Exploring Extremum Points and Mean Value Theorems via Gâteaux Derivatives in Linear 2-Normed Spaces

Abstract

This paper explores differentiation within 2-normed spaces, which extend traditional normed spaces to provide deeper insights into linear structures related to real-valued functions. Building on the foundational contributions of Gähler and Iseki, we focus on Gâteaux and Fréchet derivatives and their essential roles in differentiability, particularly in identifying extremum points and developing mean value theorems that enable optimal solutions in variational problems. We also review recent works by Liu and Zhao, along with Patel et al., which clarify the practical applications of these differentiation concepts in optimization theory. Furthermore, we examine the relationship between Gâteaux and Fréchet derivatives, revealing conditions for their coincidence that enhance the theoretical framework for differentiability in higher dimensions. Through various theorems, we demonstrate the implications of our findings for both theoretical and practical contexts, highlighting the significance of 2-normed spaces in applied mathematics. This research contributes to ongoing discussions on differentiation and optimization while underscoring the practical importance of these mathematical structures in tackling real-world challenges, suggesting that future investigations could further solidify the role of 2-normed spaces in functional analysis.