Extremal solutions for fractional evolution equations of order $1<\gamma <2$

This manuscript considers a class of fractional evolution equations with order $1<\gamma <2$ in ordered Banach space. Based on the theory of cosine operators, this paper extends the application of monotonic iterative methods in this type of equation. This method can be applied to some physical problems and phenomena, providing new tools and ideas for academic research and practical applications. Under the assumption that the linear part is an $m$-accretive operator, the positivity of the operator families of fractional power solutions is obtained by using Mainardi's Wright-type function. By virtue of the positivity of the family of fractional power solution operators, we establish the monotone iterative technique of the solution of the equation and obtain the existence of extremal mild solutions under the assumption that the upper and lower solutions exist. Moreover, we investigate the positive mild solutions without assuming the existence of upper and lower solutions. In the end, we give an example to illustrate the applied value of our study.