Existence of generalized solutions for ordinary differential equations in Banach spaces
Let E be a real B-space, r, \(a>0\), \(x_ 0\in E\), \(D=\{x\in E:| x- x_ 0| \leq r\}\), f: [0,a[\(\times D\to E\) a Carathéodory function, let \(\gamma\) be the Kuratowski's or the Hausdorff's measure of noncompactness; let g: [0,a[\(\times [0,+\infty [\to R\) be a Carathéodory function such that \(g(t,0)=0\) for all \(t\in [0,a[\) and for each bounded subset \(B\subset [0,+\infty [\) there exists \(\alpha_ B\in L^ 1[0,a[(\) such that \(| g(t,b)| \leq \alpha_ B(t)\) for a.e. \(t\in]0,a[\) and for all \(b\in B\). Then: Theorem 1. Suppose that there exists \(M\in L^ 1([0,a[)\) such that \(| f(t,x)| \leq M(t)\) for a.e. \(t\in [0,a[\) and every \(x\in D\), \(\gamma (f(t,B))\leq g(t,\gamma (B))\) for a.e. \(t\in [0,a[\) and for all \(B\subset D\). Then \(C=\{x:[0,b]\to E\) absolutely continuous function such that \(x'(t)=f(t,x(t))\) for a.e. \(t\in [0,b]\), \(x(0)=x_ 0\}\), where \(\int^{b}_{0}M(t)dt\leq r\), is nonempty and compact if f is continuous at \((0,x_ 0)\) and the only p: [0,a[\(\to [0,+\infty [\) absolutely continuous function such that \(p(0)=p'_+(0)=0\), \(p'(t)\leq 2g(t,p(t))\) for a.e. \(t\in]0,a[\) is the zero function. Moreover if \(\gamma\) is the Hausdorff's measure of noncompactness, E is generated by a weakly compact subset and if the previous condition is satisfied with 2g substituted by g and without the hypothesis \(p'_+(0)=0\), then the thesis is also true. Theorem 2. Let E be an ordered B-space and there exist \(k,M,L>0\), \(b\in [0,a[\) such that \(| x| \leq k| y|\) if \(x\leq y\); \(f(t,x)- f(t,y)\geq -M(x-y)\) for (t,x), (t,y)\(\in [0,b]\times D\), \(y\leq x\); then exist v,w: [0,b]\(\to E\) absolutely continuous functions such that \(v\leq w\), \(v'(t)\leq f(t,v(t))\), \(w'(t)\geq f(t,w(t))\) for a.e. \(t\in [0,b]\); there exists \(m\in L^ 1([0,b])\) such that \(| f(t,x)| \leq m(t)\) for a.e. \(t\in [0,b]\) and for every \(x\in P\equiv \{x\in E:v(t)\leq x\leq w(t)\) for a \(t\in [0,b]\}\); \(\gamma\) (f(t,B))\(\leq L\gamma (B)\) for a.e. \(t\in [0,b]\) and for all \(B\subset P\); then there exist \(a,b\in C\) and two monotone sequences \((x_ n)_{n\in {\mathbb{N}}}\), \((y_ n)_{n\in {\mathbb{N}}}\) that converge uniformly at a and b respectively such that \(v\leq x_ 1\leq...\leq x_ n\leq...\leq a\leq x\leq b\leq...\leq y_ n\leq...\leq y_ 1\leq w\) for every \(x\in C\) with \(v\leq x\leq w\).
scientific article; zbMATH DE number 4089982
scientific article