In this paper we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion:and with initial data [0,d] is a family of linear operators in the Banach space E generating an evolution operator and F is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii,

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We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ * > 0 such that for λ < λ * , the problem has no positive solution; for λ = λ * , it has at least one positive solution; and for λ > λ * , it has at least two positive solutions.