System of partial differential hemivariational inequalities involving nonlocal boundary conditions

Abstract Let FPT, MNC, HVI, SEPDE, SMHVI, PGCDD, and NLBC represent the fixed point theorem, measure of noncompactness, hemivariational inequality, system of nonlinear evolutionary partial differential equations, system of mixed hemivariational inequalities, partial generalized Clarke directional derivative, and nonlocal boundary condition in Banach spaces, respectively. We present and examine a complicated system acquired by the mixture of the SEPDE and SMHVI with PGCDDs when the constraint set is not necessary to be of boundedness and the issue is driven by NLBCs, that is termed as an system of partial differential hemivariational inequalities (SPDHVI) with PGCDDs. Also, it is shown that the set of solutions of SMHVI with PGCDDs implicated in the SPDHVI is the nonempty bounded set of both closedness and convexity. Besides, the measurable and upper semicontinuous natures for multivalued operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">U</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>×</m:mo> <m:mi>V</m:mi> <m:mo>→</m:mo> <m:mi>C</m:mi> <m:mi>b</m:mi> <m:mi>v</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>U</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathcal{U}}:\left[0,T]\times V\to Cbv\left(U) (see (3.20)) get validated. Finally, certain existence outcomes of SPDHVI get established with the help of the properties of PGCDD, the FPT of condensing multivalued mappings, and the technique of MNC.