On the Picard problem for hyperbolic differential equations in Banach spaces

Antoni Sadowski

ArticleOriginal scientific text

Title

On the Picard problem for hyperbolic differential equations in Banach spaces

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Affiliations

Institute of Mathematics and Physics, Technical University of Warsaw, Branch Płock, ul. Łukasiewcza 17, 09-400 Płock, Poland

Abstract

B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation

z_{x y} = f (x, y, z, z_{x y})

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_{x y} = f (x, y, z, z_{x}, z_{x y})

using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

Keywords

ENG

boundary value problem, fixed point theorem, functional-integral equation, hyperbolic equation, measure of noncompactness

A. Ambrosetti, Un teorema di essistenza per le equazioni differenziali nagli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349-360.
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.
P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I. (5) 17-A (1980), 156-160.
B. Rzepecki, Measure of Non-Compactness and Krasnoselskii's Fixed Point Theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math. 24 (1976), 861-866.
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).
B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144.

Title

On the Picard problem for hyperbolic differential equations in Banach spaces

Affiliations

Abstract

Keywords

Bibliography