Renewal-type and renewal equations usually do not have analytical solutions, and hence bounds for the functions satisfying such equations have a great practical importance. In this paper, sequences of monotone non-decreasing general lower bounds and sequences of monotone non-increasing general upper bounds for a general renewal-type equation converging to the function under interest, are given. Similar sequences of such two-sided bounds are given for the renewal function of an ordinary renewal process which converge to the renewal function and are improvements of the famous corresponding bounds of Marshall (1973). Also, such sequences of bounds converging to the ordinary renewal function, are obtained for several reliability classes of the lifetime distributions of the inter-arrival times. Finally, sequences of such two-sided bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time (excess lifetime).

更新型和更新方程通常没有解析解，因此满足此类方程的函数的界限具有很大的实际意义。在本文中，给出了收敛于感兴趣函数的一般更新型方程的单调非递减一般下界序列和单调非递增一般上限序列。对于收敛于更新函数的普通更新过程的更新函数，给出了类似的双侧边界序列，是对 Marshall (1973) 著名对应边界的改进。此外，这种收敛到普通更新函数的边界序列是针对到达间隔时间的寿命分布的几个可靠性等级获得的。最后，