The paper introduces a generalization of the commuting mapping concept, termed "weakened commutativity," and explores its properties. This concept is used to generalize several fixed point theorems, including those by Park and Bae, Hadžić, and others. The author defines compatible mappings, which are less restrictive than commuting mappings, and demonstrates their utility in metric fixed point theory. The paper provides examples to illustrate the extent to which the commutativity requirement has been weakened. It also introduces $(\varepsilon, \delta)$-contractibility for four functions and uses this concept to prove the existence of unique common fixed points under certain conditions. The results are shown to generalize existing theorems and provide new insights into fixed point theory.The paper introduces a generalization of the commuting mapping concept, termed "weakened commutativity," and explores its properties. This concept is used to generalize several fixed point theorems, including those by Park and Bae, Hadžić, and others. The author defines compatible mappings, which are less restrictive than commuting mappings, and demonstrates their utility in metric fixed point theory. The paper provides examples to illustrate the extent to which the commutativity requirement has been weakened. It also introduces $(\varepsilon, \delta)$-contractibility for four functions and uses this concept to prove the existence of unique common fixed points under certain conditions. The results are shown to generalize existing theorems and provide new insights into fixed point theory.