By Yeol Je Cho

This e-book discusses the swiftly constructing topic of mathematical research that bargains essentially with balance of useful equations in generalized areas. the basic challenge during this topic used to be proposed by way of Stan M. Ulam in 1940 for approximate homomorphisms. The seminal paintings of Donald H. Hyers in 1941 and that of Themistocles M. Rassias in 1978 have supplied loads of idea and information for mathematicians all over the world to enquire this huge area of research.

The publication provides a self-contained survey of contemporary and new effects on issues together with easy idea of random normed areas and similar areas; balance thought for brand spanking new functionality equations in random normed areas through mounted element strategy, less than either precise and arbitrary t-norms; balance conception of famous new useful equations in non-Archimedean random normed areas; and purposes within the type of fuzzy normed areas. It comprises important effects on balance in random normed areas, and is aimed at either graduate scholars and study mathematicians and engineers in a huge zone of interdisciplinary research.

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**Extra info for Stability of functional equations in random normed spaces**

**Example text**

4 Every RN-space (X, μ, T ) is a Hausdorff space. Proof Let (X, μ, T ) be an RN-space. Let x and y be two distinct points in X and t > 0. Then 0 < μx−y (t) < 1. Put r = μx−y (t). For each r0 ∈ (r, 1), there exists r1 such that T (r1 , r1 ) ≥ r0 . Consider the open balls Bx (1 − r1 , 2t ) and By (1 − r1 , 2t ). Then, clearly, Bx (1 − r1 , 2t ) ∩ By (1 − r1 , 2t ) = ∅. In fact, if there exists z ∈ Bx 1 − r1 , t t ∩ By 1 − r1 , , 2 2 then we have r = μx−y (t) t t , μy−z 2 2 ≥ T μx−z ≥ T (r1 , r1 ) ≥ r0 > r, which is a contradiction.

3 Random Functional Analysis 37 Let y1 = 0 and choose y2 ∈ W such that 1 Eλ,μ x1 − (x2 − y2 ), t ≤ Eλ,μ¯ (x1 − x2 ) + W + . 2 1 1 2 However, Eλ,μ ¯ ((x1 − x2 ) + W ) ≤ 2 and so Eλ,μ (x1 − (x2 − y2 )) ≤ ( 2 ) . Now, suppose that yn−1 has been chosen. Then choose yn ∈ W such that Eλ,μ (xn−1 + yn−1 ) − (xn + yn ) ≤ Eλ,μ¯ (xn−1 − xn ) + W + 2−n+1 . Hence, we have Eλ,μ (xn−1 + yn−1 ) − (xn + yn ) ≤ 2−n+2 . 15, for each positive integer m > n and λ ∈ (0, 1), there exists γ ∈ (0, 1) such that Eλ,μ (xm + ym ) − (xn + yn ) ≤ Eγ ,μ (xn+1 + yn+1 ) − (xn + yn ) + · · · + Eγ ,μ (xm + ym ) − (xm−1 + ym−1 ) m ≤ 2−i .

This completes the proof. 4 Let (X, · ) be a normed linear space. Define a ∗ b = ab or a ∗ b = min(a, b) and N (x, t) = kt n kt n + m x for all k, m, n ∈ R+ . Then (X, N, ∗) is a fuzzy normed space. In particular, if k = n = m = 1, then we have t , N (x, t) = t+ x which is called the standard fuzzy norm induced by the norm · . 5 Let (X, N, ∗) be a fuzzy normed space. If we define M(x, y, t) = N (x − y, t), then M is a fuzzy metric on X, which is called the fuzzy metric induced by the fuzzy norm N .