A Kneser-type theorem for the equation \(x^{(m)}=f(t,x)\) in locally convex spaces (Q1969400)

This study is a continuation of \textit{St. Szufla} [Math. Nachr. 118, 179-185 (1984; Zbl 0569.34052)]. It is considered the Cauchy problem \[ x^{(m)}= f(t,x),\quad x(0)= 0,\quad x'(0)= \eta_1,\dots, x^{(m-1)}(0)= \eta_{m-1},\tag{1} \] where \(m\) is a positive integer, \(\eta_1,\eta_2,\dots, \eta_{m-1}\in E\) and \(f\) is a bounded continuous function from \(I\times B\) into the quasicomplete locally convex topological vector space \(E\), \(I= [0,a]\), \(B= \{x\in E: p_i(x)\leq b\), \(i= 1,\dots, k\}\), \(p_1,\dots, p_k\in P\), \(P\) being a family of continuous seminorms generating the topology of \(E\). Let \(d\leq a\) be defined as \[ \sum^{m- 1}_{j= 1} p_i(\eta_j){d^j\over j!}+ M{d^m\over m!}\leq b,\quad i= 1,\dots, k, \] for certain constants \(M\) and \(b\) as above, and \(I= [0,d]\). If \(\beta_p(A)\) is the infimum of all \(\varepsilon> 0\) for which there exists a finite subset \(\{x_1,\dots, x_n\}\) of \(E\) such that \(A\subset\{x_1,\dots, x_n\}+ \beta_p(\varepsilon)\), with \(\beta_p(\varepsilon)= \{x\in E; p(x)\leq \varepsilon\}\), \(p\in P\), then the family \((\beta_p(A))_{p\in P}\) is called the measure of noncompactness of \(A\). The following result states the existence of solutions and the structure of the set of solutions to Cauchy problem (1): supposing that for each \(p\in P\) there exists a continuous nondecreasing function \(w_p: \mathbb{R}_+\to \mathbb{R}_+\) such that \(w_p(0)= 0\), \(w_p(r)> 0\) for \(r>0\) and \[ \int_{0+} {dr\over \root m\of{r^{m-1} w_p(r)}}= \infty, \] if \(\beta_p(f(t, X))\leq w_p(\beta_p(X))\) for \(p\in P\), \(t\in I\), and bounded subsets \(X\) of \(E\), then the set \(S\) of all solutions to problem (1) defined on \(I\) is nonempty, compact and connected in \(C(I,E)\).