Abstract. We obtain new results on the existence and multiplicity of fixed points of Hammerstein equations in very general cones. In order to achieve this, we combine a new formulation of cones in terms of continuous functionals with fixed point index theory. Many examples and an application to boundary value problems are also included. Key Words and Phrases: Cones, fixed points, Hammerstein equations. 2010 Mathematics Subject Classification: 37C25, 47H30, 34B15.
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We prove a new fixed point theorem of Schauder type, which applies to discontinuous operators in noncompact domains. In order to do so, we present a modification of a recent Schauder-type theorem of Pouso. We apply our result to second-order boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory. Primary 47H10; secondary 34A36; 34B15
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We introduce a new fixed point theorem of Krasnoselskii type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and discontinuous nonlinearities. 2010 MSC: 34A12; 34A36; 58J20.A classical problem [9,10,12] is that of the existence of positive solutions for the differential equation(1.1) along with suitable boundary conditions (BCs).This problem arises in the study of radial solutions in R n , n ≥ 2 for the partial differential equationwith the appropriate boundary conditions, see [4,9,10].Recently, in the paper [8], the authors study the existence of non trivial radial solutions for a system of PDEs of the previous type. First, they turn the former problem into a system of ordinary differential equations similar to (1.1).The main novelty in this paper is that we will let f to be discontinuous.The classical compression-expansion fixed point theorem of Krasnoselskii (see [2] or [13]) is a wellknown tool of nonlinear analysis and it has proved very useful to deduce existence of solutions forThe previous definition was formulated for open subsets of a cone, but it works for arbitrary nonempty subsets of a Banach space (see [11]).Closed-convex envelopes (cc-envelopes, for short) need not be upper semicontinuous (usc, for short), see [3, Example 1.2], unless some additional assumptions are imposed on T .Proposition 2.2 Let T be the cc-envelope of an operator T : U −→ K. The following properties are satisfied:1. If T maps bounded sets into relatively compact sets, then T assumes compact values and it is usc;2. If T U is relatively compact, then T U is relatively compact.
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