zbMATH — the first resource for mathematics

On the existence of continuous solutions of operator equations in Banach spaces. (English) Zbl 0609.34065
We consider the existence problem for the equation (1) $$u(x)=(Fu)(x)$$, $$x\in G$$, where $$F: C(G,B)\to C(G,B)$$ is a continuous operator, G is a compact metric space with a metric d, B is a Banach space with a norm $$\| \cdot \|$$. It is proved that if $\| (Fw)(x)-(Fw)(y)\| \leq \Omega (d(x,y),\| w(\beta (\cdot,x))-w(\beta (\cdot,y))\|)$ and $$d(\beta (\cdot,x),\beta (\cdot,y))\leq \omega (\cdot,d(x,y)),$$ then the existence of a solution of (1) often follows from the existence of a continuous solution of the functional inequality $$\gamma (r)\geq \Omega (r,\gamma (\omega (\cdot,r))),$$ $$r\in [0,\delta]$$ such that $$\gamma (0)=0$$, $$\delta >0$$. The general results are applied to solving functional-differential and functional equations with unknown function of one and several variables.
MSC:
 34G10 Linear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 39B99 Functional equations and inequalities 45G10 Other nonlinear integral equations 47E05 General theory of ordinary differential operators
Full Text: