Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

Abstract

Abstract We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales ( a ( t ) x Δ 2 ( t ) ) Δ 2 = b ( t ) f ( x ( t ) ) + c ( t ) , {\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right), such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour x ( t ) = y ( t ) + o ( 1 t β ) x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right) . The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.