On Extremal I-limit Points of Double Sequences

After F a s t [6] introduced the theory of statistical convergence of a real sequence, it has become popular among mathematicians ([2], [7]–[9], [17]). The ideas of statistical limit superior and limit inferior were first extensively studied by F r i d y and O r h a n [9]. After K o s t y r k o et al. [10] extended the idea of statistical convergence to I-convergence using the concept of an ideal I of the set of positive integers, much work has been done on different aspects of this convergence including I-limit points, I-cluster points, I-limit superior and limit inferior ( see [2], [4], [10]–[13]). Recently M u r s a l e e n and E d e l y [14] have introduced the concept of statistical convergence of double sequences and proved several basic properties. This was followed by D a s , K o s t y r k o , W i l c z y ń s k i and M a l i k [3] who introduced I and I-convergence of double sequences. As a natural consequence, in this paper, we introduce the concepts of I-limit points, I-cluster points, I-limit superior and limit inferior (automatically including the corresponding ideas with respect to statistical convergence) for double sequences, and we prove several results.