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

Abstract

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;

$\xi \left(\tau \right)=\mathrm{g}\left(t\right)+\rho {\int }_0^{\tau }{K}_1(\tau ,\wp ){ϝ}_1(\wp ,\xi (\wp \left)\right)d\wp +\varrho {\int }_0^1{K}_2(\tau ,\wp ){ϝ}_2(\wp ,\xi (\wp \left)\right)d\wp ,$(0.1) such that $\xi ={\xi }_1+{\xi }_2,{\xi }_1,{\xi }_2\in \left(\mathrm{C}\right([0,1]),R)$ $\mathrm{g}={\mathrm{g}}_1+{\mathrm{g}}_2,{\mathrm{g}}_l:[0,1]\to R,l=1,2,$ ${ϝ}_l(\wp ,\xi (\wp \left)\right)={ϝ}_{l1}^{*}(\wp ,{\xi }_1^{*})+i{ϝ}_{l2}^{*}(\wp ,{\xi }_2^{*}),$ ${ϝ}_{\mathit{lj}}^{*}:[0,1]\times R\to R\mathrm{for}l,j=1,2,\mathrm{and}{\xi }_1^{*},{\xi }_2^{*}\in \left(\mathrm{C}\right([0,1]),R)$ $K_l(t,\wp )=K_{l1}^{*}(t,\wp )+{\mathit{iK}}_{l2}^{*}(t,\wp ),\mathrm{for}l,j=1,2\mathrm{and}{K}_{\mathit{lj}}^{*}:[0,1{]}^2\to R,$ 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.