A generalization of the Krasnoselskii-Petryshyn compression and expansion theorem: An essential map approach (Q2731062)

Let \(E\) be a Banach space and \(P_B(E)\) the bounded subsets of \(E\). The Kuratowski measure of non-compactness is the map \(\alpha: P_B(E)\to [0,\infty))\) defined by \(\alpha(X)= \{\varepsilon> 0: X\subset \bigcup X_i\), \(i= 1,\dots, n\) and \(\text{diam}(X_i)\leq\varepsilon\}\); here \(X\in P_B(E)\). Let \(Z\) be a nonempty subset of \(E\) and \(F:Z\to 2^E\). \(F\) is called:NEWLINENEWLINENEWLINE(i) countably \(k\)-set contractive \((k\geq 0)\) if \(F(Z)\) is bounded and \(\alpha(F(Y))\leq k\alpha(Y)\) for all countably bounded set \(Y\) of \(Z\);NEWLINENEWLINENEWLINE(ii) countably condensing if \(F\) is countably \(1\)-set and \(\alpha(F(Y))< \alpha(Y)\) for all countably bounded set \(Y\) of \(Z\) with \(\alpha(Y)\neq 0\).NEWLINENEWLINENEWLINEIf \(C\) is a closed, convex subset of \(E\), and \(U\) is an open subset of \(C\), \(D(\overline U, C)\) denotes the set of all upper semicontinuous, countably \(k\)-set contractive \((0\leq k<1)\) maps \(F:\overline U\to CK(C)\), where \(\overline U\) denotes the closure of \(U\) in \(C\) and \(CK(C)\) denotes the family of nonempty, convex, compact subsets of \(C\). \(D_{\partial U}(\overline U, C)\) denotes the set of all maps \(F\in D(\overline U,C)\) with \(x\not\in F(x)\) for \(x\in\partial U\), where \(\partial U\) denotes the boundary of \(U\) in \(C\). The authors introduce the notion of essential and inessential maps for countably condensing maps.NEWLINENEWLINENEWLINEA map \(F\in D_{\partial U}(\overline U,C)\) is essential in \(D_{\partial U}(\overline U,C)\) if for every map \(G\in D_{\partial U}(\overline U,C)\) with \(G/\partial U= F/\partial U\) we have that there exists \(x\in U\) with \(x\in G(x)\). Otherwise \(F\) is inessential in \(D_{\partial U}(\overline U,C)\) i.e. there exists a map \(G\in D_{\partial U}(\overline U,C)\) with \(G/\partial U= F/\partial U\) and \(x\not\in G(x)\) for \(x\in\overline U\).NEWLINENEWLINENEWLINESome properties of essential and inessential maps are established. In particular, the authors prove that if a map \(G\) is essential and \(G\cong F\) then \(F\) is essential.NEWLINENEWLINENEWLINELet \(C\) be a closed, convex subset of \(E\) with \(\alpha u+\beta v\in C\) for all \(\alpha\geq 0\), \(\beta\geq -\) and \(u,v\in C\). Let \(\rho> 0\) with \(B_\rho= \{x: x\in C\) and \(\|x\|< \rho\}\), \(S_\rho= \{x: x\in C\) and \(\|x\|= \rho\}\) and of course \(\overline B_\rho= B_\rho\cup S_\rho\). In Section 3 the authors present following generalization of the Krasnosel'skij-Petryshin compression and expansion theorem in a cone for countably \(k\)-set contractive maps.NEWLINENEWLINENEWLINETheorem: Let \(E\) and \(C\) be as above and \(r\), \(R\) the constanta with \(0< r< R\). Suppose \(F\in D(\overline B_R, C)\) and assume the following conditions hold: \(x\not\in \lambda F(x)\) for \(x{\i}S_r\) and \(\lambda\in (0,1)\) and \(\exists v\in C-\{0\}\) with \(x\overline{\in} F(x)+\delta v\) for all \(\delta> 0\) and \(x\in S_R\). Then \(F\) has a fixed point in \(\{x\in C: r\leq \|x\|\leq R\}\).