Solutions of a quadratic Volterra–Stieltjes integral equation in the class of functions converging at infinity

On a set-valued functional integral equation of Volterra-Stieltjes type

“…A set-valued functional equation has been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [15-18, 20, 23, 24, 27]). The interest in the study of Volterra-Stieltjes integral equations was initiated mainly by the papers (see [4,12,13,21]).…”

“…s g(t, s) = t 0 p(s, y)dy, and in this case integral inclusion of Volterra-Stiltjes type (1.1) has the formx(t) ∈ p(t) + F 1 t, t (s, x(ϕ(s)) ( t (s, y)dy) ds ,(7 4). …”

Open mappings: The case for a new direction in fixed point theory

“…However, when v is not locally Lipschitz (with respect to x), then Theorem 6.3 cannot be applied. But the result in [1] may be still valid, the cost now being rather expected: loss of uniqueness. To illustrate this, take a look at the equation…”

“…The changes are that, now, we have |v(t, s, x)| = (t 2 + 1)te −t + |x(s)| p and Φ(x) = |x| p . As the function v is no more Lipschitz in x, thus Theorem 6.3 cannot be applied, however, the existence result in [1] can yield existence of solutions in B 1 . Note that in view of continuity we immediately see that for any solution x it holds x(0) = 0, a fact that is also implied by Lemma 6.1, as one may easily observe that (A1)-(a 4 ) are still valid.…”