Abstract We consider ℓ p -direct sums (1 ≤ p < ∞ ) and c 0 -direct sums of countably many normed spaces and find the dual of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us answer the Chmieliński, Khurana, and Sain question raised in [4] on the existence of a non-approximately smooth normed space whose every element is smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.
The main aim of this paper is to use the k-Hilfer fractional derivative to derive certain P & oacute;lya-Szeg & ouml; fractional integral inequalities. Further fractional integral inequalities are obtained. The results presented here extend and generalize various existing inequalities associated with the Riemann-Liouville, Caputo, Saigo, and Hilfer fractional operators. The findings contribute to the growing theory of fractional calculus and offer potential tools for the analysis of fractional differential and integral equations.
In this manuscript, we accelerate the local convergence of a third-order biparametric iterative method in & Ropf; or & Copf; by assuming that the first-order Fr & eacute;chet derivative satisfies the Generalized continuity condition. We extend this analysis by using the H & ouml;lder continuity condition, which allows us to solve more numerical problems. Our study also shows the sizes of the convergence balls, the smallest error bounds that can be computed, and the fact that the answer is unique. Several math tests show that this third-order method gives better results than the midpoint method established by I.K. Argyros and S. George [4]. This method solves problems that earlier studies have not been able to solve.
The present paper investigates the asymptotic behavior of positive generalized self-similar solutions for a nonlinear hybrid problem involving nth-order derivative porous medium equations. We provide sufficient conditions for the existence and uniqueness of weak solutions that have compact support and dynamic characteristics. Furthermore, we establish the behavior of these solutions by examining a specific set of variables and their signs, which must meet certain conditions to determine whether the solutions exist globally or locally in time.
This article presents a unified framework that extends the scope of two existing theorems on multivariate Hardy-Hilbert-type integral inequalities. Key to this extension is the use of two additional adjustable parameters that increase flexibility and generality. The framework also has the originality of including the incomplete lower gamma function in the integral definitions governed by a parameter. Detailed proofs are given, mainly based on the Laplace transform, the generalized Young inequality, the generalized H & ouml;lder integral inequality and changes of variables. This article thus provides a new comprehensive foundation for future research in generalized multivariate integral inequalities.
