Hyers-Ulam-Rassias stability of Volterra integral equations with delay within weighted spaces

Abstract

We obtain weak conditions to guarantee the Hyers-UlamRassias stability of (nonlinear) Volterra integral equations with delay. In particular, this leads to a generalization of some results previously known. Basically, this is done by using certain weight functions in the framework of the space of continuous functions. Indeed, the method consists in a convenient combination of the classical Banach fixed point theorem together with a consideration of a weighted metric. Therefore, we avoid the use of the strict successive approximation method and also the consideration of generalized metrics (which need to be typically combined with a consequent fixed point alternative theorem). Some concrete examples are presented at the end of the paper.

Hyers‐Ulam‐Rassias stability of nonlinear integral equations through the Bielecki metric

“…where σ is a nondecreasing continuous function σ:[a, b]→(0,∞). We recall that C p ð½a; bÞ; d p À Á and Cð½a; bÞ; d ð Þare complete metric spaces (cf previous studies 39,40 Under the present conditions, we will deduce that the operator T is strictly contractive with respect to the metric (7). Indeed, for all u, v ∈ C([a, b]), we have Due to the fact that M 2 þ Lη ð Þ<1, it follows that T is strictly contractive.…”

“…We will consider the operator T:C p ([a, b])→C p ([a, b] Theorem, which ensures that we have the σ-semi-Hyers-Ulam stability for the Volterra integral equation (2) with (39) being obtained by using the definition of the metric d p , (5) and (38). C([a, b]), we conclude that T is strictly contractive with respect to the metric (7) due to the fact that M 2 þ Lη ð Þ<1. Thus, we can again apply the Banach Fixed Point Theorem, which ensures that we have the σ-semi-Hyers-Ulam stability for the Volterra integral equation (2) with (47) being obtained by the definition of the metric d and using (5)…”

“…In this section, we will derive sufficient conditions for the Hyers-Ulam-Rassias stability of the Volterra integral equation (2). We will continue to use the metrics (6) and (7). Thus, we will now be dealing with the integral equation C([a, b]) and, under the present conditions, similarly with what was done in the proof of Theorem 2, we are able to deduce that the operator T is strictly contractive with respect to the metric (7).…”

“…We will continue to use the metrics (6) and (7). Thus, we will now be dealing with the integral equation C([a, b]) and, under the present conditions, similarly with what was done in the proof of Theorem 2, we are able to deduce that the operator T is strictly contractive with respect to the metric (7). Besides this, from the definition of the metric d and by (29) (35) for all x ∈ [a, b].…”

Gecikmeli lineer olmayan Volterra integral denklemlerin iterasyon çözümleri

“…In 2013, Castro and Guerra [2] proved the existence and uniqueness of the solution of a nonlinear Volterra integral equation with delay as follows. In addition, suppose that there are constants 2, 3 ( 0,1) such that 5 ( , 6) (6) 6 ≤ 2 ( ) (1) for all ∈ .…”