Fixed Point Theorems and Asymptotically Regular Mappings in Partial Metric Spaces

Abstract

The notion of asymptotically regular mapping in partial metric spaces is introduced, and a fixed point result for the mappings of this class is proved. Examples show that there are cases when new results can be applied, while old ones (in metric space) cannot. Some common fixed point theorems for sequence of mappings in partial metric spaces are also proved which generalize and improve some known results in partial metric spaces.

Fixed points of terminating mappings in partial metric spaces

“…More on applications of fixed point theory can be found in [3][4][5][6][7][8][9]. Furthermore, spaces and mappings are very important when studying fixed points; metric spaces, partial metric spaces, fuzzy spaces, smooth spaces, contractive mappings, monotone mappings and so on, see [1,2,4,[10][11][12][13][14][15][16].…”

“…In 2004, Oltra and Valero [19] also generalized the Matthews's fixed point theorem in a complete partial metric space, in the sense of O'Neill. In 2013, Shukla et al [16] introduced the notion of asymptotically regular mappings in a partial metric space, and established some fixed point results. Recently, Onsod et al [8] established some fixed point results in a complete partial metric space endowed with a graph.…”

Generalized Cyclic p-Contractions and p-Contraction Pairs Some Properties of Asymptotic Regularity Best Proximity Points, Fixed Points

“…In this case, T : X → X is a so-called p-contraction that was proposed and discussed in [1] as an expansion of Banach's contraction principle. See also [2] for the conceptual extensions to the proposed p-contraction pairs as well as for the discussion of their properties. Other typical contractive mappings are also included as particular cases of ( 1) satisfying the general given constraints.…”

“…The precise formal definition is recalled later on in the next section. Basically, the well-known Banach's contraction principle is weakened by incorporating the absolute value of the difference between the distances of any involved points x, y ∈ X to their images through T. It is well-known that an important support as an "a priori" property for the Cauchyness of sequences and their convergence to fixed points is that the mappings be asymptotically regular; basically, the sequences of distances between points and their images through the involved mapping converge asymptotically to zero [2]. On the other hand, in the rich existing background literature on fixedpoint theory and its applications, special attention is paid to q-cyclic contractions on the union of q(≥ 2) subsets in metric spaces.…”

Some Common Fixed Point Theorems in 0-σ-Complete Metric-Like Spaces

“…Further, Matthews showed that the Banach contraction principle is valid in partial metric spaces and can be applied in program verification. Later, several authors generalized the result of Metthews (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31]). O'Neill [22] generalized the concept of partial metric space a bit further by admitting negative distances.…”