(Last up-dated: December 11, 2014)

Grant PN-II-RU-TE-2011-3-0239
Funded by the Ministry of Education and Research
Title of the project:
Fixed Point Theorems for Mappings defined on Cartesian Product with applications to solving nonlinear functional equations
(Teoreme de punct fix pentru operatori definiti pe produs cartezian cu aplicatii la rezolvarea ecuatiilor functionale neliniare)
Abstract. The main aim of the project is to perform a complex study of various classes of Presic type operators (including new classes that we have introduced in some recent papers), having in view several topics of research which are currently very dynamic, but which have not covered this general type of operators yet. Among these topics we mention: fixed points and common fixed points for multi-valued Presic type operators, fixed points for non-self Presic type operators, fixed points and common fixed points for Presic type operators in partially ordered metric spaces, coupled and tripled fixed points for Presic type operators, fixed point for cyclic Presic type operators. As a characteristic of Presic type operators, the constructive theorems involve k-step iterative methods. The results expected to be obtained while studying the aforementioned topics should have a wide range of applications mainly in solving nonlinear functional equations.
A secondary aim of the project is to cover some of these topics for the class of almost contractions, a very interesting and currently intensively studied type of generalized contraction operators, which has been subject to our study so far.
The outcomes of the project, along with the results due to other authors and published by the moment, shall find a proper place in a planned monograph on K-Step Fixed Point Iteration Methods and Applications, the first of its kind in literature so far.
Research Team members
Assoc. Prof. Dr. Madalina PACURAR, project director
Prof. Dr. Vasile BERINDE, member
Assist. Prof. Dr. Mihaela PETRIC
Dr. Marin BORCUT

Research Grant PN-II-RU-TE-2011-3-0239 (2012-2014)

Funded by CNCS-UEFISCDI

Title of the project: Fixed point theorems for mappings defined on cartesian product with applications to solving nonlinear functional equations (Teoreme de punct fix pentru operatori definiti pe produs cartezian cu aplicatii la rezolvarea ecuatiilor functionale neliniare)

 Program type / Project type: IDEI / Proiecte de cercetare exploratorie - PCE Project Code: PN-II-RU-TE-2011-3-0239 Contract number: 48/5.10.2011

Abstract (Rezumatul rapoartelor de activitate in limba engleza)

Main Objectives:

1. Obtaining fixed point theorems (FPT) and common fixed point theorems (CFPT) for Prešić type operators combined with conditions of Ciric, Kannan, Rus etc. type;
2. Obtaining fixed point theorems (FPT) and common fixed point theorems (CFPT)  for cyclical operators of this type;
3. Obtaining fixed point theorems (FPT) and common fixed point theorems (CFPT) for non self operators of this type;
4. Establishing connexions between FPTs and coupled and triple fixed point theorems for related classes of operators to the ones under study at 1-3;
5. Organising a Workshop on Fixed Point Theory and Applications, in the last year of running the project, to ensure large spreading of the project's results;
6. Writing a monograph on the topic of the project ("k-step fixed point iterative methods").

Main results reported:

1. The obtained results were more comprehensive than the ones initially planned, as we have obtained not only FPTs and CFPTs and connexions between them and coupled and triple fixed point theorems, but also coupled and tripled fixed point theorems and coincidence point theorems, thus covering all initial plans for items 1, 2 and 4 in the list of Objectives.

2. We also succeeded to solve the most difficult problem, stated in Objective 3, i.e., obtaining FPTs for non self operators. The obtained results are concretised in 39 published papers published, accepted for publication or submitted for publication, of which 28 papers are published in Web of Science (ISI) journals and 11 papers in BDI journals (indexed in international databases), and in 20 conference presentations and  8 invited conferences. Note that 5 of these ISI articles were included in the list of awarded articles by UEFISCDI on 2013 and 2014 (PN-II-RU-PRECISI-2014-8-6711, PN-II-RU-PRECISI-2014-8-6710, PN-II-RU-PRECISI-2014-8-6709, PN-II-RU-PRECISI-2013-7-3677, PN-II-RU-PRECISI-2013-7-3676) and another one (article no. 6 in the list on the web page of the project) is eligible for awarding but, although published in the first part of 2014, it is not yet indexed in Web of Science.

3. We organised the 5th Minisymposium on Fixed Point Theory and Applications (1-7 June 2014) to which were represented some of the most active research groups in fixed point theory from around the world (32 presentations were included in the final programme), see the web site of the conference http://icam.ubm.ro/?icam=ICAM10, for more details.

