In the present paper, we refine the notion of the partial modular metric defined by Hosseinzadeh and Parvaneh to eliminate the occurrence of discrepancies in the non-zero self-distance and triangular inequality. In support of this, we discuss non-trivial examples. Finally, we prove a common fixed-point theorem for four self-mappings in partial modular metric space and an application to our result; the existence of a solution for a system of Volterra integral equations is discussed.

In the present paper, we introduce the notion of C*-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space, C*-algebra-valued partial metric space, and C*-algebra-valued modular metric space and discuss it with examples. Some fixed point results using (ϕ,MF)-contraction mapping are discussed in such space. In addition, we study the stability of obtained results in the spirit of Ulam and Hyers. As an application, we also provide the existence and uniqueness of the solution for a system of Fredholm integral equations.

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