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In this paper, we establish new results on the existence, uniqueness, and convergence for a one-parameter family of mappings, where each member has a unique fixed point that is also the unique common fixed point of the entire family.These results apply to both contractive-type mappings and those that do not adhere to any contractive conditions.Our theorem includes, as special cases, the results of [B.D. Gel'man, Caristi's inequality and -contraction mappings,
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