Fixed points and fuzzy stability of an additive-quadratic functional equation
The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. Using fixed point method, we prove the Hyers-Ulam stability of the functional equation
Stability of cubic and quartic rho-functional inequalities in fuzzy normed spaces
In this paper, we solve the following cubic ρ-functional inequality N(f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x) (1) −ρ ( 4f ( x+ y 2 ) + 4f ( x− y 2 ) − f(x+ y)− f(x− y)− 6f(x) ) , t) ≥ t t+ φ(x, y) and the following quartic ρ-functional inequality N(f(2x+ y) + f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x) + 6f(y) (2) −ρ ( 8f ( x+ y 2 ) + 8f ( x− y 2 ) − 2f(x+ y)− 2f(x− y)− 12f(x) + 3f(y) ) , t) ≥ t t+ φ(x, y) in fuzzy normed spaces, where ρ is a fixed real number with ρ 6= 2. Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional inequality (1) and the quartic ρ-functional inequality (2) in fuzzy Banach spaces. c ©2016 All rights reserved.
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 ...