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Nonordered discontinuous upper and lower solutions for first-order ordinary differential equations. (English) Zbl 1001.34001

The author studies the first-order equations

x ' (t)=f(t,x(t)),fora.e.tI=[0,1],x(0)=x 0 ,


x ' (t)=f(t,x(t)),fora.e.tI=[0,1],x(1)=y 0 ,

where f:I× is a Carathéodory function.

He proves that if there exist αBV - (I) and βBV + (I) such that α ' (t)f(t,α(t)) and β ' (t)f(t,β(t)) for a. e. tI then such problems have extremal solutions in [m,M] for all x 0 [α(0),β(0)] and y 0 [β(1),α(1)]. Here, m(t)=min{α(t),β(t)}, M(t)=max{α(t),β(t)} for all tI, and BV + (I) (BV - (I)) denotes the set of functions of bounded variation in I with nondecreasing (nonincreasing) singular part.

Moreover, if α(0)β(0) and α(1)β(1) then, for every x 0 [α(0),β(0)] and y 0 [β(1),α(1)], the problem x ' (t)=f(t,x(t)) for a.e. tI=[0,1], x(0)=x 0 , x(1)=y 0 , has extremal solutions in [m,M].

These results are extended to the discontinuous problem x ' (t)=q(x(t))f(t,x(t)), with f a Carathéodory function and q:(0,+) such that q, 1/qL loc ().

Finally, some examples of functions f that do not satisfy the Carathéodory conditions and for which there is no solution in [m,M] are presented in the paper.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A36Discontinuous equations