On quaternion valued rectangular S-metric space

Abstract

A metric space can be seen as a distance space having a geometric structure, with only a few axioms. In this paper we introduce the concept of quaternion valued rectangular S metric spaces. The paper treats material concerning quaternion valued rectangular S metric spaces that is important for the study of fixed point theory in Clifford analysis. We introduce the basic ideas of quaternion valued rectangular S metric spaces and Cauchy sequences and discuss the completion of a quaternion valued rectangular S metric space. In this work, we will work on H, the skew field of quaternions. This means we can write each element q ∈ H in the form q = a + bi + cj + dk where a, b, c, d ∈ R and i, j, and k are the fundamental quaternion units. For these elements we have the multiplication rules I 2 = j 2 = k 2 = −1, ij = −ji = k, kj = −jk = −i and ki = −ik = j. The conjugate element is given by q = a−bi−cj −dk. The quaternion modulus has the form of |q|= √a 2 + b 2 + c 2 + d 2. Quaternions can be defined in several different equivalent ways. Quaternion is non commutative in multiplication. There is also more abstract possibilty of treating quaternions as simply quadruples of real numbers [a, b, c, d], with operation of addition and multiplication suitably defined. The components naturally group into the imaginary part (b, c, d), for which we take this part as a vector and the purely real part, a, which called a scalar. Sometimes, we write a quaternion as [a, V ] with V = (b, c, d). For more information about metric spaces, its generalization and quaternion analysis