Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

The weakly perturbed linear nonhomogeneous impulsive systems in the form ̇ ð ‘¥ = ð ´ ( ð ‘¡ ) ð ‘¥ + 𠜀 ð ´ 1 ( ð ‘¡ ) ð ‘¥ + ð ‘“ ( ð ‘¡ ) , ð ‘¡ ∈ â„ , ð ‘¡ ∉ ð ’¯ ∶ = { ð œ ð ‘– } ℤ , Δ ð ‘¥ | ð ‘¡ = ð œ ð ‘– = ð ›¾ ð ‘– + 𠜀 ð ´ 1 ð ‘– ð ‘¥ ( ð œ ð ‘– − ) , ð œ ð ‘– ∈ ð ’¯ ⊂ â„ , ð ›¾ ð ‘– ∈ â„ ð ‘› , and ð ‘– ∈ ℤ are considered. Under the assumption that the generating system (for 𠜀 = 0 ) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter 𠜀 with finitely many terms with negative powers of 𠜀 , and we suggest an algorithm of construction of these solutions.