Extremality results for first-order discontinuous functional differential equations

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.…”

“…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.…”

On the Cauchy Problem for First Order Discontinuous Ordinary Differential Equations

“…This equivalence is established by means of a change of variable developed in Section 4 in [6] and, independently, in [3]. In this sense, we can say that Theorem 3.1 in [6] also generalizes some existence results within the pioneer work developed in [4,[7][8][9][10] on equations of the form of (2). On the other hand, Theorem 3.1 in [6] also improves the main result in [13].…”

Existence theory for first order discontinuous functional differential equations

“…In [1] the authors also consider functional-boundary conditions. However, the functional periodic conditions that we consider in this example are not covered by the main result in [1] because the assumptions that are required on the corresponding function B are rather strong. In particular, the reader can check that condition (B1) in [1] does not hold when trying to apply Theorem 3.1 in [1] to (4.1)-(4.2).…”