In this note we prove that some recent results on an implicit variational inequality problem for multivalued mappings, which seem to extend and improve some well-known and celebrated results, are not correct.

We deal with the integral equation $u\left(t\right)=f(t,{\int}_{I}g(t,z)u\left(z\right)\phantom{\rule{0.166667em}{0ex}}dz)$, with $t\in I:=[0,1]$, $f:I\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in {L}^{s}(I,{\mathbb{R}}^{n})$, $s\in \phantom{\rule{0.166667em}{0ex}}]1,+\infty ]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.

Let E be a Banach space. We consider a Cauchy problem of the type
⎧ ${D}_{t}^{k}u+{\sum}_{j=0}^{k-1}{\sum}_{\left|\alpha \right|\le m}{A}_{j,\alpha}\left({D}_{t}^{j}{D}_{x}^{\alpha}u\right)=f$ in ${\mathbb{R}}^{n+1}$,
⎨
⎩ ${D}_{t}^{j}u(0,x)={\phi}_{j}\left(x\right)$ in ${\mathbb{R}}^{n}$, j=0,...,k-1,
where each ${A}_{j,\alpha}$ is a given continuous linear operator from E into itself. We prove that if the operators ${A}_{j,\alpha}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in {C}^{\infty}({\mathbb{R}}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${\mathbb{R}}^{n+1}$.

We consider a multifunction $F:T\times X\to {2}^{E}$, where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

We consider the integral equation $h\left(u\left(t\right)\right)=f\left({\int}_{I}g(t,x)\phantom{\rule{0.166667em}{0ex}}u\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\right)$, with $t\in [0,1]$, and prove an existence theorem for bounded solutions where $f$ is not assumed to be continuous.

We deal with the integral equation $u\left(t\right)=f\left({\int}_{I}g(t,z)\phantom{\rule{0.166667em}{0ex}}u\left(z\right)\phantom{\rule{0.166667em}{0ex}}dz\right)$, with $t\in I=[0,1]$, $f:{\mathbf{R}}^{n}\to {\mathbf{R}}^{n}$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in {L}^{\infty}(I,{\mathbf{R}}^{n})$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.

We show that a recent existence result for the Nash equilibria of generalized games with strategy sets in $H$-spaces is false. We prove that it becomes true if we assume, in addition, that the feasible set of the game (the set of all feasible multistrategies) is closed.

We deal with the implicit integral equation $$h\left(u\left(t\right)\right)=f(\phantom{\rule{0.166667em}{0ex}}t\phantom{\rule{0.166667em}{0ex}},{\int}_{I}g(t,z)\phantom{\rule{0.166667em}{0ex}}u\left(z\right)\phantom{\rule{0.166667em}{0ex}}dz)\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.a.}\phantom{\rule{4.0pt}{0ex}}t\in I,$$
where $I:=[0,1]$ and where $f:I\times [0,\lambda ]\to \mathbb{R}$, $g:I\times I\to [0,+\infty [$ and $h:\phantom{\rule{0.166667em}{0ex}}]\phantom{\rule{0.166667em}{0ex}}0,+\infty \phantom{\rule{0.166667em}{0ex}}[\phantom{\rule{0.166667em}{0ex}}\to \mathbb{R}$. We prove an existence theorem for solutions $u\in {L}^{s}\left(I\right)$ where the contituity of $f$ with respect to the second variable is not assumed.

We establish a fixed point theorem for a continuous function $f:X\to E$, where $E$ is a Banach space and $X\subseteq E$. Our result, which involves multivalued contractions, contains the classical Schauder fixed point theorem as a special case. An application is presented.

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