Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞

It is well known in functional analysis that construction of \(k\)-order derivative in Sobolev space \(W_p^k\) can be performed by spreading the \(k\)-multiple differentiation operator from the space \(C^k.\) At the same time there is a definition of \((k,p)\)-differentiability of a function at an individual point based on the corresponding order of infinitesimal difference between the function and the approximating algebraic polynomial \(k\)-th degree in the neighborhood of this point on the norm of the space \(L_p\). The purpose of this article is to study the consistency of the operator and local derivative constructions and their direct calculation. The function \(f\in L_p[I], \;p>0,\) (for \(p=\infty\), we consider measurable functions bounded on the segment \(I\) ) is called \((k; p)\)-differentiable at a point \(x \in I\;\) if there exists an algebraic polynomial of \(\;\pi\) of degree no more than \(k\) for which holds \( \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), \) where \(\;J_h=[x_0-h; x_0+h]\cap I.\) At an internal point for \(k = 1\) and \(p = \infty\) this is equivalent to the usual definition of the function differentiability. The discussed concept was investigated and applied in the works of S. N. Bernshtein [1], A. P. Calderon and A. Sigmund [2]. The author's article [3] shows that uniform \((k, p)\)-differentiability of a function on the segment \(I\) for some \(\; p\ge 1\) is equivalent to belonging the function to the space \(C^k[I]\) (existence of an equivalent function in \(C^k[I]\)). In present article, integral-difference expressions are constructed for calculating generalized local derivatives of natural order in the space \(L_1\) (hence, in the spaces \(L_p,\; 1\le p\le \infty\)), and on their basis - sequences of piecewise constant functions subordinate to uniform partitions of the segment \(I\). It is shown that for the function \( f \) from the space \( W_p^k \) the sequence piecewise constant functions defined by integral-difference \(k\)-th order expressions converges to \( f^{(k)} \) on the norm of the space \( L_p[I].\) The constructions are algorithmic in nature and can be applied in numerical computer research of various differential models.