Functional Equation fx pfx − 1 − qfx − 2 and Its Hyers-Ulam Stability
2.Moreover,asubstantialpartofproofofTheorem 2.1canbederivedfromtheoremspresentedinthebooks 19,20 .However,the theorems in 19, 20 deal with solutions of the linear equation under some regularityconditions, for example, the continuity, convexity, diﬀerentiability, analyticity and so on,while Theorem 2.1 deals with the general solution of 1.7 without regularity conditions.
A NOTE ON STABILITY OF THE LINEAR FUNCTIONAL EQUATIONS OF HIGHER ORDER AND FIXED POINTS OF AN OPERATOR
We prove two general theorems, which appear to be very useful in investigation of the Hyers-Ulam stability of a higher order linear functional equation in single variable, with constant coecients. We give several examples of their applications. In particular we show that we obtain in this way several fixed point results for a particular operator. The main tool in the proofs is a complexification of a real normed (or Banach) space X, which can be described as the tensor product X R 2 endowed with the Taylor norm.
Sharp discontinuity tracking applied to explosion problems
Abstract Explosion dynamics problems involve the propagation and interaction of shocks and contact discontinuities, usually in several space dimensions. Since standard finite difference methods which smear these discontinuities do not give sufficiently accurate results, a sharp discontinuity tracking method which explicitly follows the surfaces of discontinuity has been developed. This method is applied to explosion problems in one and two space dimensions.