# On the Picard problem for hyperbolic differential equations in Banach spaces

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

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

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topAntoni 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

top- [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] 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] P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I. (5) 17-A (1980), 156-160.
- [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] 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] B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144.