Existence theorems for weak solutions to \(n\)th-order differential equations in Banach spaces
The existence of weak solutions to the de La Vallee-Poussin boundary value problem \[ x^{(n)}=f(t,x,x' \dots, x^{(n-1)}), \quad x^{(i)}(a_k)=0, \tag{1} \] \[ k=1, \dots p, \;i=0,1, \dots i_k-1, \;a<a_1<a_2< \dots < a_p<b,\;1\leq p\leq n, \;\sum_{k=1}^pi_k=n, \] or to the equivalent integral equation \[ x(t)=\int_a^bG(s,x)f(s,x(s), \dots, x^{(n-1)}(s))ds \] is studied (where \(f:J\times E^n\to E\) is a weakly-weakly continuous function, \(J=[a,b], E\) is a sequentially weakly complete Banach space, and \(G(t,x)\) is the Green function of the problem \(x{(n)}=0, \;x^{(i)}(a_k)=0\)). Weak compactness type conditions are imposed in terms of the measure of weak noncompactness \(\beta\). It is proved that if \(p_1P_1+ \dots + p_nP_n<1,\) for \(P_i\) such that \[ \int_a^b| G^{(i-1)}(s,x)| ds\leq P_i \text{ and }\beta(f(J\times X_1 \times \dots \times X_n)) \leq p_1\beta(X_1) + \dots + p_n \beta(X_n) \] for any bounded subsets \(X_1, \dots X_n\) of \(E\), then there exists a weak solution to problem (1).
scientific article; zbMATH DE number 1251495