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Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations. (English) Zbl 1257.45004

The quadratic Urysohn integral equation of weakly singular type \[ x(t)=a(t)+f(t,x(t))\int_0^\infty(t-s)^{\alpha-1}u(t,s,x(s))ds \] and the corresponding Volterra (fractional integral) equation \[ x(t)=a(t)+f(t,x(t))\int_0^t(t-s)^{\alpha-1}u(t,s,x(s))ds \] are considered. More precisely, under a rather restrictive Lipschitz condition for \(f\) with a sufficiently small constant and some other growth conditions, it is shown that the first problem has a solution, and that the second problem has a solution with \(x(t)\to0\) as \(t\to\infty\) resp. (under slightly different assumptions) that it has a bounded solution \(x\) such that all bounded solutions \(y\) with not too large norm satisfy \(| x(t)-y(t)|\to0\) as \(t\to\infty\) (somewhat misleading, this uniqueness type property is called “local stability”).
The method of proof is an application of a fixed point theorem of Darbo type in locally convex space (the fixed point theorem is implicitly shown so that formally only the Schauder-Tychonoff fixed point theorem is used).

MSC:

45G05 Singular nonlinear integral equations
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
45M10 Stability theory for integral equations
26A33 Fractional derivatives and integrals
47H10 Fixed-point theorems
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