A Partial metric space(G,ρ) gives a generalized mathematical basis that coordinates well with the approximate theory and invariant elements. It serves as an essential tool in theoretical computer science due to its applications in domain theory, recursion theory, denotation semantics, and distributed computing. This advantage prompted the current study to support the theoretical aspect of the subject. Here, there are two cases of work. The first one is finding a common fixed point in an ordered complete partial metric space (G,ρ,≼)for a (Μ,ψ,φ)-contractive map Γ Μ: G→G w.r.t. a map Μ: G→G and a sufficient contractive condition if Γ,Μ are two weakly increasing maps where ψ,φ are altering distance functions. The second is finding a coincidence point of a pair (Γ,f) and a coincidence point of a pair (Μ,m), where Γ,Mare multi-valued maps, f,m are single-valued maps. All these maps are involved in setting up a contractionary integral condition adopted in this part of the paper. The pair (Γ,f) is the strongest straight relying on (Μ,m). For this, the partial Hausdorff distance, as proposed by H. Aydi et al., was recalled. Numerical examples were also presented to illustrate these results and support our work.