Gülen Başcanbaz-Tunca

Bivariate Bernstein type operators

In this paper, we introduce bivariate extension of Bernstein type operators defined in 11. We show that these operators preserve some properties of the original function f, such as Lipschitz constant and monotonicity. Furthermore, we present the monotonicity of the sequence of bivariate Bernstein type operators for n when f is ?-convex.

Some preservation properties of MKZ-Stancu type operators

In this work, we construct Stancu type modification of the generalization of Meyer-Konig and Zeller operators (MKZ) defined in [12]. We show that the Lipschitz constant of a Lipschitz continuous function and the properties of the function of modulus of continuity can be retained by these operators.

A SPECTRAL EXPANSION FOR SCHRÖDINGER OPERATOR

In this paper we consider the Schrodinger operator L generated in L 2 (R+) by y 00 + q(x)y = µy, x ∈ R+ := (0,∞) subject to the boundary condition y 0 (0) − hy (0) = 0, where,q is a complex valued function summable in (0,∞ and h 6 is a complex constant, µ is a complex parameter. We have assumed that sup x∈R+ {exp(e √ x)|q(x)|} 0, holds which is the minimal condition that the eigenvalues and the spec- tral singularities of the operator L are finite with finite multiplicities. Under this condition we have given the spectral expansion formula for the operator L using an integral representation for the Weyl func- tion of L. Moreover we also have investigated the convergence of the spectral expansion.

SPECTRAL PROPERTIES OF A NON SELFADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS WITH A SPECTRAL PARAMETER IN THE BOUNDARY CONDITION

and the boundary condition y(0) = 0 as Ly= ly,whereqis a complex-valued function. The spectral analysis of Lhas been studied by Naimark[7]. Naimark has proved that there are some poles of resolvent’s kernelwhich are not the eigenvalues of the operator L.(Schwartz [8] named thesepoints as spectral singularities of L).Moreover Naimark has proved thatspectral singularities are on the continuous spectrum, he has also shownthat Lhas a finite number of eigenvalues and spectral singularities withfinite multiplicities if the condition

SPECTRAL PROPERTIES OF THE KLEIN-GORDON s-WAVE EQUATION WITH SPECTRAL PARAMETER-DEPENDENT BOUNDARY CONDITION

We investigate the spectrum of the differential operator Lλ defined by the Klein-Gordon s-wave equation y′′ +(λ−q(x))2y = 0, x ∈R+ = [0,∞), subject to the spectral parameterdependent boundary condition y′(0)−(aλ+b)y(0)= 0 in the space L(R+), where a≠±i, b are complex constants, q is a complex-valued function. Discussing the spectrum, we prove that Lλ has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions limx→∞q(x) = 0, supx∈R+{exp(ε √ x)|q′(x)|} <∞, ε > 0, hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.

Spectral properties of a Schrödinger equation with a class of complex potentials and a general boundary condition

In this paper we investigate the spectrum and the spectral singularities of an operator L generalized in L2(R+) by the differential expression l(y)=y″−∑k=0n−1λkqk(x)y,x∈R+=[0,∞), and the boundary condition ∫0∞K(x)f(x)dx+αf′(0)−βf(0)=0, where λ is a complex parameter, qk, k=0,1,…,n−1, are complex valued functions, q0,q1,…,qn−1 are differentiable on (0,∞), K∈L2(R+), and α,β∈C with |α|+|β|≠0. Discussing the spectrum we obtain that L has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions limx→∞qk(x)=0,supx∈R+eex∑k=0n−1q′k(x)+K(x)<∞ hold, where k=0,1,…,n−1 and e>0.