Existence of Solutions of Periodic Boundary Value Problems for Impulsive Functional Duffing Equations at Nonresonance Case. (10th September 2008) - Liu, Xingyuan; Liu, Yuji
Abstract : This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations:x ′ ′ ( t ) + α x ′ ( t ) + β x ( t ) = f ( t, x ( t ), x ( α 1 ( t ) ), …, x ( α n ( t ) ) ),   a.e.     t ∈ [ 0, T ],   Δ x ( t k ) = I k ( x ( t k ), x ′ ( t k ) ),   k = 1, …, m,   Δ x ′ ( t k ) = J k ( x ( t k ), x ′ ( t k ) ),   k = 1, …, m,   x ( i ) ( 0 ) = x ( i ) ( T ),   i = 0, 1 . Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.