$I_{\lambda}$-convergence in intuitionistic fuzzy $n$-normed linear space
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The notion of lacunary ideal convergence in intuitionistic fuzzy normed linear space (IFNLS) was introduced by the present corresponding author [P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (2012), 708-715] and an open problem in that paper was whether every lacunary $I$-convergent sequence is lacunary $I$-Cauchy. Further, a new concept of convergence of sequences in an intuitionistic fuzzy $n$-normed linear space (IFnNLS) was given in [M. Sen, P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy $n$-normed linear spaces, Math. Comput. Modelling, 54 (2011), 2978-2985]. With the help of this new definition of convergence, the main aim of this paper is to introduce the concept of $I_{\lambda}$-convergence in an IFnNLS, where $I$ is an ideal of a family of subsets of positive integers $\mathbb{N}$. We also define $I_{\lambda}$-limit points and $I_{\lambda}$-cluster points and establish relations between them. Finally we introduce the notion of $I_{\lambda}$-Cauchy sequence in IFnNLS. We improve and extend some existing results and give a positive answer to the open problem mentioned above in the setting of an IFnNLS.
Abstract
The notion of lacunary ideal convergence in intuitionistic fuzzy normed linear space (IFNLS) was introduced by the present corresponding author [P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (2012), 708-715] and an open problem in that paper was whether every lacunary $I$-convergent sequence is lacunary $I$-Cauchy. Further, a new concept of convergence of sequences in an intuitionistic fuzzy $n$-normed linear space (IFnNLS) was given in [M. Sen, P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy $n$-normed linear spaces, Math. Comput. Modelling, 54 (2011), 2978-2985]. With the help of this new definition of convergence, the main aim of this paper is to introduce the concept of $I_{\lambda}$-convergence in an IFnNLS, where $I$ is an ideal of a family of subsets of positive integers $\mathbb{N}$. We also define $I_{\lambda}$-limit points and $I_{\lambda}$-cluster points and establish relations between them. Finally we introduce the notion of $I_{\lambda}$-Cauchy sequence in IFnNLS. We improve and extend some existing results and give a positive answer to the open problem mentioned above in the setting of an IFnNLS.
참고문헌 (38)
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