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Let $X$ be a non-empty set. We say that an element $x\in X$ is a $\varphi$-fixed point of $T$, where $\varphi: X\to [0,\infty)$ and $T: X\to X$, if $x$ is a fixed point of $T$ and $\varphi(x)=0$. In this paper, we establish some existence results of $\varphi$-fixed points for various classes of operators in the case, where $X$ is endowed with a metric $d$. The obtained results are used to deduce some fixed point theorems in the case where $X$ is endowed with a partial metric $p$

Fixed point theory in partial metric spaces via φ-fixed point’s concept in metric spaces