On a Volterra Integral Equation with Delay, via w-Distances


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.


Veronica Ilea
Department of Mathematics, Babes-Bolyai University, Romania

Diana Otrocol
Department of Mathematics, Technical University of Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


Volterra integral equation with delay; w-distance; weakly Picard operator; abstract Gronwall lemma



Cite this paper as:

V. Ilea and D. Otrocol, On a Volterra integral equation with delay, via w-distances, Mathematics, 9 (2021), art. id. 2341. https://doi.org/10.3390/math9182341

About this paper



Publisher Name


Print ISSN

Not available yet.

Online ISSN


Google Scholar Profile

[1] Burton, T.A., Volterra Integral and Differential Equations; Academic Press: New York, NY, USA, 1983.
[2] Corduneanu, C., Integral Equations and Stability of Feedback Systems; Academic Press: New York, NY, USA, 1973.
[3] Gripenberg, G., Londen, S.O., Staffans, O. Volterra Integral and Functional Equations; Encyclopedia of Mathematics and its Applications 34; Cambridge University Press: Cambridge, UK, 1990.
[4] Guo, D., Lakshmikantham, V., Liu X., Nonlinear Integral Equations in Abstract Spaces; Kuwer Academic Publishers: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 1996.
[5] Sidorov, D. Integral Dynamical Models: Singularities, Signals and Control; World Scientific Publishing Company: Singapore, 2014.
[6] Olmstead, W.E., Roberts, C.A., Deng K., Coupled Volterra equations with blow-up solutions. J. Integral Equ. Appl. 1997, 7, 499–516. [CrossRef]
[7] Panin, A.A., On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation. Math. Notes 2015, 97, 892–908. [CrossRef]
[8] Sidorov, D.,  Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations. Diff. Equ. 2014, 50, 1217–1224. [CrossRef]
[9] Ilea, V.A.; Otrocol, D. An application of the Picard operator technique to functional integral equations. J. Nonlinear Convex Anal.
2017, 18, 405–413.
[10] Ilea, V.A.; Otrocol, D. On the Burton method of progressive contractions for Volterra integral equations. Fixed Point Theory 2020,
21, 585–594. [CrossRef]
[11] Serban, M.A., Data dependence for some functional-integral equations. J. Appl. Math. 2008, 1, 219–234.
[12] Dobritoiu, M., An integral equation with modified argument. Studia Univ. Babes-Bolyai Math. 2004, 49, 27–33.
[13] Ilea, V.A., Otrocol, D. Existence and uniqueness of the solution for an integral equation with supremum, via w-distances. Symmetry 2020, 12, 1554 [CrossRef]
[14] Marian, D., Ciplea, S., Lungu, N., Ulam-Hyers stability of Darboux-Ionescu problem. Carpathian J. Math. 2021, 37, 211–216.
[15] Aguirre Salazar, L., Reich, S. A remark on weakly contractive mappings. J. Nonlinear Conv. Anal. 2015, 16, 767–773.
[16] Dobritoiu, M., An application of the w-weak generalized contractions theorem. J. Fixed Point Theory Appl. 2019, 21, 93. [CrossRef]
[17] Kada, O., Suzuki, T., Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44, 381–391.
[18]  Suzuki, T., Takahashi, W.,  Fixed points theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. J. Juliusz Schauder Cent. 1996, 8, 371–382. [CrossRef]
[19] Takahashi, W.,  Wong, N.C., Yao, J.C., Fixed point theorems for general contractive mappings with w-distances in metric spaces. J. Nonlinear Conv. Anal. 2013, 14, 637–648.
[20] Wongyat, T., Sintunavarat, W., The existence and uniqueness of the solution for nonlinear Fredholm and Volterra integral equations together with nonlinear fraction differential equations via w-distances. Adv. Diff. Equ. 2017, 2017, 211. [CrossRef]
[21] Wongyat, T.,  Sintunavarat, W., On new existence and uniqueness results concerning solutions to nonlinear Fredholm integral equations via w-distances and weak altering distance functions. J. Fixed Point Theory Appl. 2019 21, 7. [CrossRef]
[22] Rus, I.A., Generalized Contractions and Applications; Cluj University Press: Cluj-Napoca, Romania, 2001.
[23] Rus, I.A., Picard operators and applications. Sci. Math. Jpn. 2003, 58, 191–219.
[24] Rus, I.A., Fixed points, upper and lower fixed points: Abstract Gronwall lemmas. Carpathian J. Math. 2004, 20, 125–134.
[25] Marian, D.,  Ciplea, S., Lungu, N., On a Functional Integral Equation. Symmetry 2021, 13, 13–21. [CrossRef]
[26] Marian, D., Ciplea, S.A., Lungu, N., Optimal and Nonoptimal Gronwall Lemmas. Symmetry 2020, 12, 17–28. [CrossRef]
[27] Reich, S., Zaslavski, A.J., Almost all nonexpansive mappings are contractive. C. R. Math. Rep. Acad. Sci. Can. 2000, 22, 118–124.
[28] Reich, S., Zaslavski, A.J., The set of noncontractive mappings is sigma-porous in the space of all nonexpansive mappings. C. R. Acad. Sci. Paris Ser. I Math. 2001, 333, 539–544. [CrossRef].


Related Posts