On the Balanced Pantograph Equation of Mixed Type

We consider the balanced pantograph equation (BPE) \(y{\prime}\left(x\right)+y\left(x\right)={\sum }_{k=1}^{m}{p}_{k}y\left({a}_{k}x\right)\) $y\prime \left(x\right)+y\left(x\right)={\sum }_{k=1}^m{p}_{k}y\left(a_k,x\right)$ , where a_k, p_k > 0 and \({\sum }_{k=1}^{m}{p}_{k}=1\) ${\sum }_{k=1}^m{p}_k=1$ . It is known that if \(K={\sum }_{k=1}^{m}{p}_{k}{\text{ln}}{a}_{k}\le 0\) $K={\sum }_{k=1}^m{p}_k\text{ln}{a}_k\le 0$ 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., a₁ < 1 < a_m, and prove that, in this case, the BPE has a nonconstant solution y and that y(x) ~ cx^σ as x → ∞, where c > 0 and σ is the unique positive root of the characteristic equation \(P\left(s\right)=1-\sum_{k=1}^{m} {p}_{k}{a}_{k}^{-s}=0\) $P\left(s\right)=1-{\sum }_{k=1}^m{p}_k{a}_k^{-s}=0$ . We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞.