A glance into the beauty of Fixed Point Theory: Professor Emeritus Ioan A. Rus on his 80th anniversary

Author: Vasile Berinde and Mădălina Păcurar

Abstract: Professor Emeritus Ioan A. Rus, a prominent and unfatiguable mathematician with an impressive oeuvre in the field of Nonlinear Analysis, particularly in Fixed Point Theory, will be 80 on 28th of August 2016. On this occasion, we would like to congratulate him and wish him good health, happiness, the same impressive physical and mental vigour, as well as new bright results in mathematics.
We also take this opportunity to present some information on the visibility and impact of Professor Rus' main scientific contributions, which would thus complement the information given in the following anniversary articles [Berinde, V. and Petrușel, A., Professor Ioan A. Rus on his 70th birthday: a complete scientist, an accomplished mathematician, Fixed Point Theory 7 (2006), No. 2, 167--174; Berinde, V. and Păcurar, M., The joy of doing mathematics. Interview with Professor Ioan A. Rus, Eur. Math. Soc. Newsl., 76 (2010), 47--50; Petrușel, A., Professor Ioan A. Rus on his 75th birthday, Fixed point theory and its applications, 33--38, Casa Cărții de Știință, Cluj-Napoca, 2013]. PDF |

Retraction method in fixed point theory for monotone nonexpansive mappings

Author: M. R. Alfuraidan

Abstract: In this paper we study the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings in Banach spaces endowed with a graph. In particular, we prove that under certain conditions, this set is a monotone nonexpansive retract. PDF |

A comparison of some fixed point iteration procedures by using the basins of attraction

Author: Gheorghe Ardelean, Ovidiu Cosma and Laszlo Balog

Abstract: Several iterative processes have been defined by researchers to approximate the fixed points of various classes operators. In this paper we present, by using the basins of attraction for the roots of some complex polynomials, an empirical comparison of some iteration procedures for fixed points approximation of Newton's iteration operator. Some numerical results are presented. The Matlab $m$-files for generating the basins of attraction are presented, too. PDF |

Caristi's random fixed point theorem for generalized distance on Polish spaces

Author: Areerat Arunchai and Somyot Plubtieng

Abstract: In this paper, we present the random version of generalized Caristi's fixed point theorem for generalized distance on Polish spaces. Moreover, we prove some Caristi's random fixed point theorems for multi-valued mappings on Polish spaces. Our results in this paper extend and improve some known results in the literature. PDF |

Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph

Author: Laszlo Balog and Vasile Berinde

Abstract: Let $K$ be a non-empty closed subset of a Banach space $X$ endowed with a graph $G$. The main result of this paper is a fixed point theorem for nonself Kannan $G$-contractions $T: K \rightarrow X$ that satisfy Rothe's boundary condition, i.e., $T$ maps $\partial K$ (the boundary of $K$) into $K$. Our new results are extensions of recent fixed point theorems for self mappings on metric spaces endowed with a partial order and also of various fixed point theorems for self and nonself mappings on Banach spaces or convex metric spaces. PDF |

Fixed point theorems for Prešić almost contraction mappings in orbitally complete metric spaces endowed with directed graphs

Author: Pornpimon Boriwan, Narin Petrot and Suthep Suantai

Abstract: The main aim of this paper is to introduce a class of generalized contractions in product spaces in the sense of Prešić. Some examples and fixed point theorems for such introduced mappings in the setting of orbitally complete metric spaces are proved. The results presented here extend and include many existing several results in the literature. PDF |

Fixed point approximation of Prešić nonexpansive mappings in
product of $\mathbf{CAT}\left( 0\right) $ spaces

Author: Hafiz Fukhar-ud-din, Vasile Berinde and Abdul Rahim Khan

Abstract: We obtain a fixed point theorem for Prešić
nonexpansive mappings on the product of $CAT\left( 0\right) $ spaces and
approximate this fixed points through Ishikawa type iterative algorithms under
relaxed conditions on the control parameters. Our results are new in the
literature and are valid in uniformly convex Banach spaces. PDF |

Abstract: We present a new iterative method, based on the so called $\alpha$-dense curves, to approximate coupled fixed points of nonexpansive mappings. Compactness condition on the mapping or its domain of definition is necessary. As application, we construct a sequence which converges to a solution of certain system of integral equations of Volterra type. PDF |

Abstract: In this paper, we give examples of cyclic operators defined on various types of sets, in order to illustrate some results in the extremely rich literature following the seminal paper [Kirk, W. A., Srinivasan, P. S. and Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), No. 1, 79 -- 89]. All examples which are presented enrich the list of cyclic operators and give a subject to future studies of this type of operators. PDF |

An improvement result concerning fixed point theory for cyclic contractions

Author: Mohamed Jleli and Bessem Samet

Abstract: In this note, we obtain an improvement result for cyclic
contractions by weakening the closure assumption that is usually supposed in the literature. We present some applications of the obtained result to prove the existence of solutions for a system of functional equations. PDF |

A study of a system of operator inclusions via a fixed point approach and applications to functional-differential inclusions

Author: Adrian Petrușel, Gabriela Petrușel and Jen-Chih Yao

Abstract: In this paper, some existence results for a system of operator inclusions are presented. Qualitative properties of the solution set are also discussed. The method is based on the application of a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces. The approach is new even for the case of the metric spaces. As an application, an existence result for a mixed boundary and initial value problem for a system of second order differential inclusions is
given. PDF |

On the approximation of fixed points for a new class of generalized Berinde mappings

Author: Bessem Samet

Abstract: In this paper, we introduce a new class of operators, for which a fixed point theorem is proven. This class of mappings is very large and unifies several classes of contractive type operators from the literature, including Berinde mappings. Such fact is proven via a comparison with various metrical contractive type mappings. PDF |

Fixed point theorems for nonself $G$-almost contractive mappings in Banach spaces endowed with graphs

Author: Jukrapong Tiammee, Yeol Je Cho and Suthep Suantai

Abstract: In this paper, we prove some fixed point theorems for non-self $G$-almost contractive mappings in Banach spaces with a directed graph and give some examples to illustrate our main results. The main results in this paper extend and generalize many known results in the literature therein. PDF |