## Abstract

It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.

## Authors

**C. D. Alecsa**

Babes-Bolyai University, Mihail Kogalniceanu street no. 1, Cluj-Napoca, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Cluj-Napoca ,Romania

## Keywords

Banach algebras; (G)-convergence; (H)-convergence; differential equations; fixed points; sequeneces of contractions

## Paper coordinates

Cristian-Daniel Alecsa, *Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations, *International J. Nonlin. Anal. Appl., 10 (2019) 2, pp. 227-254,

DOI: 10.22075/ijnaa.2019.18884.2040

## About this paper

##### Journal

International J. Nonlin. Anal. Appl.

##### Publisher Name

Semnan Univ., Iran

##### DOI

10.22075/ijnaa.2019.18884.2040 (not working yet)

##### Print ISSN

##### Online ISSN

google scholar link

[1] L. Barbet, K. Nachi, *Sequences of contractions and convergence of fixed points*, Monografias del Seminario Matemático García de Galdeano 33(2006), 51–58.

[2] F.F. Bonsall, *Lectures on Some Fixed Point Theorems of Functional Analysis,* Tata Institute of Fundamental Research, Bombay, 1962.

[3] H. Huang, G. Deng, S. Radenovic, *Some topological properties and fixed point results in cone metric spaces over Banach algebras*, Positivity, 2019, https://doi.org/10.1007/s11117-018-0590-5.

[4] H. Huang, S. Radenovic, *Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications*, J. Nonlinear Sci. Appl. 8 (2015), 787–799.

[5] H. Huang, S. Hu, B.Z. Popovic, S. Radenovic, *Common fixed point theorems for four mappings on cone b-metric spaces over Banach algebras*, J. Nonlinear Sci. Appl. 9 (2016), 3655–3671.

[6] L. Huang, X. Zhang, *Cone metric spaces and fixed point theorems of contractive mappings*, J. Math. Anal. Appl. 332(2), 2007, 1468-1476.

[7] S. Jankovic, Z. Kadelburg, S. Radenovic, *On cone metric spaces* : A survey, Nonlinear Anal. 74 (2011), No. 7, 2591–2601.

[8] A. Khamsi, *Remarks on cone metric spaces and fixed point theorems of contractive mappings*, Fixed Point Theory Appl. (2010), Article ID 2010:315398, 7 pages.

[9] B. Li, H. Huang, *Fixed point results for weak ϕ-contractions in cone metric spaces over Banach algebras and applications,* Journal of Function Spaces (2017), Article ID 5054603, 6 pages.

[10] H. Liu, S. Xu, *Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings*, Fixed Point Theory and Appl. (2013), Article ID 2013:320, 1–10.

[11] S.N. Mishra, R. Pant, R. Panicker, *Sequences of (ψ, φ)-weakly contractive mappings and stability of fixed points*, Int. Journal of Math. Analysis 7 (2013), No. 22, 1085–1096.

[12] S.N. Mishra, S.L. Singh, R. Pant, *Some new results on stability of fixed points*, Chaos, Solitons & Fractals 45 (2012), 1012–1016.

[13] S.B. Nadler Jr., *Sequences of contractions and fixed points*, Pacific J. Math. 27 (1968), 579–585.

[14] M. Pacurar , *Sequences of almost contractions and fixed points*, Carpathian J. Math. 24 (2008), No. 2, 101–109.

[15] W. Rudin, *Functional Analysis* (2nd edition), McGraw-Hill, New York, 1991.

[16] S.P. Singh, W. Russell, *A note on a sequence of contraction mappings*, Can. Math. Bull. 12 (1969), 513–516.

[17] S. Xu, S. Radenovic, *Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality*, Fixed Point Theory and Appl. (2014), Article ID 2014:102, 1–12.

[18] P. Yan, J. Yin, Q. Leng, *Some coupled fixed point results on cone metric spaces over Banach algebras and applications*, J. Nonlinear Sci. Appl. 9 (2016), 5661–5671.