Volume 67 , issue 4 ( 2015 ) | back |

A note on generalized Whitney maps | 233$-$245 |

**Abstract**

For metrizable continua, there exists the well-known notion of a Whitney map. Garcı a-Velazquez extends the definition of Whitney map for $C(X)$, where $X$ is an arbitrary continuum (not necessarily metrizable). He also shows that the examples he considers do not admit such generalized Whitney maps. In this paper we shall investigate the properties of continua which admit such generalized Whitney maps.

**Keywords:** Generalized Whitney map; hyperspace.

**MSC:** 54B20; 54F15

Generalized cone b-metric spaces and contraction principles | 246$-$257 |

**Abstract**

The concept of generalized cone b-metric space is introduced as a generalization of cone metric space, cone b-metric space and cone rectangular metric space. An analogue of Banach contraction principle and Kannan's fixed point theorem is proved in this space. Our result generalizes many known results in fixed point theory.

**Keywords:** Fixed point; cone rectangular metric space; cone b-metric space.

**MSC:** 47H10; 54H25

On solving parabolic equation with homogeneous boundary and integral initial conditions | 258$-$268 |

**Abstract**

In this paper we consider the second order parabolic partial differential equation with constant coefficients subject to homogeneous Dirichlet boundary conditions and initial condition containing nonlocal integral term. We derive first and second order finite difference schemes for the parabolic problem, combining implicit and Crank-Nicolson methods with two discretizations of the integral term. One numerical example is presented to test and illustrate the proposed algorithm.

**Keywords:** Finite difference method; stability estimate; parabolic equation; non-local condition; second-order of convergence.

**MSC:** 65M12; 65M06, 65M22

Partial isometries and norm equalities for operators | 269$-$276 |

**Abstract**

Let $H$ be a Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper we shall show that if $A ın B(H)$ is a nonzero closed range operator, then the injective norm $\Vert A^{*}\otimes A^{+}+A^{+}\otimes A^{*}\Vert_{\lambda}$ attains its minimal value 2 if and only if $A/\Vert A\Vert$ is a partial isometry. Also we shall give some characterizations of partial isometries and normal partial isometries in terms of norm equalities for operators. These characterizations extend previous ones obtained by A. Seddik in [On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities, J. Math. Anal. Appl. 351 (2009), 277--284], and by M. Khosravi in [A characterization of the class of partial isometries, Linear Algebra Appl. 437 (2012)].

**Keywords:** Closed range operator; Moore-Penrose inverse; injective norm; partial isometry; normal operator; EP operator; operator equality.

**MSC:** 47A30; 47A05, 47B15

Some spectral properties of generalized derivations | 277$-$287 |

**Abstract**

Given Banach spaces $\Cal{X}$ and $\Cal{Y}$ and Banach space operators $Aın L(\Cal{X})$ and $Bın L(\Cal{Y})$, the generalized derivation $\delta_{A,B} ın L(L(\Cal{Y},\Cal{X}))$ is defined by $\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concerned with the problem of transferring the left polaroid property, from operators $A$ and $B^{*}$ to the generalized derivation $\delta_{A,B}$. As a consequence, we give necessary and sufficient conditions for $\delta_{A,B}$ to satisfy generalized a-Browder's theorem and generalized a-Weyl's theorem. As an application, we extend some recent results concerning Weyl-type theorems.

**Keywords:** Left polaroid; elementary operator; finitely left polaroid.

**MSC:** 47A10; 47A53, 47B47

Korovkin type approximation theorem in $A^{\Cal I}_{2}$-statistical sense | 288$-$300 |

**Abstract**

In this paper we consider the notion of $A^\Cal {I}_2$-statistical convergence for real double sequences which is an extension of the notion of $A^\Cal {I}$-statistical convergence for real single sequences introduced by Savas, Das and Dutta. We primarily apply this new notion to prove a Korovkin type approximation theorem. In the last section, we study the rate of $A^\Cal {I}_2$-statistical convergence.

**Keywords:** Ideal; $A^\Cal {I}_2$-statistical convergence; positive linear operator; Korovkin type approximation theorem; rate of convergence.

**MSC:** 40A35; 47B38, 41A25, 41A36

A common fixed point theorem for new type compatible maps on partial metric spaces | 301$-$308 |

**Abstract**

In this paper, we introduce the concept of compatible maps of type $(I)$ and of type $(II)$ in partial metric space and prove a common fixed point theorem for four such maps on complete partial metric space.

**Keywords:** Fixed point; partial metric space; compatible maps of type (I) and of type (II).

**MSC:** 54H25; 47H10