Wijsman statistical, strong cesaro, and ideal convergence of closed sets in idempotent bicomplex metric spaces

Abstract

This paper studies generalized Wijsman convergence for sequences of nonempty closed subsets of an idempotent bicomplex metric space. Let rho = d(1)e(1) + d(2)e(2), where d(1) and d(2) are metrics on the same underlying set X. For a nonempty subset A of X, we define the bicomplex point-to-set distance by rho(x, A) = d(1)(x, A)e(1) + d(2)(x, A)e(2). This definition retains the idempotent decomposition of the metric and allows the convergence of closed sets to be studied through the two component distance functions. For an exponent sequence p = (p(k)) satisfying 0 < inf(k)p(k) <= sup(k)p(k) < infinity, we introduce Wijsman statistical convergence, Wijsman strong Cesaro convergence of power type p, Wijsman I -convergence, and Wijsman I K -convergence. We prove that each of these notions is equivalent to the simultaneous validity of the corresponding Wijsman convergence conditions with respect to the component metrics d(1) and d(2). As consequences, ordinary Wijsman convergence implies Wijsman statistical convergence, while Wijsman strong Cesaro convergence implies Wijsman statistical convergence. The converse implication holds when the component distance deviations are pointwise bounded. For admissible ideals I and K , we show that W - I p K -convergence implies W - I p -convergence for every sequence if and only if K subset of I . We also compare the componentwise bicomplex point-to-set distance with the point-to-set distance determined by the associated real metric. Examples show that these two constructions need not agree and that the principal implications obtained in the paper cannot, in general, be reversed.