4. The monograph k-Step fixed point iterative methods is almost ready for print (112 pages are completely technically edited) and has the following content:

1. Introduction

2. Prešić Fixed Point Theorems

3. Prešić type Fixed Point Theorems

4. Common Fixed Point Theorems and coincidence Point Theorems

5. Prešić Fixed Point Theorems for multivalued mappings

6. Prešić Fixed Point Theorems in partially ordered metric spaces

7. Prešić Fixed Point Theorems for cyclical mappings

8. Nonexpansive Prešić type mappings

9. Prešić Fixed Point Theorems for single-valued and multi-valued nonself mappings

10. Applications

11. References

Research Team members

Assoc. Prof. Dr. Madalina PACURAR, project director
Prof. Dr. Vasile BERINDE, member
Assist. Prof. Dr. Mihaela PETRIC, member
Dr. Marin BORCUT, member

Published papers (2012-2014)

A. Papers published in ISI (Web of Science) journals

(1) V. Berinde, Mădălina Păcurar, A constructive approach to coupled fixed point theorems in metric spaces, Carpathian J. Math. 31 (2015), No. 3 (in print)

(2) V. Berinde and Khan, A.R., Convergence theorems for admissible perturbations of generalised pseudocontractive operators, J. Nonlinear Convex Anal. 16 (2015), No. 1 (in print)

(3) Fukhar-ud-din, H., Berinde, V., Iterative methods for the class of quasi-contractive type

operators and comparsion of their rate of convergence in convex metric spaces, Filomat (accepted)

(4) M. Borcut, V. Berinde and Mădălina Păcurar, Tripled fixed point theorems for mixed monotone Chatterjea type contractive operators, J. Computational Analysis and Applications 2015 (accepted)

(5) V. Berinde, Pacurar, M. and Rus, I.A., From a Dieudonne theorem concerning the Cauchy problem to an open problem in the theory of weakly Picard operators, Carpathian J. Math. 30 (2014), No. 3, 283-292

(6) V. Berinde, Khan, A.R. and Pacurar, M., Coupled solutions for a bivariate weakly nonexpansive operator  by iterations,  Fixed Point Theory Appl., 2014:149

(7) V. Berinde and Mădălina Păcurar, Stability of $k$-step fixed point iterative methods for some Pre\v si\' c type contractive mappings, J. Ineq. Appl. 2014, 2014:149

(8) M. Borcut, V. Berinde and Mădălina Păcurar, Tripled fixed point theorems for mixed monotone Kannan type contractive operators, J. Applied Math. Volume 2014, Article ID 120203, 8 pages

(9) V. Berinde, St. Maruster and I. A. Rus, An abstract point of view on iterative approximation of fixed points of nonself operators, J. Nonlinear Convex Anal., 15 (2014), No. 5,  851-865

(10) Berinde, V., Khan, A.R. and Păcurar, M., Convergence theorems for admissible perturbations of $\phi$-pseudocontractive operators, Miskolc Math. Notes, 15 (2014), No. 1, 27-37

(11) Maryam A. Alghamdi, V. Berinde and N. Shahzad, Fixed points of non-self almost contractions, Carpathian J. Math. 30 (2014), No. 1, 7-14

(12) Maryam A. Alghamdi, V. Berinde and Naseer Shahzad, Fixed points of multi-valued non-self almost contractions, J. Appl. Math. 2013  2013: 621614

(13) V. Berinde, F. Vetro, Fixed point for cyclic weak $\Psi,C$-contractions in 0-complete partial metric spaces, Filomat 27 (2013), No. 8, 1405-1413

(14) V. Berinde and Madalina Pacurar, Fixed point theorems for nonself single-valued almost contractions, Fixed Point Theory 14 (2013), no. 2, 301-311

(15) Vasile Berinde, Convergence theorems for ﬁxed point iterative methods deﬁned as admissi-ble perturbations of a nonlinear operator, Carpathian J. Math., 29 (2013), No. 1, 9-18

(16) Vasile Berinde, C. Mortici, New sharp estimates of the generalized Euler-Mascheroni constant, Mathematical Inequalities & Applications 16  (2013), No. 1, 279–288

(17) O. Acar, V. Berinde and I. Altun, Fixed Point Theorems for Ciric Strong Almost Contractions in Partial Metric Spaces, J. Fixed Point Theory Appl. 12 (2012), no. 1-2, 247-259 DOI 10.1007/s11784-013-0104-9

(18) Vasile Berinde, Madalina Pacurar, Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory and Applications 2012, 2012:115 doi:10.1186/1687-1812-2012-115

(19) Vasile Berinde, F. Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory and Applications 2012, 2012:105

(20) Vasile Berinde, Coupled coincidence point theorems for mixed monotone nonlinear operators, Computers and Mathematics with Applications 64 (2012), No. 6, 1770-1777

(21) Vasile Berinde, Approximating fixed points of implicit almost contractions, Hacet. J. Math. Stat. 40 (2012) No. 1 93-102

(22) Vasile Berinde, Coupled fixed point theorems for $\phi$-contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 75 (2012) 3218–3228 doi 10.1016/j.na.2011.12.021

(23) M. Borcut, Vasile Berinde, Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput. 218 (2012) 5929–5936

(24) E. Karapinar, Vasile Berinde, Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces, Banach J. Math. Anal. 6 (2012), No. 1, 74-89

(26) Borcut, Marin, Tripled fixed point theorems for monotone mappings in partially ordered metric spaces, Carpathian J. Math. 28 (2012), no. 2, 215--222.

