bel and Euler Summation Formulas for SBV(R)

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

Asymptotic (statistical) periodicity in two-dimensional maps

<p style='text-indent:20px;'>In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.</p>

New generalized trapezoid type inequalities for complex functions defined on unit circle and applications

UDC 517.5 We establish new generalized trapezoid type inequalities for complex functions defined on unit circle via the function of bounded variation and the functions satisfying H¨older type condition. Using these results, quadrature rule formula is also provided.

Approximation of the Mumford–Shah functional by phase fields of bounded variation

In this paper, we introduce a new phase field approximation of the Mumford–Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an [Formula: see text]-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio–Tortorelli approximation, where the phase field is an [Formula: see text]-function, shows that the new model leads to sharper phase fields.

System of nonlocal resonant boundary value problems involving p-Laplacian

Our aim is to study the existence of solutions for the following system of nonlocal resonant boundary value problem $$\begin{array}{} \displaystyle (\varphi (x'))' =f(t,x,x'),\quad x'(0)=0, \quad x(1)=\int\limits_{0 }^{1}x(s){\rm d} g(s), \end{array}$$ where the function ϕ : ℝn → ℝn is given by ϕ (s) = (φp1(s1), …, φpn(sn)), s ∈ ℝn, pi > 1 and φpi : ℝ → ℝ is the one dimensional pi -Laplacian, i = 1,…,n, f : [0,1] × ℝn × ℝn → ℝn is continuous and g : [0,1] → ℝn is a function of bounded variation. The proof of the main result is depend upon the coincidence degree theory.

Generalized Baskakov Kantorovich operators

In this paper, we construct generalized Baskakov Kantorovich operators. We establish some direct results and then study weighted approximation, simultaneous approximation and statistical convergence properties for these operators. Finally, we obtain the rate of convergence for functions having a derivative coinciding almost everywhere with a function of bounded variation for these operators.