Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations (Q430315)

The quadratic Urysohn integral equation of weakly singular type NEWLINE\[NEWLINEx(t)=a(t)+f(t,x(t))\int_0^\infty(t-s)^{\alpha-1}u(t,s,x(s))dsNEWLINE\]NEWLINE and the corresponding Volterra (fractional integral) equation NEWLINE\[NEWLINEx(t)=a(t)+f(t,x(t))\int_0^t(t-s)^{\alpha-1}u(t,s,x(s))dsNEWLINE\]NEWLINE 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'').NEWLINENEWLINEThe 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).