transformations Ponctuelles}, S. M. F. Bull. 34, 71-84 (1906). \bibitem{ioffee} A. D. Ioffe; \emph{Global surjection and global inverse mapping theorems in Banach spaces}, Rep. Moscow Refusnik Semin., Ann. N. Y. Acad. Sci. \textbf{491} (1987), 181-188. \bibitem{SIW} D. Idczak, A. Skowron, S. Walczak; \emph{On the diffeomorphisms between Banach and Hilbert spaces}, Adv. Nonlinear Stud. \textbf{12} (2012), no. 1, 89-100. \bibitem{idczakIFT} D. Idczak; \emph{A global implicit function theorem and its applications to functional equations}, Discrete Contin. Dyn. Syst., Ser. B \textbf{19} (2014), no. 8, 2549-2556. \bibitem{IDCZAK_Gen_GIFT} D. Idczak; \emph{On a generalization of a global implicit function theorem}, Adv. Nonlinear Stud. \textbf{16} (2016), no. 1, 87-94. \bibitem{jabri} Y. Jabri; \emph{The mountain pass theorem. Variants, generalizations and some applications}, Encyclopedia of Mathematics and its Applications, 95. Cambridge University Press, Cambridge, 2003. \bibitem{katriel} G. Katriel; \emph{Mountain pass theorems and global homeomorphism theorems}, Ann. Inst. Henri Poincar\'{e}, Anal. Non Lin\'{e}aire, \textbf{11 }(1994), no. 2, 189-209. \bibitem{Levy} P. L\'{e}vy; \emph{Sur les fonctions de lignes implicites}, Bull. Soc. Math. France, \textbf{48} (1920), 13--27. \bibitem{Mahwin} J. Mawhin; \emph{Problemes de Dirichlet Variationnels Non Lin\'{e}aires}, S\'{e}minaire de Math\'{e}matiques Sup\'{e}rieures, \textbf{104}, Montreal 1987. \bibitem{majewski} M. Majewski; \emph{Control system defined by some integral operator}, Opuscula Math. \textbf{37} (2017), no. 2, 313-325. \bibitem{add4} S. A. Marano, G. Molica Bisci, D. Motreanu; \emph{Multiple solutions for a class of elliptic hemivariational inequalities}, J. Math. Anal. Appl., \textbf{337} (2008), 85-97. \bibitem{add2} N. Marcu, G. Molica Bisci; \emph{Existence and multiplicity results for nonlinear discrete inclusions}, Electron. J. Differential Equations, \textbf{2012} (2012), p. 1-13. \bibitem{add1} G. Molica Bisci, D. Repov\v{s}; \emph{Nonlinear Algebraic Systems with discontinuous terms}, J. Math. Anal. Appl. \textbf{398} (2013), 846--856. \bibitem{phelps} R. R. Phelps; \emph{Convex functions, monotone operators and differentiability}, 2nd ed., Lecture Notes in Mathematics. 1364. Berlin: Springer-Verlag 1993. \bibitem{plastock} R. Plastock; \emph{Homeomorphisms Between Banach Spaces}, Trans. Amer. Math. Soc. \textbf{200} (1974), 169--183. \bibitem{pucci} P. Pucci, J. Serrin; \emph{Extensions of the mountain pass theorem}, J. Funct. Anal. \textbf{59} (1984), 185-210. \bibitem{rad} M. R\u{a}dulescu, S. R\u{a}dulescu; \emph{Local inversion theorems without assuming continuous differentiability}, J. Math. Anal. Appl. \textbf{138} (1989), no. 2, 581-590. \bibitem{zampieri} G. Zampieri; \emph{Diffeomorphisms with Banach space domains}, Nonlinear Anal., Theory Methods Appl. \textbf{19} (1992), no. 10, 923-932. \bibitem{integro1} J. R. Wang, W. Wei; \emph{Nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces}, Ann. Polon. Math., \textbf{105} (2012), no. 3, 209--223. \bibitem{integro2} J. Wang, W. Wei; \emph{An application of measure of noncompactness in the study of integrodifferential evolution equations with nonlocal conditions}, Proc. A. Razmadze Math. Inst. \textbf{158} (2012), 135--148. \bibitem{zeidler} E. Zeidler; \emph{Applied Functional Analysis. Main Principles and Their Applications}, Applied Mathematical Sciences. 109. New York, NY: Springer-Verlag. xvi, 404 p. (1995). \end{thebibliography} \end{document}