Some fixed point results on weak partial metric spaces

Abstract

The concept of partial metric p on a nonempty set X was introduced by Matthews [13] and it was slightly modified by Heckmann [11] as weak partial metric. In [12], the authors studied fixed point result of new extension of Banach's contraction principle to partial metric space and give some generalized versions of the fixed point theorem of Matthews. In the present paper, we extend and generalize the previous results to weak partial metric spaces.

Generalized Hyers-Ulam stability of cubic functional inequality

“…A classical Banach contraction principle is one of the fundamental results in the fixed point theory, and then several authors have studied to generalize and improve the fixed point theory by applying new contractive conditions for operators and by replacing complete metric spaces with various abstract spaces (see e.g. [1], [7], [18], [22]).…”

Common fixed point of four self maps on dislocated metric spaces

“…Further, Matthews showed that the Banach Contraction principle is valid in partial metric spaces and can be applied in program verification. After that, many authors studied and generalized the results of Matthews (see, for example, [1,3,5,6]). …”

“…For more details, we can refer to [1,6]. (pm1) x = y if and only if p(x, x) = p(y, y) = p(x, y), (pm2) p(x, x) p(x, y),…”

“…It is clear that the partial metric space is a weak partial metric space, but the converse may not be true; see [6,Example 12]. (i) a sequence {x n } in X converges to x ∈ X if and only if p(x n , x) → p(x, x) as n → ∞; (ii) a sequence {x n } in X is called a Cauchy sequence if and only if lim m,n→∞ p(x n , x m ) exists (and are finite); (iii) a sequence {x n } in X is called a 0-Cauchy sequence if and only if p(x n , x m ) → 0 as m, n → ∞; (iv) the space (X, p) is said to be complete if every Cauchy sequence {x n } in X converges; (v) the space (X, p) is said to be 0-complete if every 0-Cauchy sequence {x n } in X converges to a point…”

Fixed Points for Two Pairs of Absorbing Mappings in Weak Partial Metric Spaces

“…In 1999, Heckmann [10] introduced the notion of weak partial metric spaces, which is a generalization of partial metric spaces. Some results for mappings in weak partial metric spaces have been recently obtained by [2] and [4].…”