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

O’Regan, Donal

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 \[ y(t)= h(t)+\int^t_0 k_1(t,s)f_1(s,x(s))ds+ \int^\infty_0 k_2(t,s)f_2(s,x(s))ds,\tag{E} \] 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).
For the entire collection see [Zbl 0838.00012].

Reviewer: C.Corduneanu (Arlington)

MSC:

45G05

Singular nonlinear integral equations

Keywords:

nonlinear integral equations on the half line; Schauder-Tikhonov fixed point theorem; bounded solutions

Citations:

Zbl 0784.58050