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 Kl (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.