Fixed Points of Nonlinear and Asymptotic Contractions in the Modular Space

Abstract

A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.

Fixed point theorems for contraction mappings in modular metric spaces

“…Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [2][3][4][5][6][7][8][9][10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11][12][13] and others.…”

“…Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14][15][16][17][18] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4,[19][20][21][22][23][24][25][26].…”

Fixed points in modular spaces with new type contractivity

“…In this section, by using some ideas from [14,18] we will prove a fixed point theorem for a new type of contractivity as follows.…”

“…Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively by many mathematicians such as Arandelović, [8], Edelstein [9],Ćirić [10], Rakotch [11], Reich [12], Kirk [13]. In addition, Razani et al [14] proved some fixed point theorems of non linear and asymptotic contractions in modular spaces. Also, quasi-contraction mappings in modular spaces without ∆ 2 -condition were investigated by Khamsi [15].…”

Some fixed point theorems in cone modular spaces with a graph

“…These spaces were developed following the successful theory of Orlicz spaces [17]. Recently, several authors reformulated many problems of fixed point theory in modular spaces (see [3,[13][14][15]22] and references therein). In 2005, Echenique [7] studied fixed point theory by using graphs.…”