On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term ( [ x ″ ( t ) ] α ) ′ + q ( t ) x α ( σ ( t ) ) + f ( t ) = 0 , t ≥ t 0 . Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when α is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case α = 1 , a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when α = 1 . Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions.