By Bernard R. Gelbaum

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3] W. Fechner, On some inequalities connected with the exponential function, Arch. Math. (Brno) 44 (2008), No. 3, 217–222. [4] W. Fechner, A functional characterization of two inequalities between means, (submitted). [5] Z. Gajda, On stability of the Cauchy equation on semigroups, Aequationes Math. 36 (1988), No. 1, 76–79. [6] Z. Gajda, Invariant means and representations of semigroups in the theory of func´ ¸ skiego w Katowicach [Scientific tional equations, Prace Naukowe Uniwersytetu Sla ´ ¸ ski, Katowice, 1992.

Y−x (16) Then the estimate (15) holds true. Proof. It suffices to apply Proposition 2, then [3, Theorem 2], Remark 1 and the Hahn– Banach Theorem. One may obtain analogous results for equation (13). It suffices to apply [4, Theorem 7] and repeat the reasoning used previously. Proposition 3. Given an ε ≥ 0, let f0 : I → R satisfy f0 (y) − f0 (x) ≤ 4f0 y−x 6 x+y 2 + f0 (x) + f0 (y) + ε, for each x, y ∈ I such that x < y. Then f0 (x) = f (x) − 61 ε for x ∈ I, where f is a solution of (5). Theorem 4.

Capture Conditions in a Pursuit–Evasion Game between Players with Biproper Dynamics, Journal of Optimization Theory and Applications , 126:3, 503– 528, 2005. , A Folk Theorem for Repeated Sequential Games, The Review of Economic Studies, Volume 69:2 493–512(20), 2002. In: Functional Equations, Difference Inequalities... ISBN 978-1-60876-461-7 c 2010 Nova Science Publishers, Inc. 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.