On characterization of non-Newtonian superposition operators in some sequence spaces

Abstract

In this paper, we define a non-Newtonian superposition operator N P f where f : N × R(N) α → R(N) β by N P f (x) = f (k, x k) ∞ k=1 for every non-Newtonian real sequence x = (x k). Chew and Lee [4] have characterized P f : p → 1 and P f : c 0 → 1 for 1 ≤ p < ∞. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus. We characterize N P f : ∞ (N) → 1 (N) , N P f : c 0 (N) → 1 (N) , N P f : c (N) → 1 (N) and N P f : p (N) → 1 (N), respectively. Then we show that such N P f : ∞ (N) → 1 (N) is *-continuous if and only if f (k, .) is *-continuous for every k ∈ N.

ON *-BOUNDEDNESS AND *-LOCAL BOUNDEDNESS OF NON-NEWTONIAN SUPERPOSITION OPERATORS IN c_(0,α) AND c_α TO l_(1,β)

“…After, some properties of superposition operator, such as boundedness, continuity, were studied by Tainchai [3], Sama-ae [4], Sagır and Güngör [6] and many others. Non-Newtonian superposition operator was defined and characterized in some non-Newtonian sequence spaces by Sagır and Erdogan in [9]. In this article, we define *-boundedness and *-locally boundedness of operator.…”

“…) is β−bounded on every α−bounded subset of R α for all k ∈ N. Sagır and Erdogan [9] have characterized the non-Newtonian superposition operators N P f on c 0,α and c α as the following.…”

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Many investigations have been made about of non-Newtonian calculus and superposition operators until today. Non-Newtonian superposition operator was defined by Sagır and Erdogan in [9]. In this study, we have defined *-boundedness and *-locally boundedness of operator.

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On *-Continuity and *-Uniform Continuity of Some Non-Newtonian Superposition Operators

“…After, some properties of superposition operator, such as boundedness, continuity, were studied by Thuangoon [3], Sağır and Güngör [5] and many others [6,12]. Non-Newtonian superposition operator was defined and characterized in some non-Newtonian sequence spaces by Sağır and Erdoğan [8] . Also Erdoğan and Sağır [10] worked on *-boundedness and *-locally boundedness of some non-Newtonian superposition operators.…”