On the Picard problem for hyperbolic differential equations in Banach spaces

Antoni Sadowski

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2003)

  • Volume: 23, Issue: 1, page 31-37
  • ISSN: 1509-9407

Abstract

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B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

How to cite

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Antoni Sadowski. "On the Picard problem for hyperbolic differential equations in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 23.1 (2003): 31-37. <http://eudml.org/doc/271552>.

@article{AntoniSadowski2003,
abstract = {B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_\{xy\} = f(x,y,z,z_\{xy\})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_\{xy\} = f(x,y,z,z_x,z_\{xy\})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].},
author = {Antoni Sadowski},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {boundary value problem; fixed point theorem; functional-integral equation; hyperbolic equation; measure of noncompactness; functional integral equation},
language = {eng},
number = {1},
pages = {31-37},
title = {On the Picard problem for hyperbolic differential equations in Banach spaces},
url = {http://eudml.org/doc/271552},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Antoni Sadowski
TI - On the Picard problem for hyperbolic differential equations in Banach spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2003
VL - 23
IS - 1
SP - 31
EP - 37
AB - B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_{xy} = f(x,y,z,z_{xy})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_{xy} = f(x,y,z,z_x,z_{xy})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].
LA - eng
KW - boundary value problem; fixed point theorem; functional-integral equation; hyperbolic equation; measure of noncompactness; functional integral equation
UR - http://eudml.org/doc/271552
ER -

References

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  1. [1] A. Ambrosetti, Un teorema di essistenza per le equazioni differenziali nagli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349-360. 
  2. [2] K. Goebel, W. Rzymowski, An existence theorem for the equations x' = f(t,x) in Banach space, Bull. Acad. Polon. Sci., Sér. Sci. Math. 18 (1970), 367-370. Zbl0202.10003
  3. [3] P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I. (5) 17-A (1980), 156-160. 
  4. [4] B. Rzepecki, Measure of Non-Compactness and Krasnoselskii's Fixed Point Theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math. 24 (1976), 861-866. Zbl0341.47039
  5. [5] B. Rzepecki, On the existence of solution of the Darboux problem for the hyperbolic partial differential equations in Banach Spaces, Rend. Sem. Mat. Univ. Padova 76 (1986). Zbl0656.35087
  6. [6] B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144. 

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