Functional differential equations with maxima, via step by step contraction principle

Abstract

T. A. Burton presented in some examples of integral equations a notion of progressive contractions on C([a, ∞[). In 2019, I. A. Rus formalized this notion (I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019) no. 3, 111–120), put ”step by step” instead of ”progressive” in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on C([a, b], B) and C([a, ∞[, B) where B is a Banach space. In this paper we use the abstract result given by I. A. Rus, to study some classes of functional differential equations with maxima.

Authors

Veronica Ilea
Department of Mathematics Babes-Bolyai University , Faculty Mathematics and Computer Science, Cluj-Napoca, Romania


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

Keywords

G-contraction; step by step contraction; Picard operator; weaakly Picard operator; generalized fibre contraction theorem; functional differential equation; functional integral equation; equation with maxima.

References

PDF

Cite this paper as:

V. Ilea, D. Otrocol, Functional differential equations with maxima, via step by step contraction principle, Carpathian J. Math., 37 (2021) no. 2, pp. 195-202, DOI: 10.37193/CJM.2021.02.05

About this paper

Journal

Carpathian J. Mathematics

Publisher Name
Print ISSN

1584 – 2851

Online ISSN

1843 – 4401

Google Scholar Profile

soon

[1] Bainov, D. D. and Hristova, S., Differential equations with maxima, Chapman & Hall/CRC Pure and Applied Mathematics, 2011
[2] Berzig, M., Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications, J. Fixed Point Theory Appl., 12 (2012), No. 1-2, 221–238
[3] Burton, T. A., Integral equations, transformations, and a Krasnoselskii–Schaefer type fixed point theorem, Electron. J. Qual. Theory Differ. Equ., (2016), No. 66, 1–13; doi: 10.14232/ejqtde.2016.1.66
[4] Burton, T. A., Existence and uniqueness results by progressive contractions for integro- differential equations, Nonlinear Dyn. Syst. Theory, 16 (2016), No. 4, 366–371
[5] Burton, T. A., An existence theorem for a fractional differential equation using progressive contractions, J. Fract. Calc. Appl, 8 (2017), No. 1, 168–172
[6] Burton, T. A., A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions, Fixed Point Theory, 20 (2019), No. 1, 107–112
[7] Corduneanu, C., Abstract Volterra equations: a survey, Math. and Computer Model., 32 (2000), No. (11–13) 1503–1528
[8] Halanay, A., Differential Equations: Stability, Oscillations, Time Lags, Acad. Press, New York, 1966
[9] Ilea, V. and Otrocol, D., On the Burton method of progressive contractions for Volterra integral equations, Fixed Point Theory, 21 (2020), No. 2, 585–594
[10] Marian, D. and Lungu, N., Ulam-Hyers-Rassias stability of some quasilinear partial differential equations of first order, Carpatian J. Math., 35 (2019), No. 2, 165–170
[11] Marian, D., Ciplea, S. A. and Lungu, N., On the Ulam-Hyers stability of biharmonic equation, U. P. B. Sci. Bull., Series A, 82 (2020), No. 2, 141–148
[12] Marian, D., Ciplea, S. A. and Lungu, N., Optimal and nonoptimal Gronwall lemmas, Symmetry, 12 (2020), No. 10, 1728, 1–10
[13] Otrocol, D., Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Funct. Anal. Appl., 15 (2010), No. 4, 613–619
[14] Otrocol, D. and Rus, I. A., Functional-differential equations with “maxima” via weakly Picard operators theory, Bull. Math. Soc. Sci. Math. Roumanie (N. S) , 51 (99) (2008), No. 3, 253–261
[15] Otrocol, D. and Rus, I. A., Functional-differential equations with maxima of mixed type argument, Fixed Point Theory, 9 (2008), No. 1, 207–220
[16] Rus, I. A., Generalized contractions and applications, Cluj University Press, 2001
[17] Rus, I. A., Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003), No. 1, 191–219
[18] Rus, I. A., Cyclic representations and fixed points, Ann. T. Popoviciu Seminar of Functional Eq. Approx. Convexity, 3 (2005), 171–178
[19] Rus, I. A., Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9 (2008), No. 1, 293–307
[20] Rus, I. A., Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey, Carpathian J. Math., 26 (2010), No. 2, 230–258
[21] Rus, I. A., Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019), No. 3, 111–120
[22] Samet, B. and Turinici, M., Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal., 13 (2012), No. 2, 82–97

2021

Related Posts