We provide quantitative results on the asymptotic behavior of Dykstra’s algorithm with Bregman projections, a combination of the well-known Dykstra’s algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-orthogonal setting. The results we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence, which, in particular, contain the case of polyhedra in Euclidean spaces. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra’s method with Bregman projections in suitable infinite-dimensional spaces.
In this paper, we generalize the identities related to Steffensen's inequality using the Taylor formula by introducing a positive weight. For these generalized identities, we obtain bounds by applying the weighted Hermite-Hadamard inequality for a convex or concave function of the form |f((n))|(q). Furthermore, we generalize known weighted Hermite-Hadamard-type inequalities by using new identities associated with Steffensen's inequality.
The aim of this paper is twofold. First, we develop the notion of decay characters and evaluate these characters in important data spaces that often appear in problems related to fluid mechanics. Next, we apply our evaluation for the decay character of the initial datum r .u0/ to study the decay of solution and its derivatives to n-dimensional generalized Navier-Stokes equations with fractional-order Laplace operator .(-Delta) (alpha) when 0 < alpha < n+2 /4 . We establish the optimal uniform decay estimates for the L-2 -norm of solution u, as well as decay estimates ||Lambda(m)u||(2)(L) and||partial derivative(k)(t) Lambda (m)u||L-2 for positive integer k and non-negative real number m.
In this paper, we study positive stable weak solutions to the double phase problem {div(w(x)|del u|(p-2)del u + |del u|(r-2)del u) = f(x) u(-q) in R-+(N), {(w(x)|del u|(p-2) + |del u|(r-2)) partial derivative u/partial derivative eta = -u(-theta) on (sic)R-+(N) where p >= r >= 2, theta, q > 0, a < b+ p, and w; f is an element of L- loc(1).(R-+(N)) are nonnegative functions such that w(x)<= C-1|x|(a) and f(x) >= C-2|x|(b) for large |x|, where C-1; C-2 > 0. Under some appropriate assumptions on p, q, r, theta, a, b, and N , we prove the non-existence of positive stable weak solutions.
We consider the Dirichlet problem associated with a degenerate elliptic equation featuring both a singular lower-order term and a singular right-hand side. Despite the presence of degeneracy and singularities, we establish some existence results for bounded solutions, as well as existence results for distributional solutions under minimal assumptions on the data.
