By R. Shankar
In line with path fabric utilized by the writer at Yale collage, this sensible textual content addresses the widening hole discovered among the arithmetic required for upper-level classes within the actual sciences and the data of incoming scholars. This awesome publication bargains scholars a very good chance to reinforce their mathematical talents via fixing a number of difficulties in differential calculus. by means of protecting fabric in its least difficult shape, scholars can watch for a delicate access into any path within the actual sciences.
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Editor: John Michael Rassias, pp. 37-46 Chapter 3 S OME S TABILITY R ESULTS FOR E QUATIONS AND I NEQUALITIES C ONNECTED WITH THE E XPONENTIAL F UNCTION Włodzimierz Fechner and Roman Ger∗ Institute of Mathematics, Silesian University Bankowa 14, 40-007 Katowice, Poland Abstract We generalize some earlier results connected with the Hyers-Ulam stability of functional equations and inequalities related to the exponential function. In our first theorem we prove the stability of equation f = f ′ in reflexive normed spaces.
References [1] Altman, E. , A hybrid (differential-stochastic) zero-sum game with a fast stochastic part, 46–59, in New Trends in Dynamic Games and Applications (ed. ), Birkh¨auser, 1995. , Commitment and Observability in Games, Games and Economic Behavior, Volume 8, Issue 2, 271–280, 1995. [3] Brandts, J. , Reference Points and Negative Reciprocity in Simple Sequential Games, Games and Economic Behavior, Volume 36, Issue 2, 138–157, 2001. G. and Lygeros, J. ’s), Stochastic Hybrid Systems. CRC Press/Taylor and Francis Group, 2007.
That implies Q(2n ) = 4n Q(1), n ∈ N and for x = 1 2n we have: Q(1) = 4n Q(2−n ) ⇐⇒ Q(2−n ) = 4−n Q(1). For α ≤ 2 : sup x=0 |f (2n ) − Q(2n )| |f (x) − Q(x)| ≥ sup |x|α 2nα n∈N |4n n ln 2 − 4n Q(1)| 2nα n∈N = sup = sup 2n(2−α) |n ln 2 − Q(1)| = ∞. n∈N For α > 2 : sup x=0 |f (x) − Q(x)| |f (2−n ) − Q(2−n )| ≥ sup |x|α 2−nα n∈N |4−n n ln 2 − 4−n Q(1)| 2−nα n∈N = sup = sup 2n(α−2) |n ln 2 − Q(1)| = ∞. n∈N 51 52 L. G˘avrut¸a and P. G˘avrut¸a References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J.