Abstract
In this paper, we introduce the concept of a generalized weak (phi,mathcal{R})-contraction and employ this to prove some fixed point results for self-mappings in partial metric spaces endowed with a binary relation mathcal{R}. We also establish some consequences in ordered partial metric spaces and metric spaces with a binary relation and exemplify that our results are a sharpened version of results of Zhiqun Xue (Nonlinear Funct. Anal. Appl. 21(3):497–500, 2016) and Alam and Imdad (J. Fixed Point Theory Appl. 17(4):693–702, 2015). Finally, we provide the existence of a solution for integral and fuzzy partial differential equations.
Highlights
The very first contribution to fixed point theory was due to Banach [3] in 1922
We introduce the notions, e.g. R-precompleteness, ρ-self-closedness and R-continuity in the setting of partial metric spaces endowed with a binary relation R and establish fixed point results for generalized weak (φ, R)-contraction mappings
Motivated by Alam and Imdad [18], we present the notion of R-continuity in the setting of partial metric spaces as follows
Summary
The very first contribution to fixed point theory was due to Banach [3] in 1922. He conferred the celebrated result in his thesis, namely the Banach contraction principle. We introduce the notions, e.g. R-precompleteness, ρ-self-closedness and R-continuity in the setting of partial metric spaces endowed with a binary relation R and establish fixed point results for generalized weak (φ, R)-contraction mappings.
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