Symmetry Methods and Fixed Point Theory for Positive Solutions of a Twelfth-Order Boundary Value Problem with Applications

In this paper, we investigate the existence and positivity of solutions for a class of twelfth-order nonlinear boundary value problems that naturally arise in the mathematical modeling of elastic and micro-mechanical systems. The considered model incorporates higher-order derivatives to account for nonlocal and gradient effects that commonly appear in the analysis of micro- and nano-scale elastic structures. By employing the Leray–Schauder nonlinear alternative and fixed point theorems, we establish sufficient conditions for the existence of at least one positive solution. The analysis relies on the explicit construction and properties of the associated Green’s function, which plays a fundamental role in deriving upper and lower bounds for the nonlinear term. The obtained results extend and generalize earlier works on sixth, eighth and tenth-order problems to the twelfth-order case. Finally, numerical examples are presented to illustrate the applicability and accuracy of the theoretical findings. The results provide a rigorous analytical foundation for the study of high-order elastic models and micro-scale structural stability.