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∫0τK1(τ,℘)ϝ1(℘,ξ(℘))d℘+ϱ∫01K2(τ,℘)ϝ2(℘,ξ(℘))d℘, such that ξ=ξ1+ξ2,ξ1,ξ2∈(C([0,1]),R) g=g1+g2,gl:[0,1]→R,l=1,2, ϝl(℘,ξ(℘))=ϝl1*(℘,ξ1*)+iϝl2*(℘,ξ2*), ϝlj*:[0,1]×R→Rforl,j=1,2,andξ1*,ξ2*∈(C([0,1]),R) Kl(t,℘)=Kl1*(t,℘)+iKl2*(t,℘),forl,j=1,2andKlj*:[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.

中文翻译：

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复值模糊度量空间中非线性混合Volterra Fredholm-Hammerstein积分方程的唯一解的存在性

本文的目的是研究在复杂平面中考虑的以下混合非线性Volterra Fredholm-Hammerstein积分方程的唯一解的存在；（0.1）ξ（τ）= g（t）+ρ∫0τK1（τ，℘）ϝ1（℘，ξ（℘））d℘+ ϱ∫01K2（τ，℘）ϝ2（℘，ξ（℘）） d℘，ξ=ξ1+ξ2，ξ1，ξ2∈（C（[0,1]），R）g = g1 + g2，gl：[0,1]→R，l = 1,2， （℘，ξ（℘））=ϝl1*（℘，ξ1*）+iϝl2*（℘，ξ2*），ϝlj*：[0,1]×R→Rforl，j = 1,2，ξ1*，ξ2 *∈（C（[0,1]），R）Kl（t，℘）= Kl1 *（t，℘）+ iKl2 *（t，℘），forl，j = 1,2和Klj *：[0,1 ] 2→R，其中ρ和ϱ是常数g（t），核K l（τ，℘）和非线性函数ϝ1（℘，ξ（℘）），ϝ2（℘，ξ（℘））是在区间0≤τ≤1上的连续函数。在这个方向上，在复值模糊度量空间的设置中，我们使用（ψ，ϕ）压缩条件的概念对自映射应用定点结果。