(27) Borcut, Marin, Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput. 218 (2012), no. 14, 7339--7346.

(28) Petric, Mihaela Ancuţa; Zlatanov, Boyan, Best proximity points and fixed points for $p$-summing maps. Fixed Point Theory Appl. 2012, 2012:86, 12 pp.

B. Papers published in journals indexed in international databases

(1) Păcurar, M., Berinde, V. Borcut, M. and Petric, M., Triple fixed point theorems for mixed monotone Pre\v{s}i\' c-Kannan and Pre\v{s}i\' c-Chatterjea mappings in partially ordered metric spaces, Creat. Math. Inform. 23 (2014), no. 2, 217-228

(2) Berinde, V. and Kovacs, G., Controlling autonomous scalar discrete dynamical systems generated by non self Lipschitzian functions, Creat. Math. Inform. 23 (2014), no. 2, 141-150

(3) Bozantan, A., Berinde, V., About the implementation and some applications of the FIXPOINT software minipackage, Creat. Math. Inform. 23 (2014), no. 1, 41-50

(4) Berinde, V., A man who had known his star in the sky. A homage to Professor Eugen Grebenikov (1932-2013), Creat. Math. Inform. 23 (2014), no. 1, 15-30

(5) Berinde, V., Păcurar, M., The role of the Pompeiu-Hausdorff metric in fixed point theory, Creat. Math. Inform. 22 (2013), no. 2, 143-150

(6) Berinde, V., Cioban, M., Generalized distances and their associate metrics. Impact on ﬁxed point theory, Creat. Math. Inform. 22 (2013), no. 1, 23 - 32

(7) Bozantan, A., Berinde, Vasile, Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems, Creative Math. Inform. 22 (2013), No. 1, 11-18

(8) Berinde, V., A generalization of Mortici lemma, Creat. Math. Inform. 21 (2012), no. 2, 129-134

(9) Can, N.V., Berinde, V., Luong, N.V., N.X. Thuan, A coupled coincidence point theorem in partially ordered metric spaces, Kragujevac J. Math. 37 (2013), No. 1, 103–119

(10) Bumbariu, Oana, Berinde, V., An empirical study of the E-algorithm for accelerating numerical sequences, Appl. Math. Sciences, 6 (2012), Issue 21-24, 1181-1190

(11) Borcut, Marin, Tripled coincidence theorems for monotone mappings in partially ordered metric spaces. Creat. Math. Inform. 21 (2012), no. 2, 135--142.

Conference presentations (2012-2014)

A. Invited Conferences

(1) V. Berinde, Some numerical aspects of ﬁxed point iterative methods for solving nonlinear optimization problems, Galatasaray University, 5 November 2012
(2) V. Berinde, Constructive Fixed Point Theory and Applications, Invited Seminar, King Abdulaziz University, Jeddah, Saudi Arabia, 12 February 2013
(3) V. Berinde, Recent trends in metrical fixed point theory, 21 noiembrie 2013, Naresuan University, Phitsanulok, Thailanda
(4) V. Berinde, Approximating fixed points and common fixed points of almost contractions, 22 noiembrie 2013, Naresuan University, Phitsanulok, Thailand
(5) V. Berinde, Stability of fixed point iteration procedures, 22 noiembrie 2013, Naresuan University, Phitsanulok, Thailand
(6) V. Berinde, Approximating coupled and tripled fixed points and applications, 26 noiembrie 2013, Chiang Mai University, Chiang Mai, Thailand
(7) V. Berinde, On the role of the Pompeiu-Hausdorff metric in mathematics and multidisciplinary fields of research, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, 29th of January, 2014
(8) V. Berinde, Mathematical olympiads and their impact on the Romanian mathematics education system, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, 10th of February, 2014

