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Differential equations with maxima-differential equations that contain the maximum of the unknown function over a previous interval-adequately model real-world processes whose present state significantly depends on the maximum value of the state on a past time interval. This book guides readers toward further investigations and applications of these equations for a more accurate study of real-world problems. It explains how to find solutions to initial value and boundary value problems as well as how to study stability and oscillatory properties of solutions. It also presents a systematic development of the averaging method.