Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces.

The purpose of this article is to investigate the existence of unique solution for the following mixed nonlinear Volterra Fredholm-Hammerstein integral equation considered in complex plane; (0.1) ξ (τ) = g (t) + ρ ∫ 0 τ K 1 (τ , ℘) ϝ 1 (℘ , ξ (℘)) d ℘ + ϱ ∫ 0 1 K 2 (τ , ℘) ϝ 2 (℘ , ξ (℘)) d ℘ , such that ξ = ξ 1 + ξ 2 , ξ 1 , ξ 2 ∈ (C ([ 0 , 1 ]) , R) g = g 1 + g 2 , g l : [ 0 , 1 ] → R , l = 1 , 2 , ϝ l (℘ , ξ (℘)) = ϝ l 1 * (℘ , ξ 1 *) + i ϝ l 2 * (℘ , ξ 2 *) , ϝ lj * : [ 0 , 1 ] × R → R for l , j = 1 , 2 , and ξ 1 * , ξ 2 * ∈ (C ([ 0 , 1 ]) , R) K l (t , ℘) = K l 1 * (t , ℘) + iK l 2 * (t , ℘) , for l , j = 1 , 2 and K lj * : [ 0 , 1 ] 2 → R , where ρ and ϱ are constants, g (t), the kernels K_l (τ, ℘) and the nonlinear functions ϝ₁ (℘, ξ (℘)), ϝ ₂ (℘, ξ (℘)) are continuous functions on the interval 0 ≤ τ ≤ 1. In this direction we apply fixed point results for self mappings with the concept of (ψ, ϕ) contractive condition in the setting of complex-valued fuzzy metric spaces. This study will be useful in the development of the theory of fuzzy fractional differential equations in a more general setting. [ABSTRACT FROM AUTHOR]