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On local convexity of nonlinear mappings between Banach spaces

100%
Banakh I., Banakh T., Plichko A., Prykarpatsky A.
Open Mathematics
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2012
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tom 10
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nr 6
2264-2271
EN
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
2
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The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension

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Samoilenko A., Prykarpatsky Y., Prykarpatsky A.
Open Mathematics
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2005
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tom 3
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nr 3
529-557
EN
The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.
3
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The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

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Lustyk M., Janus J., Pytel-Kudela M., Prykarpatsky A.
Open Mathematics
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2009
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tom 7
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nr 4
775-786
EN
The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained.
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