Nonlinear higher-order dynamic equation with polynomial growth and mixed boundary conditions

This paper investigates the existence of solutions for a class of non-linear higher-order dynamic equations subject to mixed boundary conditions. We consider boundary value problems in which the non-linear reaction functions satisfy polynomial growth conditions both in the interior of the domain and on the boundary. Our analysis employs a systematic approach based on fixed-point theorems for expansive mappings combined with completely continuous operators to establish stronger existence results. Under appropriate growth conditions on the non-linear terms, we first prove the existence of at least one classical solution, which is not guaranteed to be non-negative. We then strengthen our hypotheses to establish the existence of at least three non-negative solutions. The theoretical framework relies on cone theory and carefully constructed open bounded subsets within function spaces equipped with appropriate norms. Our methodology provides a unified approach to multiple solution problems for higher-order dynamic equations.