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In this article, we prove the generalized Hyers-Ulam stability for the following additive-quarticfunctional equation: f(x + 3y) + f(x - 3y) + f(x + 2y) + f(x - 2y) + 22f(x) + 24f(y) = 13[ f(x + y) + f(x - y)] + 12f(2y), where f maps from an additive group to a complete non-Archimedean normed space.
Content available remote Stability of an AQCQ functional equation in non-Archimedean (n, β)-normed spaces
In this paper, we adopt direct method to prove the Hyers-Ulam-Rassias stability of an additive-quadratic-cubic-quartic functional equation f(x+2y)+f(x-2y) = 4f(x+y)+4f(x-y)-6f(x)+f(2y)+f(-2y)-4f(y)-4f(-y) in non-Archimedean (n,β)-normed spaces.
Content available remote Hyers-Ulam stability of quadratic forms in 2-normed spaces
In this paper, we obtain Hyers-Ulam stability of the functional equations f(x+y, z+w) + f(x-y, z-w) = 2f(x, z) + 2f(y, w), f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) + 2f(y, w) and f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) - 2f(y, w) in 2-Banach spaces. The quadratic forms ax2+bxy+cy2, ax2+by2 and axy are solutions of the above functional equations, respectively.
H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.
In this paper, we examine the stability of Kirk-Ishikawa and Kirk-Mann iteration processes for nonexpansive and quasi-nonexpansive operators in uniformly convex Banach space. To the best of our knowledge, apart from the results of Olatinwo [19], stability of fixed point iteration processes has not been investigated in uniformly convex Banach space. Our results generalize, extend and improve some of the results of Harder and Hicks [11], Rhoades [26, 27], Osilike [23], Berinde [2, 3] as well as Imoru and Olatinwo [12].
Content available remote On the Torricellian point in inner product spaces
The concept of Torricellian point related to a set of n vectors in normed linear spaces is introduced and the general properties obtained. The existence and uniqueness of the Torricellian point in inner product spaces are established.
We establish coincidence point and common fixed point results for multivalued f-weak contraction mappings which assume closed values only. As an application, related common fixed point and invariant approximation are obtained in the setup of certain metrizable topological vector spaces. Our results provide extensions as well as substantial improvements of several well known results in the literature.
Content available remote A closedness theorem for normed spaces
For spaces X, Y, for which some algebraic operations are defined and in some cases topologies for X, Y are defined too, we define for the space X a dual space Xd with respect to the space Y. If [..] is a topology for Y (compatible with the algebraic operations of Y), then the pointwise topology rp for Yx is defined. We show that Xd is (algebraically)rp-closed in Yx. For normed spaces is shown that suitable subspaces of Xd are rp-closed in a product space K C Yx. As a corollary we obtain a generalization of Alaoglu's theorem.
The paper gives two theorems for the asymptotic correctness of an evolutionary algorithm (EA) that processes chromosomes from an arbitrary Banach space. It is argued that even if the mutation cannot yield an arbitrarily far offspring, the EA may be asymptotically correct provided that the selection is nonelitist and each feasible individual may reproduce with nonzero probability. An illustrative example accompanies the paper.
Content available remote Stability of the Euler-Lagrange-Rassias functional equation
Let F be a field, a1, a2 is an element of F, K is an element of {R, C}, s an element of K\{0,1}, X be a linear space over F, S C is contained in X be nonempty, and Y be a Banach space over K. Under some additional assumptions on S we show some stability results for the functional equation Q (a1x + a2y) + Q (a2X - a1y) = s[Q{x) + Q{y)} in the class of function Q : S -> Y.
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