Abstract
We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u↦fl(t,x,u),l=1,2 and no compact property of C0-semigroups eAl(t),l=1,2.
Highlights
Let the EPs, VIs, EFs, HVIs, CGS, DVIs, DHVIs, DMVIs, DMHVIs, EE, PDEs and PCGDDs represent the equilibrium problems, variational inequalities, energy functionals, hemivariational inequalities, Clarke’s generalized subdifferential, differential variational inequalities, differential hemivariational inequalities, differential mixed variational inequalities, differential mixed hemivariational inequalities, evolution equation, partial differential equations and partial Clarke’s generalized directional derivatives, respectively.To the best of our knowledge, the theory of VIs, which was first extended to treat the EPs, is intently relevent to the convexity of EFs, and is the basis of various arguments of monotonicity
It is worth mentioning that, for the first time, the Rothe rule was used in [18] to study the parabolic-type HVI driven by the abstract EE
There have been a few papers devoted to the Rothe rule for HVIs, see [21]
Summary
Let the EPs, VIs, EFs, HVIs, CGS, DVIs, DHVIs, DMVIs, DMHVIs, EE, PDEs and PCGDDs represent the equilibrium problems, variational inequalities, energy functionals, hemivariational inequalities, Clarke’s generalized subdifferential, differential variational inequalities, differential hemivariational inequalities, differential mixed variational inequalities, differential mixed hemivariational inequalities, evolution equation, partial differential equations and partial Clarke’s generalized directional derivatives, respectively. Up to now, only one reference (i.e., [11]), investigated the DHVI in Banach spaces consisting of the EE and elliptic-type HVI instead of the parabolic-type It was assumed in [11] that the constraint set K is of boundedness, the function u → f (t, x, u) maps convex subsets of K to convex sets and the C0-semigroup eA(t) is of compactness. If the mappings in the method based on the KKM approach are not the KKM ones, there are several possibilities which happen in the demonstration process, e.g., in particular, whenever studying the parabolic-type SHVI This might result in an unsuccessful continuation of the demonstration. This is precisely the shortcoming of the KKM-based approach
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