Some Remarks on the Spectrum of Nonlinear Continuous Operators

In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study, this question is used a different approach that allows the studying of all eigenvalues of the nonlinear operator relative to another nonlinear operator. Here shows that in nonlinear operators, case is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found are provided. Moreover, the nonlinear problems including parameters are studied.Comment: arXiv admin note: substantial text overlap with arXiv:2201.0224

Coupled systems of nonlinear variational inequalities and applications

In this paper we investigate the existence of solutions for a system consisting of two inequalities of variational type. Each inequality is formulated in terms of a nonlinear bifunction $\chi$ and $\psi$, respectively and a coupling functional $B$. We consider two sets of assumptions ${\bf (H_\chi^i)}$, ${\bf (H_\psi^j)}$ and ${\bf (H_B^k)}$, $i,j,k\in\{1,2\}$ and we show that, if the constraints sets are bounded, then a solution exists regardless if we assumed the first or the second hypothesis on $\chi$, $\psi$ or $B$, thus obtaining eight possibilities. When the constraint sets are unbounded a coercivity condition is needed to ensure the existence of solutions. We provide two such conditions. We consider nonlinear coupling functionals, whereas, in all the papers that we are aware of that dealing with such type of inequality systems the coupling functional is assumed bilinear and satisfies a certain "inf-sup" condition. An application, arising from Contact Mechanics, in the form of a partial differential inclusion driven by the $\Phi$-Laplace operator is presented in the last section