A Novel Integral-Type CLR Condition for Common Fixed Points of Weakly Compatible Mappings in Neutrosophic Metric Spaces with Applications to Dynamic Programming

This chapter establishes a new common fixed-point theorem for two pairs of weakly compatible mappings in neutrosophic metric spaces. The mappings are studied under an integral-type contractive condition formulated through the common limit range property. By combining altering-distance functions, compatibility conditions, and neutrosophic metric structures, the proposed theorem generalizes and unifies several earlier results from fuzzy, intuitionistic fuzzy, and classical fixed-point theory. The theoretical framework does not require some of the restrictive assumptions commonly imposed in related fixed-point results, thereby broadening its applicability. The principal theorem is applied to functional equations arising in multistage decision processes within dynamic programming. The application proves both the existence and uniqueness of solutions under the stated conditions. A nontrivial illustrative example supports the validity of the main findings. The results provide a foundation for further investigations involving multivalued mappings, differential and integral equations, optimization, control theory, and computational methods under neutrosophic uncertainty.