and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half–integer k and even dimensions, reproducing integer–order Sobolev spaces in even dimensions, but they turn out to have two additional non–polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the “dimensi...

We investigate the functions spaces on R for which the generalized partial derivatives Dk k f exist and belong to different Lorentz spacesΛkk w , where pk > 1 andw is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces, the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincr...

We consider pseudodifferential Douglis-Nirenberg systems on Rn with components belonging to the standard Hörmander class S∗ 1,δ(R n×Rn), 0 ≤ δ < 1. Parameter-ellipticity with respect to a subsector Λ ⊂ C is introduced and shown to imply the existence of a bounded H∞-calculus in suitable scales of Sobolev, Besov, and Hölder spaces. We also admit non pseudodifferential perturbations. Applications...

In this manuscript, the existence, uniqueness, and stability of solutions to terminal value problem Riemann-Liouville fractional equations are established in variable exponent Lebesgue spaces L p ( . ) $$ {L}^{p(.)} We convert using generalized intervals piece-wise constant function. Further, Banach contraction principle is used, Ulam-Hyers-stability examined, finally, we construct an example.

In this paper we present several extensions of theoretical tools for the analysis of Discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1,∞), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization pro...

In this work a dual-mixed approximation of a nonlinear generalized Stokes problem is studied. The problem is analyzed in Sobolev spaces which arise naturally in the problem formulation. Existence and uniqueness results are given and error estimates are derived. It is shown that both lowest-order and higher-order mixed finite elements are suitable for the approximation method. Numerical experime...