A long-standing question in the theory of measures noncompactness is that for Kuratowski measure α defined on a metric space M, and every bounded subset B⊂M, there countable B0⊂B such α(B0)=α(B)? In this paper, we give an affirmative answer to above. It done by showing each nonempty set B Banach space, so strongly finitely representable B0, free ultrafilter U affinely isometric ultrapower [co(B...

We consider the second-order nonlinear difference equations of the form ∆(rn−1∆xn−1) + pn f (xn−k) = hn. We show that there exists a solution (xn), which possesses the asymptotic behaviour ‖xn − a ∑n−1 j=0 (1/r j) + b‖ = o(1), a,b ∈ R. In this paper, we extend the results of Agarwal (1992), Dawidowski et al. (2001), Drozdowicz and Popenda (1987), M. Migda (2001), and M. Migda and J. Migda (1988...

Abstract In this paper we estimate the Kuratowski and Hausdorff measures of noncompactness bounded subsets spaces vector-valued functions differentiable functions. To end, use a quantitative characteristic modeled on new equicontinuity-type concept classical characteristics related to pointwise relative compactness. We obtain regular in taken into consideration. The established inequalities red...

We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in R2. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent syst...

A method is proposed for detecting whether two CCITT group 4 images were scanned from the same document. Features are extracted from rectangular patches of text and compared with a modified Hausdorff distance measure. Two images are said to be ‘‘equivalent’’ (i.e., they were scanned from the same document) if the Hausdorff measure finds that a specified number of features are located within a g...