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Existence results for nonlinear integral equations on the half line. (English) Zbl 0862.45006
Corduneanu, 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

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

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 $\left[0,\infty \right)$ into ${ℝ}^{n}$, 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).

##### MSC:
 45G05 Singular nonlinear integral equations