A Cauchy problem on time scales with applications (Q2919597)

By considering non-absolutely convergent delta-integrals, the author proved the existence of continuous solutions for a nonlocal Cauchy problem on time scales in Banach spaces. His main result is:NEWLINENEWLINE A time scale \(\mathbb T\) is a nonempty closed set of real numbers \(\mathbb R\), with the subspace topology inherited from the standard topology of \(\mathbb R\). Let \(X\) be a Banach space and denote by \(C(\mathbb T,X)\) the space of \(X\)-valued continuous functions on \(\mathbb T\) and by \(B_R\) its closed ball of radius \(R\) centered in the null element of this space, while \(\left\|\cdot\right\|_C\) stands for the suppremum norm. Then holds:NEWLINENEWLINELet \(b:\mathbb T\to R^+\) be a continuous function and \(f:\mathbb T\times X\to X\) satisfy:NEWLINENEWLINE \noindent i) for every continuous \(x:\mathbb T\to X\), the function \(t\in\mathbb T\mapsto f(t;x(t))\) is HL-\(\Delta\)-integrable;NEWLINENEWLINE \noindent ii) if \(\{x_n\}_n\) converges to \(x\) with respect to the weak topology of \(C(T,X)\), then \((f(\cdot,x_n(\cdot)))_n\) pointwisely weakly converges to \(f(\cdot;x(\cdot))\);NEWLINENEWLINE \noindent iii) for every \(R>0\), the set \(\{(HL)\int_0^\cdot f(s;x(s))\Delta s,\| x\|_C\leq R\}\subset C(T,X)\) is:NEWLINENEWLINE 1) equicontinuous and pointwisely relatively weakly compact;NEWLINENEWLINE 2) weakly uniformly \(ACG_*\), i.e. \(\{(HK)\int_0^\cdot\langle x^*,f(s;x(s))\rangle\Delta s,\| x\|_C\leq R\}\) is uniformly \(ACG_*\), for all \(x^*\in X^*\);NEWLINENEWLINE \noindent iv) \(\lim\sup_{R\rightarrow\infty}\left( \frac 1R\sup_{\| x\|_C\leq R} \| f(\cdot,x(\cdot))\|_A\right) <\frac 12\);NEWLINENEWLINE \noindent v) \(\| b\|_C\leq\frac 1{2T}\).NEWLINENEWLINEThen the differential problem possess global continuous solutions.