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Two theorems for multivalued condensing mapping in wedges. (English) Zbl 1258.47067

Summary: We discuss fixed-point and approximation results for multivalued condensing mappings in wedges. Let \(E\) be a Hausdorff locally convex space and \(W\) be a wedge in \(E\). Let \(D\) be an open subset of \(E\) and \(\theta\in D\) such that the closure \(\overline D\) of \(D\) is convex. Put \(D_W:= D\cap W\) and denote by \( CK(W)\) the family of all nonempty convex compact subsets of \(W\). We show that, if \(f\colon\overline D_W\to CK(W)\) is a continuous condensing mapping, then there exists \(e_0\in\overline D_W\) such that \[ d_{P_{D_W}}(f(e_0),e_0)=d_{P_{D_W}}(f(e_0),\overline D_W), \] where \(P_{D_W}\) is the Minkowski function of \(D_W\) in \(E\). Moreover, if \(d_{P_{D_W}}(f(e_0),\overline D_W)>0\), then \(e_0\in\partial D_W\).

MSC:

47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
46A03 General theory of locally convex spaces
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