Abstract
Abstract We introduce ordered quasicontractions and g-quasicontractions in partially ordered metric spaces and prove the respective coincidence point and (common) fixed point results. An example shows that the new concepts are distinct from the existing ones. We also prove fixed point theorems for mappings satisfying so-called weak contractive conditions in the setting of partially ordered metric space. Hence, generalizations of several known results are obtained. Mathematics Subject Classification (2010): 47H10; 47N10.
Highlights
It is well known that the Banach contraction principle has been generalized in various directions
The existence of fixed points in partially ordered metric spaces was first investigated by Ran and Reurings [5], and by Nieto and Lopez [6,7]
Remark 4 If the four-term set in condition (3.2) is replaced by any of the following sets d(gx, gy), d(gx, gy), d(gx, fx), d(gy, fy), d(gx, gy), d(gx, fx), d(gy, fy), (d(gx, fy) + d(gy, fx)), similar conclusions for the existence of common fixed points of mappings f and g can be obtained
Summary
It is well known that the Banach contraction principle has been generalized in various directions. Theorem 1 Let (X, d, ≼) be a partially ordered metric space and let f, g : X ® X be two self-maps on X satisfying the following conditions: (i) fX ⊂ gX; (ii) gX is complete; (iii) f is g-nondecreasing; (iv) f is an ordered g-quasicontraction; (v) there exists x0 Î X such that gx0 ≼ fx0; (vi) if {gxn} is a nondecreasing sequence that converges to some gz Î gX gxn ≼ gz for each n Î N and gz ≼ g(gz).
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