Geodesic
On the Picard problem for hyperbolic differential equations in Banach spaces
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 23 (2003) no. 1, pp. 31-37.

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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].
Keywords: boundary value problem, fixed point theorem, functional-integral equation, hyperbolic equation, measure of noncompactness 
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Sadowski, Antoni. On the Picard problem for hyperbolic differential equations in Banach spaces. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 23 (2003) no. 1, pp. 31-37. http://geodesic.mathdoc.fr/item/DMDICO_2003_23_1_a2/

[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.

[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.

[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).

[6] B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144.