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Extremal solutions of a discontinuous scalar differential equation. (English) Zbl 0949.34005
The authors prove the existence of minimal and maximal absolutely continuous solutions x:[0,1] to the initial value problem x ' (t)=f(t,x(t)), x(0)=0. Here, f:[0,1]× ¯ satisfies standard “measurability” assumptions and a nonstandard “continuity” assumption: for every x, the function tf(t,x) is Lebesgue measurable; there exists a Lebesgue integrable function M:[0,1] ¯ such that |f(t,x)|M(t) for almost all t[0,1] and for all x; lim sup yx f(t,y)f(t,x)lim inf yx f(t,y) for almost all t[0,1] and for all x. Applications concern the scalar equation x ' (t)=q(x(t))f(t,x(t)) where q is Lebesgue measurable as well as the vector equation x ' (t)=f(t,x(t)) where x k f i (t,,x k-1 ,x k ,x k+1 ,) is nondecreasing for each ik [cf. J. Szarski, Differential inequalities, Warszawa: PWN-Polish Scientific Publishers (1967; Zbl 0171.01502)].

34A36Discontinuous equations