JournalszaaVol. 6, No. 2pp. 121–132

On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain

  • Zdzisław Kamont

    University of Gdansk, Poland
  • Katarzyna Prządka

    University of Gdansk, Poland
On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain cover

Abstract

Under the assumptions of continuity and the existence of first-order partial derivatives of the solutions it is proved that the Cauchy problem

zx(i)(x,y)=f(i)(x,y,z(x,y),z,zy(i)(x,y))z_x^{(i)} (x, y) = f^{(i)} (x, y, z (x, y), z, z_y^{(i)} (x, y))
z(i)(x,y)=φi(x,y)for(x,y)[τ0,0]×Rn    (i=1,,m)z^{(i)} (x, y) = \varphi_i (x, y) for (x, y) \in [–\tau_0, 0] \times \mathbb R^n \;\; (i= 1, \dots, m)

admits at most one solution if the function f=(f(i),,f(m)f = (f^{(i)}, \dots, f^{(m)} of the variables (x,y,p,z,q)(x, y, p, z, q) satisfies a Lipschitz condition with respect to (p,z,q)(p, z, q), or a Lipschitz condition with respect to (p,z)(p, z) and a Hölder condition with respect to qq.

Cite this article

Zdzisław Kamont, Katarzyna Prządka, On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain. Z. Anal. Anwend. 6 (1987), no. 2, pp. 121–132

DOI 10.4171/ZAA/235