@article{Mesmouli2026, 
author = {Mouataz Billah Mesmouli and Yasir A. Madani and Ioan-Lucian Popa and Taher S. Hassan},
title = {On nonlinear    (  p  ,  q  )-fractional hybrid pantograph equations: Existence, and uniqueness},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {11546-11558},
keywords = {(p, q)-fractional derivative, hybrid pantograph equation, fixed point methods, Banach contraction principle, Dhage's theorem, proportional delays},
url = {https://www.sciopen.com/article/10.3934/math.2026475},
doi = {10.3934/math.2026475},
abstract = {We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type    (  p  ,  q  )-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the    (  p  ,  q  )-fractional setting.}
}