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On solutions of an integral equation related to traffic flow on unbounded domains. (English) Zbl 1059.45003

The following integral equation is studied: \[ x(t)= h(t)+ f(t, x(t)) \int^\infty_0 u(t,s,x(s))\,ds.\tag{1} \] Using the technique of measures of noncompactness it is shown that the equation (1) is solvable in the space of continuous and bounded functions on \(\mathbb{R}_+\). That result is proved, among others, under the following conditions: (i) There exist \(r> 0\) and \(k_r> 0\) such that \(| f(t,x)- f(t,y)|\leq k_r| x-y|\) for \(t\in \mathbb{R}_+\) and \(x,y\in \mathbb{R}\) such that \(| x|\leq r\), \(| y|\leq r\), (ii) \(u: \mathbb{R}_+\times \mathbb{R}_+\times \mathbb{R}\to \mathbb{R}\) is continuous and there exist continuous functions \(a,b,\psi: \mathbb{R}_+\to \mathbb{R}_+\) with \(\psi\) being nondecreasing on \(\mathbb{R}_+\) and \(a(t)\to 0\) as \(t\to\infty\). Moreover, it is assumed that \(b\) is bounded and integrable on \(\mathbb{R}_+\) and \(| u(t,s,x)|\leq a(t) b(s)\psi(| x|)\) for all \(t,s\in \mathbb{R}_+\), \(x\in \mathbb{R}\) with \(| x|\leq r\). Apart from this there are also assumed some extra conditions.

MSC:

45G10 Other nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
90B20 Traffic problems in operations research
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