## Abstract

In the present paper, we introduce new types of Presić operators. These operators generalize the well-known Istrăţescu mappings, known as convex contractions. Also, we study the existence and uniqueness of fixed points for this type of operators and the convergence of one-step sequence toward the unique fixed point. Also, data dependence results are presented. Finally, some examples are given, suggesting that the above mappings are proper generalizations of convex contractions of second order

## Authors

**Cristian Daniel Alecsa**

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

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

## Keywords

Convex contractions; fixed point; Presic operators; data dependence.

## Paper coordinates

C.-D. Alecsa, *Some fixed point results regarding convex contractions of Presić type, *J. Fixed Point Theory Appl., 20 (2018), art. 7,

DOI: 10.1007/s11784-018-0488-7

## About this paper

##### Journal

*Journal of Fixed Point Theory and Applications*

##### Publisher Name

Springer

##### Print ISSN

1661-7738

##### Online ISSN

google scholar link

[1] Abbas, M., Ilić, D., Nazir, T*. Iterative approximation of fixed points of generalized weak Presic type k-step iterative method for a class of operators*. Filomat 29(4), 713–724 (2015)

[2] Alghamdi, M.A., Alnafei, S.H., Radenovic, S., Shahzad, N., *Fixed point theorem for convex contraction mappings on cone metric spaces*. Math. Comput. Model. 54, 2020–2026 (2011)

[3] Alnafei, S.H., Radenovic, S., Shahzad, N., *Fixed point theorems for mappings with convex diminishing diameters on cone metric spaces*. Appl. Math. Lett. 24, 2162–2166 (2011)

[4] Banach, S., *Sur les opérations dans les ensembles abstraits et leur application aux equations integrales*. Fund. Math. 3, 133–181 (1922)

[5] Berinde, V., Păcurar, M., *Stability of k-step fixed point iterative methods for some Pres̆ić type contractive mappings*. J. Inequal. Appl. 2014:149 (2014)

[6] Ćirić, L.B., Pres̆ić, L.B., *On Presic type generalisation of Banach contraction mapping principle*. Acta Math. Univ. Comenian 76(2), 143–147 (2007)

[7] Istrăţescu, V., *Some fixed point theorems for convex contraction mappings and convex nonexpansive mappings (I)*. Lib. Math. 1, 151–164 (1981)

[8] Istrăţescu, V., *Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters – I*. Ann. Mat. Pura Appl. 130, 89–104 (1982)

[9] Istrăţescu, V., *Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. II*. Ann. Mat. Pura Appl. 134, 327–362 (1983)

[10] Khan, M.S., Berzig, M., Samet, B., *Some convergence results for iterative sequences of Pres̆ić type and applications*. Adv. Differ. Equ. 38 (2012). https://doi.org/10.1186/1687-1847-2012-38

[11] Miculescu, R., Mihail, A., *A generalization of Matkowski’s fixed point theorem and Istrăţescu’s fixed point theorem concerning convex contractions*. J. Fixed Point Theory Appl. 19, 1525–1533 (2017)

[12] Mureşan, V., Mureşan, A.S., *On the theory of fixed point theorems for convex contraction mappings*. Carpatian J. Math. 31(3), 365–371 (2015)

[13] Păcurar, M., *Approximating common fixed points of Presić–Kannan type operators by a multi-step iterative method*. An. Şt. Univ. Ovidiu Constanţa 17(1), 153–168 (2009)

[14] Păcurar, M., *A multi-step method for approximating common fixed points of Presić–Rus type operators on metric spaces*. Studia Univ. Babeş-Bolyai Math. LV(1), 149–162 (2010)

[15] Păcurar, M., *Fixed points of almost Presić operators by a k-step iterative method*. An. Ştiinţ. Univ. AI.I. Cuza Iaşi Mat. 57:199–210 (2011)

[16] Pathak, H.K., George, R., Nabwey, H.A., El-Paoumy, M.S., Reshma, K.P., *Some generalized fixed point results in b-metric space and application to matrix equations*. Fixed Point Theory Appl. 2015:101 (2015)

[17] Pres̆ić, S.B., *Sur une classe d’inéquations aux differences finies et sur la convergence de certain suites*. Pub. de l’Inst. Math. Belgrade 5(19), 75–78 (1965)

[18] Rus, I., *An iterative method for the solution of the equation x=f(x,…,x).* Anal. Numér. Thor. Approx. 10(1), 95–100 (1981)

[19] Sastry, K.P.R., Rao, ChS, Sekhar, C., Balaiah, M., *A fixed point theorem for cone convex contractions of order m≥2*. Int. J. Math. Sci. Eng. Appl. 6(1), 263–271 (2012)

[20] Shukla, S., Radenovic, S., *Presić–Maya type theorems in ordered metric spaces*. Gulf J. Math. 2(2), 73–82 (2014)

[21] Shukla, S., Radenovic, S., Pantelić, S., *Some fixed point theorems for Pres̆ić–Hardy–Rogers type contractions in metric spaces*. J. Math. (2013). https://doi.org/10.1155/2013/295093

[22] Shukla, S., Radenovic, S., *Some generalizations of Pres̆ić type mappings and applications*. An. Ştiinţ. Univ. AI.I. Cuza Iaşi Mat. (2015). https://doi.org/10.1515/aicu-2015-0026