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THE CAUCHY PROBLEM FOR CERTAIN GENERALIZED DIFFERENTIAL EQUATIONS OF FIRST ORDER WITH SINGULARITY

THE CAUCHY PROBLEM FOR CERTAIN GENERALIZED DIFFERENTIAL EQUATIONS OF FIRST ORDER WITH SINGULARITY DEMONSTRATIO MATHEMATICAVol. XXXIINo 21999Anna SzadkowskaTHE CAUCHY PROBLEM FOR CERTAIN GENERALIZEDDIFFERENTIAL EQUATIONS OF FIRST ORDERWITH SINGULARITYThe present paper is devoted to a natural generalization of differential equations for mappings from subset of a Banach space into a Banachspace.The subject matter refers to studies of generalized differential equations of the first order introduced in [7].Let X, Y be Banach spaces over the field M and let U and V be opensubsets of X and Y, respectively. Let h be a mapping from U into X andF a mapping from U x V into Y.We shall start with defining a derivative of a function / in a directionof the mapping h on U, denoted by (V^/)(x) for x € U, and generalizingthe well known notion of the directional derivative [6]. From a point of viewof differencial geometry, a directional derivative V ^ / means a derivative inthe direction of a vector field (with a singularity, because h(0) = 0). Thenwe consider the Cauchy problem(Vh)f(x)+ Af(x) = F(x,f(x)),/(0) = 0for mappings from a subset of a Banach space into a Banach space, whichare defined in C or in C»'1, with the assumption that 0 is a singular point(i.e. h(0) = 0). We also http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

THE CAUCHY PROBLEM FOR CERTAIN GENERALIZED DIFFERENTIAL EQUATIONS OF FIRST ORDER WITH SINGULARITY

Szadkowska, Anna
Demonstratio Mathematica , Volume 32 (2): 20 – Apr 1, 1999
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Publisher
de Gruyter
Copyright
© by Anna Szadkowska
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1999-0209
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXIINo 21999Anna SzadkowskaTHE CAUCHY PROBLEM FOR CERTAIN GENERALIZEDDIFFERENTIAL EQUATIONS OF FIRST ORDERWITH SINGULARITYThe present paper is devoted to a natural generalization of differential equations for mappings from subset of a Banach space into a Banachspace.The subject matter refers to studies of generalized differential equations of the first order introduced in [7].Let X, Y be Banach spaces over the field M and let U and V be opensubsets of X and Y, respectively. Let h be a mapping from U into X andF a mapping from U x V into Y.We shall start with defining a derivative of a function / in a directionof the mapping h on U, denoted by (V^/)(x) for x € U, and generalizingthe well known notion of the directional derivative [6]. From a point of viewof differencial geometry, a directional derivative V ^ / means a derivative inthe direction of a vector field (with a singularity, because h(0) = 0). Thenwe consider the Cauchy problem(Vh)f(x)+ Af(x) = F(x,f(x)),/(0) = 0for mappings from a subset of a Banach space into a Banach space, whichare defined in C or in C»'1, with the assumption that 0 is a singular point(i.e. h(0) = 0). We also

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1999

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