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Applying quantum calculus for the existence of solution of \(q\)-integro-differential equations with three criteria. (English) Zbl 1469.34103

Summary: 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 \(q\)-integro-differential equation \[ D_q^\alpha u(t) + w \left(t,u(t),u^\prime(t), D_q^\beta u(t), \int_0^t f(r)u(r) \mathrm{d}r, \varphi(u(t)) \right) = 0, \] with three criteria and under some boundary conditions which therein we use the concept of Caputo fractional \(q\)-derivative and fractional Riemann-Liouville type \(q\)-integral. New existence results are obtained by applying \(\alpha\)-admissible map. Lastly, we present two examples illustrating the primary effects.

MSC:

34K37 Functional-differential equations with fractional derivatives
39A13 Difference equations, scaling (\(q\)-differences)
45J05 Integro-ordinary differential equations
47H10 Fixed-point theorems
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