Article type: Research Article
Authors: Humaira, a | Sarwar, Muhammada | Abdeljawad, Thabetb; c; d
Affiliations:
[a] Department of Mathematics, University of Malakand, Chakdara, Pakistan
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[b] Department of Mathematics and General Sciences, Price Sultan University, Riyadh, Saudi Arabia
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[c] Department of Medical Research, China Medical University, Taichung, Taiwan
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[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: [email protected].
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
