Weak solutions for the dynamic Cauchy problem in Banach spaces (Q923954)

The following dynamic Cauchy problem in Banach spaces \[ x^\Delta(t)=f(t,x(t)), \;t\in J =\mathbb{R_+}, \quad x(0)=x_0, \] is studied, where \(x^\Delta\) denotes the weak \(\Delta\)-derivative of \(x\), \(J\) is a time scale interval and \(f\) is weakly-weakly sequentially continuous with values in a Banach space, satisfying some regularity conditions expressed in terms of the De Blasi measure of weak noncompactness. The existence of weak solutions to the above problem for both \(\mathbb{Z}\) and \(\mathbb{R}\) is considered. Moreover, the main result extends the existence of weak solutions to the discrete intervals with nonuniform step size \(\mathbb{K}_q=q^{\mathbb{Z}}\cup\{0\}\), where \(q\neq 1\) is a fixed real number. The authors establish an existence result by introducing a weakly sequentially continuous operator associated to an integral equation which is equivalent to the above problem and using the fixed point theorem for weakly sequentially continuous mappings given by \textit{I. Kubiaczyk} [Discuss. Math., Differ. Incl. 15, No.~1, 15--20 (1995; Zbl 0832.47046)].