New criteria for the existence of non-trivial fixed points in cones
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RESEARCH
Open Access
New criteria for the existence of non-trivial fixed points in cones Alberto Cabada1 , José Ángel Cid2* and Gennaro Infante3 *
Correspondence: [email protected] 2 Departamento de Matemáticas, Universidade de Vigo, Pabellón 3 (Edifico Físicas), Campus de Ourense, Ourense, 32004, Spain Full list of author information is available at the end of the article
Abstract We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers. MSC: 47H10; 34B10; 34B18 Keywords: Krasnosel’ski˘ı fixed point theorem; positive solutions; conical shells; multiplicity
1 Introduction and preliminaries The classical cone compression-expansion fixed point theorem of Krasnosel’ski˘ı (see Theorem . below) and the monotone iterative technique (see Theorem . below) are among the most popular and fruitful tools to deal with the existence of solutions for nonlinear problems. Following earlier ideas of Persson [] valid in the finite-dimensional setting, both methods were combined in [] to obtain the existence of a fixed point, assuming the operator T to be monotone non-decreasing with some conditions on the set of supersolutions. This result was improved in [] by relaxing the monotonicity condition. More recently, in [] the authors were able to present a refinement of the results of [, ], by allowing a comparison between a point and a boundary, instead of on two boundaries as in Krasnosel’ski˘ı’s theorem. This approach, which required a monotonicity assumption on the operator on a conical shell, has proved to be well suited to establish multiplicity results. Our aim in this paper is to pursue this line of research by obtaining new fixed point theorems, valid not only for non-decreasing (Section ) but also for non-increasing operators (Section ). We point out that this type of theorems can be combined in the applications to obtain the existence of multiple non-trivial solutions. This fact is illustrated in Section , where the existence of multiple positive solutions for a nonlocal boundary value problem modeling the displacement of a beam is discussed. We now recall some definitions that will be useful in the sequel. A subset K of a real Banach space N is a cone if and only if it is closed, K + K ⊂ K , λK ⊂ K for all λ ≥ and K ∩ (–K) = {}. A cone K defines the partial ordering in N given by x ≤ y if and only if y – x ∈ K . The notation x < y means x ≤ y and y = x. The cone K is called normal with a normal constant c ≥ if and only if x ≤ cy for all x, y ∈ N with ≤ x ≤ y. Whenever int(K) = ∅, the symbol x y means y – x ∈ int(K) and the cone is said to be solid. ∂K denotes the boundary of K and d(x, ∂K) is the distance of x to the boundary of K . © 2013 Cabada et al.; licensee Springer. This is an Open Access article distributed under the terms of the
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