On Ulam Stabilities of Delay Hammerstein Integral Equation

Abstract

In this paper, we consider a Hammerstein integral equation (Hammerstein IE) in two variables with two variables of time delays. The aim of this paper is to investigate the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of the considered IE via Banach’s fixed point theorem (Banach’s FPT) and the Bielecki metric. The proofs of the new outcomes of this paper are based on these two basic tools. As the new contributions of the present study, here, for the first time, we develop the outcomes that can be found in the earlier literature on the Hammerstein IE, including variable time delays. The present study also includes complementary outcomes for the symmetry of Hammerstein IEs. Finally, a concrete example is given at the end of this study for illustrations.

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

“…Furthermore, several attractive outcomes concerning applications of fixed point methods, various fixed point theorems, and EUSs, several other qualitative properties with regard to certain IEs and IDEs can be found in the books of Abbas et al [18] and Burton [19], and the papers of Chauhan et al [20], Deep et al [21], Graef et al [22], Lungu and Rus [23], Khan et al [24,25], Tunç and Tunç [26,27], Tunç et al [27,28], and their references.…”

“…To the best knowledge of the authors of this paper, according to the above literature information and relative international databases, the EUSs of HTFIE (12) in a Banach space via the technique called Burton's progressive contractions, the Chebyshev norm, and complete metric have not been discussed up to now. However, Tunç and Tunç [26] and Castro and Simões [36] obtained some new results with regard to the Ulam stabilities of certain-delay Hammerstein-type integral equations in the bounded and infinite interval cases. The considered delay of Hammerstein-type integral equations and the topic of Tunç and Tunç [26] and Castro and Simões [36] are not related to those of this paper.…”

“…However, Tunç and Tunç [26] and Castro and Simões [36] obtained some new results with regard to the Ulam stabilities of certain-delay Hammerstein-type integral equations in the bounded and infinite interval cases. The considered delay of Hammerstein-type integral equations and the topic of Tunç and Tunç [26] and Castro and Simões [36] are not related to those of this paper. Indeed, HTFIE (12) is a new mathematical model and it can be seen that HTFIE (1) is different from all the above mathematical models as IEs and IDEs and those can be found in the international data base.…”

New Results on Ulam Stabilities of Nonlinear Integral Equations

“…As for a comprehensive treatment of the subject with regard to the qualitative behaviors of certain VIEqs, VIDEqs, IDEqs and some others, we refer the readers to the following works: for the existence and Ulam stability of quadratic IEqs by Schauder's fixed point theorem, see Abbas and Benchohra [4]; the existence and asymptotic stability of nonlinear VIEqs by a fixed point theorem, see Banaś and Rzepka [5]; the stability of FDEqs by fixed point theory, see Burton [6]; the HU stability and HUR stability of VIEqs with delay, Hammerstein IEqs and IDEqs, respectively, see Castro and Ramos [7] and Castro and Simões [8,9]; the HU stability for ODEqs and partial differential equations via the Gronwall lemma, see Ciplea et al [10]; the HUR stability of Volterra-Hammerstein IEqs by the fixed point method, see Ciplea et al [11] and Tunç and Tunç [12]; the Ulam stabilities of iterative FDEqs of the first order by the fixed point method, see Egri [13]; the HU stability and HUR stability of VIDEqs by the fixed point method, see Tunç and Tunç [14] and Tunç et al [15,16]; the HU stability and HUR stability of VIEqs by the fixed point method, see Jung [17] and Ö grekçi [18]; the HUR stability of functional equations and fractional differential equations by the fixed point method, respectively, see Jung [19] and Khan et al [20]; the HU stability for operatorial equations and inclusions via nonself operators, see Petru et al [21]; the HU stability of ODEqs and differential operators, see Popa and Raşa [22]; the Ulam stability of the linear mapping in Banach spaces, see Rassias [23]; Ulam stability, see Ulam [24]; the stability of IDEqs in the sense of Lyapunov, see Bohner and Tunç [25] and Tunç and Tunç [26]; the stability of mappings of the Hyers-Ulam type, see [27]; the Ulam-type stability, see [28]; and the references of these sources.…”