In this paper, we establish some fixed point results in quasi-metric structures via modified ω-distance and using one control function, known as Jachymski function. Our results generalize the fixed point result of Alegre and Marín [Topology Appl. 203:32-41, 2016], and give an affirmative answer to the natural question of Alegre Gil et al. [Results Math. 74(4):1-9, 2019]. Apart from these, we utilize one of our results to study the solvability of a certain kind of fractal difference equation of networks communication.