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For any positive integer n and a set of positive integers m i {m}_{i} , i = 1 , 2 , … , n + 1 i=1,2,\ldots ,n+1 , we construct a class of quadratic eigenparameter-dependent boundary Sturm-Liouville problems with n transmission conditions, which have at most ∑ i = 1 n + 1 m i + n + 5 {\sum }_{i=1}^{n+1}{m}_{i}+n+5 eigenvalues. The key to this analysis is still the division of intervals and an iterative construction of the characteristic function. Further, some examples are given for a simple explanation.

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