Existence of solutions and Kamke's theorem for functional differential equations in Banach spaces
Let E be a Banach space, B a linear space of functions from \(]-\infty,0]\) into E satisfying some axioms, \(\sigma\in {\mathbb{R}}\). Let \(a,H,r\in ]0,+\infty[\), \(\phi\in B\), \(f: [\sigma,\sigma +a]\times (B\cap B(\phi,r))\to E\) a uniformly continuous function such that \(| f| \leq H\); \(U: ]\sigma,\sigma+a]\times [0,2r]\to [0,+\infty [\) a Kamke function such that \(\alpha (f(t,b))\leq U(t,\alpha (b))\) for a.e. \(t\in]\sigma,\sigma +a]\) and every \(b\subset B\cap B(\phi,r)\), where \(\alpha\) is the measure of noncompactness of Kuratowski; 0 is the unique absolutely continuous function \(u: [\sigma,\sigma +a]\to {\mathbb{R}}\) such that (*) \(u(\sigma)=u'(\sigma)=0\), \(u'(t)=KU(t,u(t))+\beta u(t)\) for a.e. \(t>\sigma\), where K is a constant which appears in the axioms of B, \[ \beta =\limsup_{t\to 0}1/t(\inf \{k\in [0,+\infty [:\quad \alpha (S(t)b)\leq k\alpha(b)\text{ for all bounded }b\subset B\}-1) \] and \((S(t)\phi)(\vartheta)= \phi(t+\vartheta)\) if \(t+\vartheta <0\), \((S(t)\phi)(\vartheta)= \phi(0)\) if \(t+\vartheta \geq 0\) for \(t\geq 0\). Then \(CP(f,\sigma,\phi)\) has solution i.e. there exist \(\gamma >0\) and \(x: ]-\infty,\sigma+\gamma]\to E\) such that \(x'(t)=f(t,x_ t)\) for \(t>\sigma\), \(x_{\sigma}=\phi\), where \(x_ t: ]-\infty,0]\to E\), \(x_ t(\vartheta)=x(t+\vartheta)\). Let \(\Omega\) be an open subset of \({\mathbb{R}}\times B\), \((\tau_ n,\phi_ n)\in {\mathbb{R}}\times B\), \(\sigma_ n\in]-\infty,\tau_ n[\), \(\sigma_ n\to \sigma\), \(\phi_ n\to \phi\) where \((\sigma,\phi)\in \Omega\), \(f,f_ n: \Omega \to E\), \(n\in {\mathbb{N}}\), for every \((\tau,\psi)\in \Omega\) there exist \(a,r>0\) and a Kamke function \(U: ]\tau,\tau+a]\times [0,2r]\to [0,+\infty [\) such that \([\tau -a,\tau +a]\times (B\cap B(\psi,r))\subset \Omega,\) f is uniformly continuous and bounded on \([\tau-a,\tau+a]\times (B\cap B(\psi,r)),\) \(\alpha(f(t,b))\leq U(t,\alpha(b))\) for a.e. \(t\in]\tau,\tau+a]\) and every subset \(b\subset B\cap B(\psi,r)\), 0 is the unique absolutely continuous function \(u: [\tau,\tau+a]\to {\mathbb{R}}\) such that it holds (*), \(CP(f_ n,\tau,\psi)\) has solution for every \((\tau,\psi)\in \Omega\), if \(\{f_ n:\) \(n\in {\mathbb{N}}\}\) is uniformly bounded on a closed bounded subset of \(\Omega\) then \(\lim \alpha (\{f_ n(t,\psi)-f(t,\psi):\) \((t,\psi)\in F,\) \(n\geq k\})=0,\) \(f_ n\to f\) uniformly on every compact subset of \(\Omega\). Then if \(x_ n: ]-\infty,\tau_ n[\to E\) is a non-continuable solution of \(CP(f_ n,\sigma_ n,\phi_ n)\) there exist a subsequence \((x_{n(i)}:\) \(i\in {\mathbb{N}})\), \(\tau\leq \liminf \tau_{n(i)}\) and \(x: ]-\infty,\tau[\to E\) is a non-continuable solution of \(CP(f,\sigma,\phi)\) such that \(\limsup\{| x_{n(i)}(t)-x(t)|:\) \(\max \{\sigma,\sigma_{n(i)}\}\leq t\leq d\}=0\) for every \(d\in]\sigma,\tau[\).
scientific article; zbMATH DE number 4132636
scientific article