The concept of measures of noncompactness for operators has been introduced and thoroughly investigated by several authors (see, for instance, [2, 4, 8]). In this work, we extend this concept to the setting of multivalued operators. We employ an axiomatic framework to define measures of noncompactness for bounded linear relations, including those associated with upper semi-Fredholm perturbations. Leveraging these measures, we derive new results for certain classes of semi-Fredholm and Fredholm multivalued operators. Lastly, we establish localization results concerning some essential spectra of bounded linear relations on Banach spaces.
We consider the numerical solution of a ordinary differential equation with a discontinuity in its derivative. Specifically, we assume that the location of the discontinuity has been detected, but that a numerical error exists in this location. We study how this error propagates in the subsequent numerical solution beyond the discontinuity. We derive an analytical expression for this error, which exposes terms that can only be controlled, if possible, prior to the advent of the discontinuity. We also find a condition number that can amplify the location error significantly. An example illustrates our findings.
The concept of a trigonometrically convex function consists in replacing the linear weights inherent in the classical concept of convexity with trigonometric ones. It emerged in 2018, combined with a framework of integral inequalities. In this article, we innovate by establishing valuable bounds for the integral of the product of two functions, including at least one such trigonometrically convex function. To derive these bounds, we use a number of analytical tools, including sharp trigonometric inequalities and the Hermite–Hadamard, Chebyshev and Hölder integral inequalities. Our results are flexible; they can accommodate different convexity conditions. Detailed proofs are given for completeness.
A computational error was found in the above-mentioned paper. Although the error seemed minor at first and did not appear to alter the main conclusions, a more careful re-evaluation has uncovered additional insights into the structure and properties of rotatable 2 × 2 matrices.
This paper presents closed-form evaluations of two new Apéry-like series of weight 4 that involve harmonic numbers of the form H_2k. Several key results are derived and subsequently used to establish connections to the main series.
