Dancer, E. N. Upper and lower stability and index theory for positive mappings and applications. (English) Zbl 0755.47038 Nonlinear Anal., Theory Methods Appl. 17, No. 3, 205-217 (1991). The author considers a Banach space \(E\) with a cone \(K\) such that \(K-K\) is dense in \(E\) and a continuously differentiable increasing order- compact mapping \(T: E\to E\). A fixed point \(z\) of \(T\) is said to be stable from above if for each \(\varepsilon>0\) there is a \(\delta>0\) such that \(\| T^ n x-z\|<\varepsilon\) whenever \(x\geq z\) and \(\| x- z\|\leq\delta\). A fixed point \(z\) of \(T\) is said to be isolated from above if there is a \(\delta>0\) such that \(Tx\neq z\) if \(x>z\) and \(\| x- z\|\leq\delta\). Under some technical assumptions on \(T\), the author obtains the following characterization of stability: Assume that \(a<b\), \(Ta=a\), and \(a\) is isolated from above as a fixed point in \([a,b]\) and that \(Tb\leq b\). Then \(a\) is stable from above in \([a,b]\) if and only if the fixed point index \(\text{ind}([a,b],T,a)=1\). Moreover, this condition in turn is equivalent to the existence of strict supersolutions arbitrarily close to \(a\). The author gives some applications to discrete dynamical systems and cones with empty interior, and he indicates an application to a boundary value problem in the case of an irregular boundary. Reviewer: C.Fenske (Gießen) Cited in 8 Documents MSC: 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H10 Fixed-point theorems Keywords:positive mapping; upper and lower stability; continuously differentiable increasing order-compact mapping; fixed point; stable from above; fixed point index; existence of strict supersolutions; discrete dynamical systems; cones with empty interior; boundary value problem; irregular boundary PDFBibTeX XMLCite \textit{E. N. Dancer}, Nonlinear Anal., Theory Methods Appl. 17, No. 3, 205--217 (1991; Zbl 0755.47038) Full Text: DOI References: [1] Dancer, E. N., On the indices of fixed points of mappings in cones and applications, J. math. Analysis, 91, 131-151 (1983) · Zbl 0512.47045 [2] Dancer, E. N., Multiple fixed points of positive mappings, J. reine angew. Math., 371, 46-66 (1986) · Zbl 0597.47034 [3] DancerTrans. Am. math. Soc.; DancerTrans. Am. math. Soc. [4] DancerHessJ. reine Angew. Math.; DancerHessJ. reine Angew. Math. [5] Henry, D., Geometric theory of semilinear parabolic equations, (Lecture Notes in Mathematics, 840 (1981), Springer: Springer Berlin) · Zbl 0456.35001 [6] Hirsch, M. W., Stability and convergence in strongly monotone dynamical systems, J. reine Angew. Math., 383, 1-53 (1988) · Zbl 0624.58017 [7] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen · Zbl 0121.10604 [8] Krein, M. G.; Rutman, M. A., Linear operators leaving invariant \(a\) cone in a Banach space, Transl. Math. Monogr., Am. math. Soc. Transl., 10, 199-325 (1962) · Zbl 0030.12902 [9] Matano, H., Existence of nontrivial for equilibriums of strongly order preserving systems, J. Fac. Sci. Tokyo, 30, 645-673 (1984) · Zbl 0545.35042 [10] Nussbaum, R., The fixed point index for locally condensing maps, Annali Mat. pura appl., 87, 217-258 (1971) · Zbl 0226.47031 [11] Schaefer, H. H., Topological Vector Spaces (1971), Springer: Springer Berlin · Zbl 0212.14001 [12] Stampacchia, G., Equations Elliptiques du Second Ordre à Coefficients Discontinues (1966), Presse Université de Montréal · Zbl 0151.15501 [13] \( \textsc{Takac}d\); \( \textsc{Takac}d\) [14] Vainberg, M. M., Variational Methods in the Study of Nonlinear Equations (1964), Holden-Day: Holden-Day San Francisco, CA · Zbl 0122.35501 [15] Vanderbauwhede, A., Invariant manifolds in infinite dimensions, (Dynamics of Infinite Dimensional Systems (1987), Springer: Springer Berlin), 409-420 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.