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Convexity of solutions for an iterative equation in Banach spaces. (English) Zbl 1296.54063

Summary: By applying Schauder’s fixed point theorem we investigate the existence of increasing (decreasing) solutions of the iterative equation \(\mathcal H(f)\circ f=F\) and further give conditions under which those solutions are convex or concave. As corollaries we obtain results on iterative equation \(G(f(x),f^{n_1},\dots,f^{n_k}(x))=F(x)\) in Banach spaces, where \(n_1,n_2,\dots,n_k\geq 2\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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