On the balanced pantograph equation of mixed type

UDC 517.9 We consider the balanced pantograph equation (BPE) y ′ ( x ) + y ( x ) = ∑ k = 1 m p k y ( a k x ) , where a k , p k > 0 and ∑ k = 1 m p k = 1.  It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist.  In the present paper, we deal with a BPE of mixed type, i.e., a 1 < 1 < a m , and prove that, in this case, the BPE has a nonconstant solution y and that y ( x ) ∼ c x σ as x → ∞ ,   where c > 0 and σ is the unique positive root of the characteristic equation P ( s ) = 1 - ∑ k = 1 m p k a k - s = 0.  We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞ .