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Abstract: Consider the higher order nonlinear partial difference equation of neutral type where , , and . In this paper, we first establish the discrete Arzela–Ascoli''s theorem. Next, we obtain some sufficient conditions for the existence of bounded and unbounded nonoscillatory solution of Eq. (∗). [Copyright &y& Elsevier]

Abstract: The g-Navier–Stokes equations in spatial dimension 2 were introduced by Roh as with the continuity equation where g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the global attractor of the g-Navier–Stokes equations converges (in the sense of upper continuity) to the global attractor of the Navier–Stokes equations as in the proper sense. In this paper, we will estimate the dimension of the global attractor , for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of the global attractors converge to the corresponding upper bounds for the global attractor as in the proper sense. [Copyright &y& Elsevier]

Abstract: In this paper we study a class of inequality problems for the stationary Navier–Stokes type operators related to the model of motion of a viscous incompressible fluid in a bounded domain. The equations are nonlinear Navier–Stokes ones for the velocity and pressure with nonstandard boundary conditions. We assume the nonslip boundary condition together with a Clarke subdifferential relation between the pressure and the normal components of the velocity. The existence and uniqueness of weak solutions to the model are proved by using a surjectivity result for pseudomonotone maps. We also establish a result on the dependence of the solution set with respect to a locally Lipschitz superpotential appearing in the boundary condition. [Copyright &y& Elsevier]