Selecting families and their applications
Deguire and Lassonde [P. Deguire, M. Lassonde, Familles selectantes, Topol. Methods Nonlinear Anal. 5 (1995) 261-269] extend the concept of continuous selection and introduce the notion of selecting family for a family of set-valued mappings. In this paper, we first establish a new existence theorem of selecting families. The existence of the selecting families will be then used in order to obtain several fixed component theorems of Fan-Browder type, an intersection result, a maximal element theorem for a family of set-valued mappings and a minimax inequality.
Collectively fixed point theorems in noncompact abstract convex spaces with applications
In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...