Some important results for the conformable fractional stochastic pantograph differential equations in the L p space

‎Important mathematical topics include existence‎, ‎uniqueness‎, ‎continuous dependency‎, ‎regularity‎, ‎and the averaging principle‎. ‎In this research work‎, ‎we establish these results for the conformable fractional stochastic pantograph differential equations (CFSPDEs) in \(\mathbf{L}^{\mathrm{p}}\) space‎. ‎The situation of \(\mathrm{p}=2\) is generalized by the obtained findings‎. ‎First‎, ‎we establish the existence and uniqueness results by applying the contraction mapping principle under a suitably weighted norm and demonstrating the continuous dependency of solutions on both the initial values and fractional exponent \(\phi\)‎. ‎The second section is devoted to examining the regularity of time‎. ‎As a result‎, ‎we find that‎, ‎for each \(\Phi\in(0,\phi-\frac{1}{2})\)‎, ‎the solution to the considered problem has a \(\Phi\)-Hölder continuous version‎. ‎Next‎, ‎we study the averaging principle by using Jensen's‎, ‎Grönwall-Bellman's‎, ‎Hölder's‎, ‎and Burkholder-Davis-Gundy's inequalities‎. ‎To help with the understanding of the theoretical results‎, ‎we provide three applied examples at the end‎. ‎