Suppose that m ≥ 2, numbers p1, …, pm ∈ (1, +∞] satisfy the inequality $$\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1$$ 1 p 1 + ... + 1 p m < 1 , and functions γ1 ∈ $${L^{{p_1}}}$$ L p 1 (ℝ1), …, γm ∈ $${L^{{p_m}}}$$ L p m (ℝ1) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγk ∈ $${L^{{p_k}}}$$ L p k (ℝ1) under which the resonance set of each function γk + Δγk coincides with that of γk for 1 ≤ k ≤ m, but $${\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty $$ ‖ ∫ 0 t ∏ k = 0 m [ γ k ( τ ) + Δ γ k ( τ ) ] d τ ‖ L ∞ ( ℝ 1 ) = ∞ . The notion of a resonance point and the resonance condition for functions in the spaces Lp(ℝ1), p ∈ (1, +∞], were introduced by the author in his previous papers.