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; x(tau) = g(t) + rho integral(tau)(0) K-1(tau, P)F-1(P, xi(P)dP + rho integral(1)(0) K-2(tau, P)F-2(P, xi(P)dP, (0.1) such that xi = xi(1) + xi(2), xi(1), xi(2) is an element of (C([0, 1]), R) g = g(1) + g(2,) g(iota) : [0, 1] -> R, l = 1,2, F-l(P, xi(P)) = F-l1*(P, xi(1)*) + iF(l2)*(P, xi(2)*) F-ij*:[0, 1] x R -> R for l, j = 1, 2, and xi(1)*(,) xi(2)* is an element of (C([0, 1]), R) K-l(t, P) = K-l1* (t, P) + iK*(12)(t, P), for l, j = 1,2 and K-ij* : [0, 1](2) -> R, where rho and rho are constants, g(t), the kernels K-l(tau, P) and the nonlinear functions F-1(P, xi(P)), F-2(P, xi(P)) are continuous functions on the interval 0 <= tau <= 1. In this direction we apply fixed point results for self mappings with the concept of (psi, phi) 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.