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Zeitschrift für Analysis und ihre Anwendungen


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Volume 39, Issue 2, 2020, pp. 151–170
DOI: 10.4171/ZAA/1655

Published online: 2020-04-07

Harnack Type Inequalities and Multiple Solutions in Cones of Nonlinear Problems

Diana-Raluca Herlea[1], Donal O'Regan[2] and Radu Precup[3]

(1) Babeș-Bolyai University, Cluj-Napoca, Romania
(2) National University of Ireland, Galway, Ireland
(3) Babeș-Bolyai University, Cluj-Napoca, Romania

The paper presents an abstract theory regarding operator equations and systems in ordered Banach spaces. We obtain existence, localization and multiplicity results of positive solutions using Krasnosel'skii's fixed point theorem in cones, and a Harnack type inequality. Concerning systems, the localization is established by the vector version of Krasnosel'skii's theorem, where the compression-expansion conditions are expressed on components. The approach is sufficiently general to cover and unify a large number of results on particular classes of problems. It also can guide future research in this direction.

Keywords: Boundary value problem, positive solution, Harnack inequality, Krasnosel'skii's fixed point theorem in cones, operator equation

Herlea Diana-Raluca, O'Regan Donal, Precup Radu: Harnack Type Inequalities and Multiple Solutions in Cones of Nonlinear Problems. Z. Anal. Anwend. 39 (2020), 151-170. doi: 10.4171/ZAA/1655