A perturbed algorithm of generalized quasivariational inclusions for fuzzy mappings (Q2757174)

Let \(M,T:H\to{\mathcal F}(H)\) be two fuzzy mappings of a real Hilbert space \(H\), i.e.\ their values are given as functions from \(H\) into \([0,1]\). Let \(\varphi:H\to(-\infty,\infty]\) be convex and lower semi-continuous with subdifferential \(\partial\varphi\), and let \(g,m:H\to H\) and \(r\in(0,1]\). The generalized quasivariational inclusion problem for fuzzy mappings is to find \(u\in H\) and \(x,y\in H\) with \(T(u)(x)\geq r\) and \(M(u)(y)\geq r\) (i.e.\ the grade of membership of \(x\) resp.\ \(y\) to the fuzzy set \(T(u)\) resp.\ \(M(u)\) is at least \(r\)) such that \(y-x\in\partial\varphi((g-m)(u))\). The problem is reformulated as a (multivalued) fixed point problem. For the latter iteration algorithms of Mann and Ishikawa type are proposed. Under some Lipschitz and monotonicity assumptions with appropriate constants the existence of a solution and convergence of the algorithms are proved.