Integral inclusions. Fixed point approaches. (English) Zbl 0991.47041

The author applies the contraction mapping principle in order to obtain existence theorems for integral inclusions of the following type: Let \(K\) be mapping from \([a,b]\times[a,b]\times\mathbb{R}^{n}\) into the nonempty subsets of \(\mathbb{R}^{n}\) such that \(K(\cdot,\cdot,u)\) is lower semicontinuous for each \(u\in\mathbb{R}^{n}\), let \(g:[a,b]\to\mathbb{R}^{n}\) be continuous and assume that there is a continuous function \(l:[a,b]\times[a,b]\to\mathbb{R}^{n}\) with \(\max\{\int_a^{b}l(t,s) ds\mid t\in[a,b]\}<1\) and \(H(K(t,s,u),K(t,s,v))\leq l(t,s)\|u-v\|\) for \(s,t\in[a,b]\) and \(u,v\in\mathbb{R}^{n}\) (where \(H\) denotes the Hausdorff distance) then the Fredholm integral inclusion \(x\in\int_a^{b}K(\cdot,s,x(s)) ds+g\) possesses a continuous solution which is stable under small perturbations of \(g\). There is a similar result for Volterra type inclusions and for delayed integral inclusions.


47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.