Extremal solutions of a discontinuous scalar differential equation

On the solvability of first-order discontinuous scalar initial and boundary value problems

“…t e / , at least between assumed lower or upper solutions (see Remark 2. Existence and comparison results derived in [1,2,[4][5][6]8,12,13,16,18,19] for initial value problems are special cases of the results derived above when <p is the identity function.…”

“…We begin with a result due to Hassan and Rzymowski, 1999, which forms a basis for the proofs of our main existence results for (2.1).…”

“…ii'(r) = g(t, «(/)), (1.1) with given initial or boundary conditions, has been proved under various kinds of hypotheses which allow g to be discontinuous in both its variables (see for example [1,2,[4][5][6][7][8][9][10][11][12][13][14]16,18,19]). A culmination in this research was achieved in [5], where existence of extremal solutions of the initial value problem (1.1) was proved for a large class of discontinuous functions g. The results of [5] were applied in [18] to prove existence results for (1.1) equipped with discontinuous functional boundary conditions.…”

“…A culmination in this research was achieved in [5], where existence of extremal solutions of the initial value problem (1.1) was proved for a large class of discontinuous functions g. The results of [5] were applied in [18] to prove existence results for (1.1) equipped with discontinuous functional boundary conditions. In this paper this research is continued as follows:…”

“…(a) Existence results of [5,18] are extended to the case when u'(t) is replaced in (1.1) by dcp(u(t))/dt, where cp : R -> R is an increasing homeomorphism.…”