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The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus ( q -calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t.  q -RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital  α -admissible and  α - ψ -contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus