The purpose of this paper is to obtain a sufficient condition for a G-Cauchy sequence to be an M-Cauchy sequence in fuzzy metric spaces. Our main result provides a partial answer to the open question posed by V. Gregori and A. Sapena. For application, we give a new fuzzy version of the Banach fixed point theorem.

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

Fixed Point Result for αPf,g–Integral Contractive Mappings with Applications