Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria

Abstract

<p style='text-indent:20px;'>Crisis intervention in natural disasters is significant to look at from many different slants. In the current study, we investigate the existence of solutions for <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula>-integro-differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ D_q^{\alpha} u(t) + w\left(t , u(t), u'(t), D_q^{\beta} u(t), \int_0^t f(r) u(r) \, {\mathrm d}r, \varphi(u(t)) \right) = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with three criteria and under some boundary conditions which therein we use the concept of Caputo fractional <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula>-derivative and fractional Riemann-Liouville type <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-integral. New existence results are obtained by applying <inline-formula><tex-math id="M5">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-admissible map. Lastly, we present two examples illustrating the primary effects.</p>