On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces. (English) Zbl 0954.34050

The authors prove the existence of a weak bounded pseudo solution to the differential equation \[ x'(t) =Ax(t)x(t)+f(t,x(t)), \tag{*} \] where \(x(t) \) belongs to the Banach space \(E\), the function \(f(\cdot,x(\cdot))\) is Pettis integrable for each strongly absolutely continuous function \(x\) and \(f(t,\cdot)\) is weakly-weakly sequentially continuous. (Recall that a function \(x: \mathbb{R}_+ \to E\) is said to be a pseudosolution to (*) if it is absolutely continuous and for every functional \(x^* \in E^*\) there exists a negligible set \(A(x^*)\) such that \((x^*x)'(t) = x^*(A(t)x(t)+f(t,x(t)))\) for every \(t\notin A(x^*)\)).


34G20 Nonlinear differential equations in abstract spaces
34A34 Nonlinear ordinary differential equations and systems
54D30 Compactness