By John Michael Rassias
This e-book is a discussion board for replacing rules between eminent mathematicians and physicists, from many components of the realm, as a tribute to the 1st centennial birthday anniversary of Stanislaw Marcin ULAM. This assortment consists of exceptional contributions in mathematical and actual equations and inequalities and different fields of mathematical and actual sciences. it really is meant to spice up the co-operation between mathematicians and physicists engaged on a wide number of natural and utilized mathematical parts. This transatlantic number of mathematical principles and techniques includes a large quarter of functions within which equations, inequalities and computational innovations pertinent to their options play a center position. Ulam's impact has been large on our way of life, simply because new instruments were constructed, and innovative study effects were completed , bringing scientists of actual sciences even nearer, via fostering the emergence of latest ways, concepts and views.
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Extra info for Functional Equations, Difference Inequalities and Ulam Stability Notions (F.U.N.) (Mathematics Research Developments)
Sample text
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.