Abstract
Existence of extremal solutions of nonlinear discontinuous integral equations of Volterra type is proved. This result is extended herein to functional Volterra integral equations (FVIEs) and to a system of discontinuous VIEs as well.
Highlights
In this work the existence of extremal solutions of nonlinear discontinuous integral as well as functional integral equations is proved by weakening all forms of Caratheodory’s condition
For all n ∈ N, i xn 1 ≤ xn, ii if x : 0, 1 → R is a continuous function, which serves as a dummy function, satisfying, x 0 ≤ u 0 and x t ≤ x s u t −ustsf t, τ, x τ dτ, for s, t ∈ 0, 1, x ≤ xn in 0, 1
Taking x : 0, 1 → R to be a continuous function, which serves as a dummy function, satisfying x 0 ≥ u 0 and x t ≥ x s u t − u s following the previous steps one can show that 1.1 t s f has t, a τ, x τ dτ, for s, t minimal solution
Summary
In this work the existence of extremal solutions of nonlinear discontinuous integral as well as functional integral equations is proved by weakening all forms of Caratheodory’s condition. Meehan and O’Regan 5 established, by placing some monotonicity assumption on a nonlinear L1Caratheodory kernel of the form k t, s, x s , existence of a C 0, T solution to 1.1 It is proven in 6 that, providing some type of discontinuous nonlinearities, 1.1 has extremal solutions. Dhage 7 proved under mixed Lipschitz, Caratheodory, and monotonicity conditions existence of extremal solutions of nonlinear discontinuous functional integral equations. The main objective in this paper is to emphasize that the kernel f is not required to be neither continuous nor monotonic in any of its arguments to establish an existence of extremal solutions for 1.1 in R which generalizes in some aspects some of the previously mentioned works.
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