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On existence theorems for differential equations in Banach spaces. (English) Zbl 0569.34053
Let $E$ be a real B-space and $\mu$ a ball measure of noncompactness on $E$, $T>0$. Let $A=\{\omega: [0,T]\times [0,\infty[\to [0,\infty[$ be continuous functions, $\omega(t,.)$ nondecreasing, $\omega(t,0)=0$ and such that 0 is the only continuous function on $[0,T]$ satisfying $u(t)\le \int\sp{t}\sb{0}\omega(s,u(s))ds$, $u(0)=0\}$, $C=\{\omega: ]0,T]\times [0,\infty[\to [0,\infty[$ such that for any $\epsilon>0$ there exist $\delta >0$ and $t\sb n\to 0$, $t\sb n>0$ and $\rho\sb n: [t\sb n,T]\to [0,\infty [$ differentiable and satisfying $\rho'\sb n(t)\ge \omega(t,\rho\sb n(t))$, $\rho\sb n(t)\ge \delta t\sb n$, $0<\rho\sb n(t)\le \epsilon$ for $t\in [t\sb n,T]$, $n\in {\bbfN}\}.$ Theorem 1: Let $\omega\sb 1: [0,T]\times [0,\infty [\to [0,\infty [$ be a continuous function, $\omega\sb 1(t,.)$ nondecreasing, $\omega\sb 1(t,0)=0$ and there exist $\omega\in C$ such that $\omega\sb 1(t,u)\le \omega (t,u)$, $(t,u)\in]0,T]\times [0,\infty [$. Then $\omega\sb 1\in A.$ Theorem 2: If ${\bar \omega}(t,u)=\sup \{\mu(f(t,X)): \mu(X)=u,\emptyset \ne X\subset \overline{B(x\sb 0,r)}\}$, where f: [0,T]$\times \overline{B(x\sb 0,r)}\to E$ is a uniformly continuous function such that $\mu(f(t,X))\le \omega (t,\mu(X))$ for $t\in [0,T]$ and for any X bounded in E, $X\ne \emptyset$ and $\omega$ is a given Kamke function. Then ${\bar \omega}\in A$. Theorem 3: Let f be a function bounded by $A>0$ and as in theorem 2 and $\omega$ is a Kamke function of C. If AT$\le r$ then there exists at least one solution of $x'(t)=f(t,x(t))$ for $t\in [0,T]$, $x(0)=x\sb 0$.
Reviewer: G.Bottaro

34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
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