Solutions to Volterra integral equations in bounded $ \Phi $-variation spaces

Mireya Bracamonte; Liliana Pérez; Mireya Bracamonte; Liliana Pérez

doi:10.3934/math.2026376

AIMS Mathematics

2026, Volume 11, Issue 4: 9126-9145. doi: 10.3934/math.2026376

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Solutions to Volterra integral equations in bounded $ \Phi $-variation spaces

Mireya Bracamonte ^,,
Liliana Pérez

Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo Km. 30.5 Vía Perimetral, 090902, Ecuador

Received: 15 December 2025 Revised: 11 March 2026 Accepted: 24 March 2026 Published: 02 April 2026
MSC : 26A45, 45D05

This article examines the existence of solutions to a Volterra-type integral equation in the space of functions of bounded $ \Phi $-variation. Conditions on the given functions are established to guarantee the existence of such solutions. Furthermore, it is shown that any two solutions are equal almost everywhere. The possibility of extending solutions to larger intervals is also explored.
- Volterra integral,
- $ \Phi $-bounded variation
Citation: Mireya Bracamonte, Liliana Pérez. Solutions to Volterra integral equations in bounded $ \Phi $-variation spaces[J]. AIMS Mathematics, 2026, 11(4): 9126-9145. doi: 10.3934/math.2026376

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Abstract

This article examines the existence of solutions to a Volterra-type integral equation in the space of functions of bounded $ \Phi $-variation. Conditions on the given functions are established to guarantee the existence of such solutions. Furthermore, it is shown that any two solutions are equal almost everywhere. The possibility of extending solutions to larger intervals is also explored.

References

[1]	J. Appell, J. Banaś, N. J. Merentes Díaz, Bounded variation and around, Berlin–Boston: De Gruyter, 2014. https://doi.org/10.1515/9783110265118
[2]	M. Bracamonte, J. Ereú, L. Marchan, L. Pérez, Existence of a locally bounded variation solution of a model with infinite delay for an isolated species, J. Nonlinear Funct. Anal., 2024 (2024), 18. https://doi.org/10.23952/jnfa.2024.18 doi: 10.23952/jnfa.2024.18
[3]	D. Bugajewska, D. Bugajewski, H. Hudzik, $BV_{\varphi}$-solutions of nonlinear integral equations, J. Math. Anal. Appl., 287 (2003), 265–278. https://doi.org/10.1016/S0022-247X(03)00550-X doi: 10.1016/S0022-247X(03)00550-X
[4]	V. V. Chistyakov, O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$, J. Dyn. Control Syst., 4 (1998), 217–247. https://doi.org/10.1023/A:1022889902536 doi: 10.1023/A:1022889902536
[5]	R. M. Dudley, R. Norvaiša, Introduction and overview, In: Concrete functional calculus, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-6950-7_1
[6]	D. Filali, M. Akram, M. Dilshad, Exploring the fractional Volterra–Fredholm integro-differential equation: An iterative approach, AIMS Mathematics, 10 (2025), 23467–23495. https://doi.org/10.3934/math.20251042 doi: 10.3934/math.20251042
[7]	K. Hamidi, E. Dahia, D. Achour, A. Tallab, Two-Lipschitz operator ideals, J. Math. Anal. Appl., 491 (2020), 124346. https://doi.org/10.1016/j.jmaa.2020.124346 doi: 10.1016/j.jmaa.2020.124346
[8]	H. A. Hammad, H. ur Rehman, M. De la Sen, A new four-step iterative procedure for approximating fixed points with application to 2D Volterra integral equations, Mathematics, 10 (2022), 4257. https://doi.org/10.3390/math10224257 doi: 10.3390/math10224257
[9]	E. Messina, M. Pezzella, A. Vecchio, Asymptotic solutions of non-linear implicit Volterra discrete equations, J. Comput. Appl. Math., 425 (2023), 115068. https://doi.org/10.1016/j.cam.2023.115068 doi: 10.1016/j.cam.2023.115068
[10]	S. Schwabik, On an integral operator in the space of functions with bounded variation, Časopis pro pěstování matematiky, 97 (1972), 297–330. http://doi.org/10.21136/CPM.1972.108677 doi: 10.21136/CPM.1972.108677
[11]	S. Schwabik, On an integral operator in the space of functions with bounded variation. Ⅱ, Časopis pro pěstování matematiky, 102 (1977), 189–202.
[12]	S. Schwabik, M. Tvrdý, O. Vejvoda, Differential and integral equations: Boundary value problems and adjoints, 1979.
[13]	M. Schramm, Functions of $\Phi$-bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math. Soc., 287 (1985), 49–63. https://doi.org/10.1090/S0002-9947-1985-0766206-3 doi: 10.1090/S0002-9947-1985-0766206-3
[14]	M. De la Sen, A study of the stability of integro-differential Volterra-type systems of equations with impulsive effects and point delay dynamics, Mathematics, 12 (2024), 960. https://doi.org/10.3390/math12070960 doi: 10.3390/math12070960
[15]	J. Zhang, J. Hou, J. Niu, R. Xie, X. Dai, A high-order approach for nonlinear Volterra–Hammerstein integral equations, AIMS Mathematics, 7 (2022), 1460–1469. https://doi.org/10.3934/math.2022086 doi: 10.3934/math.2022086

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