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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.

Generalized contraction principle under relatively weaker contraction in partial metric spaces