On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned $(p;q)$-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the $(p;q)$-operators in this regard. Application of the fixed points of the $\rho$-$\theta$-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential $(p;q)$-difference Navier problems.