Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator

A Dynamic Contact Problem with History-Dependent Operators

“…The existence result is obtained by a technique used by Migórski and Kulig in [20] who studied second order subdifferential inclusions with a Volterra-type operator. The unique solvability of the inclusion is proved by a standard fixed point argument similar to those used in many papers, for instance in [15,[19][20][21] and [23]. Furthermore, we note that the abstract convergence result of Theorem 13 is based on arguments and assumptions similar to those exploited, for instance in [2,15,25] and [27].…”

“…The unique solvability of the inclusion is proved by a standard fixed point argument similar to those used in many papers, for instance in [15,[19][20][21] and [23]. Furthermore, we note that the abstract convergence result of Theorem 13 is based on arguments and assumptions similar to those exploited, for instance in [2,15,25] and [27]. In the second part of the paper we analyze a mathematical model of a contact problem for viscoelastic materials with history-dependent operators and a slipdependent friction.…”

“…Finally, from (19), the Hölder inequality and assumptions (1)(b), (6), (7)(d), (15), and (14)(b), (d), we obtain…”

“…We recall the result which is a consequence of the Banach contraction principle (for the proof see, for instance, [15]). …”

“…The research that we present in this paper concerns the study on existence and uniqueness of solutions to the hemivariational inequalities with history-dependent operators, and the investigation on the behaviour of solutions to such inequalities with respect to perturbations of operators and functions. We note that the history-dependent operators have been considered for quasistatic and evolutionary contact problems by several authors, for example, by Sofonea et al in [8,21,[26][27][28] and [29], Migórski et al in [5,15,19] and [20], Ogorzały in [23], Yao et al in [31], and Zhu in [33]. The normal compliance contact condition was introduced in [13,14,22] and it was used in many papers, see, e.g., [3,9,25] and [29].…”