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Keywords = abstract Gronwall lemma

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8 pages, 258 KB  
Article
On a Volterra Integral Equation with Delay, via w-Distances
by Veronica Ilea and Diana Otrocol
Mathematics 2021, 9(18), 2341; https://doi.org/10.3390/math9182341 - 21 Sep 2021
Viewed by 2200
Abstract
The paper deals with a Volterra integral equation with delay. In order to apply the w-weak generalized contraction theorem for the study of existence and uniqueness of solutions, we rewrite the equation as a fixed point problem. The assumptions take into account the [...] Read more.
The paper deals with a Volterra integral equation with delay. In order to apply the w-weak generalized contraction theorem for the study of existence and uniqueness of solutions, we rewrite the equation as a fixed point problem. The assumptions take into account the support of w-distance and the complexity of the delay equation. Gronwall-type theorem and comparison theorem are also discussed using a weak Picard operator technique. In the end, an example is provided to support our results. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
9 pages, 248 KB  
Article
On a Functional Integral Equation
by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu
Symmetry 2021, 13(8), 1321; https://doi.org/10.3390/sym13081321 - 22 Jul 2021
Cited by 12 | Viewed by 2149
Abstract
In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those [...] Read more.
In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those are approached in the stability point of view. In the literature, Fredholm, Volterra and Hammerstein integrals equations with symmetric kernels are studied. Our results can be applied as particular cases to these equations. Full article
(This article belongs to the Special Issue Ulam's Type Stability and Symmetries)
10 pages, 283 KB  
Article
Optimal and Nonoptimal Gronwall Lemmas
by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu
Symmetry 2020, 12(10), 1728; https://doi.org/10.3390/sym12101728 - 19 Oct 2020
Cited by 4 | Viewed by 1975
Abstract
In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. [...] Read more.
In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one. Full article
(This article belongs to the Section Mathematics)
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