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 
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/
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[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.