Applications of double composed metric space to integral and functional equations

A double composed metric space (DCMS) ( U , ρ ) is a generalized metric-type space in which the usual triangle inequality is replaced by the composed control inequality ρ ( u , w ) ≤ φ ( ρ ( u , v ) ) + ψ ( ρ ( v , w ) ) , u , v , w ∈ U , where φ , ψ : [ 0 , ∞ ) → [ 0 , ∞ ) are suitable control functions. This structure was introduced by Ayoob et al. (2023), where fixed point results were established and the development of nontrivial applications, especially to integral equations, was suggested as an important open direction. In this paper, we address this direction by providing concrete applications of double composed metric spaces to nonlinear integral and functional equations in nonmetric settings. In particular, we consider spaces such as L p [ 0,1 ] , 0 < p < 1 , where the usual metric triangle inequality fails and the classical Banach contraction principle is not directly applicable. We show that suitable function spaces can be treated as complete double composed metric spaces and then apply the generalized Banach type fixed point theorem in this framework to obtain existence and uniqueness results for nonlinear integral and functional equations.