Existence results for nonlinear integral equations on the half line.

*(English)*Zbl 0862.45006Corduneanu, C. (ed.), Qualitative problems for differential equations and control theory. Dedicated to Aristide Halanay on occasion of his 70th birthday. Singapore: World Scientific. 121-131 (1995).

The paper deals with integral equations on the positive half-axis

$$y\left(t\right)=h\left(t\right)+{\int}_{0}^{t}{k}_{1}(t,s){f}_{1}(s,x\left(s\right))ds+{\int}_{0}^{\infty}{k}_{2}(t,s){f}_{2}(s,x\left(s\right))ds,\phantom{\rule{2.em}{0ex}}\left(\mathrm{E}\right)$$

under suitable conditions to secure the existence of at least one solution. The method is based on the Schauder-Tikhonov fixed point theorem in the space of continuous maps from $[0,\infty )$ into ${\mathbb{R}}^{n}$, with the topology of uniform convergence on finite intervals. The author also applies a continuation theorem due to *M. Furi* and *M. P. Pera* [Pac. J. Math. 160, No. 2, 219-244 (1993; Zbl 0784.58050)]. In particular, existence of bounded solutions is secured for the equation (E).

Reviewer: C.Corduneanu (Arlington)

##### MSC:

45G05 | Singular nonlinear integral equations |