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

Article type: Research Article

Authors: Humaira, ^a | Sarwar, Muhammad^a | Abdeljawad, Thabet^{b; c; d}

Affiliations: [a] Department of Mathematics, University of Malakand, Chakdara, Pakistan | [b] Department of Mathematics and General Sciences, Price Sultan University, Riyadh, Saudi Arabia | [c] Department of Medical Research, China Medical University, Taichung, Taiwan | [d] Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Correspondence: [*] Corresponding author. Thabet Abdeljawad, Department of Mathematics and General Sciences, Price Sultan University, Riyadh, Saudi Arabia, Department of Medical Research, China Medical University, Taichung, and Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan. E-mail: jgovejero@ucm.es.

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; (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 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.

Keywords: (ψ, ϕ) contraction, mixed Volterra Fredholm-Hammerstein integral equation, complex valued fuzzy metric space

Keywords: Primary 47H10, secondary 54H25

DOI: 10.3233/JIFS-200459

Journal: Journal of Intelligent & Fuzzy Systems, vol. 40, no. 3, pp. 4065-4074, 2021