B. Regular presentations

(1) V. Berinde, 50 years of Higher Mathematics Education in Baia Mare
History of Mathematics and Teaching of Mathematics, Sarospatak, 23-27 May 2012
(2) V. Berinde, Stability of $k$-step fixed point iterative methods (cu Madalina Pacurar)
The Fifth International Workshop Constructive Methods for Nonlinear Boundary Value Problems, Tokaj, 28 June-1 July 2012
(3) V. Berinde, Elementary applications of some Presic type fixed point theorems (cu Madalina Pacurar)
CEEPUS Summer School 2012 Nature in Mathematics Kosice-High Tatra Mountains, 6-21 July 2012
(4) V. Berinde, Madalina Pacurar, On the stability of multi step fixed point iteration procedures[Plenary conference]
20th Conference on Applied and Industrial Mathematics. Dedicated to acad. Mitrofan M. Ciobanu, 22-25 august 2012, Chisinau, R. Moldova
(5) Madalina Pacurar, V. Berinde, Some fixed point theorems for contractive type mappings defined on product spaces, SYNASC2012 (Intern. Symp. On Symb. And Numeric Algorithms for Scientific Computing, Timisoara, 26-29 sept 2012
(6) V. Berinde, St. Maruster, I. A. Rus, An abstract point of view on iterative approximation of fixed points of nonself operators, SYNASC2012 (Intern. Symp. On Symb. And Numeric Algorithms for Scientific Computing, Timisoara, 26-29 sept 2012
(7) V. Berinde, Madalina Pacurar, Single-valued and multi-valued almost contractions. An early survey, [Plenary conference]
International Workshop on Fixed Point Theory and Applications, October 11-14, 2012, Galatasaray University, İstanbul, Turkey

(8) V. Berinde, Madalina Pacurar, On some fixed point theorems for contractive type mappings defined on product spaces, The Tenth International Conference on Fixed Point Theory and its Applications, Cluj-Napoca, 9-15 July 2012

(9) Petric M., Berinde, V., The contraction mapping principle for nonself mappings on Banach spaces endowed with a graph, 14th Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 23-26 September 2013, Univ. de Vest Timişoara

(10) Berinde, V. and Pacurar, M., Fixed point theorems for nonself single-valued cyclic contractions, 14th Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 23-26 September 2013, Univ. de Vest Timişoara

(11) Borcut, M., Pacurar M. and Berinde, V., Tripled fixed point theorems for Kannan type mixed monotone mappings, 14th Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 23-26 September 2013, Univ. de Vest Timişoara

(12) Bozantan A. and Berinde, V., The mini software package FIXPOINT for approximating fixed points of real functions, 14th Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 23-26 September 2013, Univ. de Vest Timişoara

(13) Berinde, V., Fixed point theorems for nonself multi-valued almost contractions satisfying

property (M), ICAM9 (9th INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS). Dedicated to Professor Emeritus Constantin Corduneanu on the occasion of his 85th birthday, 25-28 September 2013, Baia Mare

(14) Pacurar, M., Fixed point theorems for nonself Presic type operators, ICAM9 (9th INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS). Dedicated to Professor Emeritus Constantin Corduneanu on the occasion of his 85th birthday, 25-28 September 2013, Baia Mare

(15) Petric, M., Remarks on cyclic coupled fixed point theorems, ICAM9 (9th INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS). Dedicated to Professor Emeritus Constantin Corduneanu on the occasion of his 85th birthday, 25-28 September 2013, Baia Mare

(16) Berinde, V.,  Borcut, M., Tripled coincidence theorems for φ-contractive type mappings in partially ordered metric spaces, 5th Minisymposium of Fixed Point Theory and Applications (ICAM10), Baia Mare, 1-7 June 2014

(17) Berinde,V., Pacurar, M., A constructive approach to coupled fixed point theorems in metric spaces, 5th Minisymposium of Fixed Point Theory and Applications (ICAM10), Baia Mare, 1-7 June 2014

(18) Berinde, V., Petric, M. A., Fixed Point Theorems for Cyclic Nonself Single-Valued Almost Contractions, 5th Minisymposium of Fixed Point Theory and Applications (ICAM10), Baia Mare, 1-7 June 2014
(19) Berinde, V., Pacurar, M:, The Contraction Principle for Nonself Mappings on Banach Spaces Endowed with a Graph, The 16th International Symposium on Symbolic and Numerical Algorithms for Scientific Computing, 22nd – 25th September 2014, Timisoara

(20) Petric, M., Remarks on coupled fixed points for cyclic operators, The 16th International Symposium on Symbolic and Numerical Algorithms for Scientific Computing, 22nd – 25th September 2014, Timisoara

Conferences organized
(2012-2014)

3) 5th Minisymposium on Fixed Point Theory and Applications, Baia Mare and Turist Suior Resort, June 1-7, 2014
organized in the framework of 10th INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS (http://icam.ubm.ro)

2) Minisymposium on Fixed Point Theory and Applications, in the framework of 9th INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS (http://icam.ubm.ro/?icam=ICAM9/minisymposia)