Recall that a modular form f of weight 0 is a holomorphic function that is invari. Arabian journal of mathematics best proximity points in. Our results can be, in particular, applied to l1, showing that asymptotically nonexpansive mappings have a. Hanebaly, a fixed point theorem and its application to integral equations in modular function spaces, proceedings of the american mathematical society, vol. Iterative approximation of fixed point of multivalued quasi. Riemann surfaces, conformal classes of, when the genus and the euler characteristic. Pdf we prove the existence of fixed points of monotone. In geometric questions it is often more convenient to take the unit disc as the domain of the modular functions. The modular function conformally maps the curvilinear triangle fig.
We also discuss the concept of modular function and prove fixed point results for weakly modular contractive maps in modular function spaces. Dening a norm, particular banach spaces of functions can be consid. Following the same direction, one may think of a nonlinear version of modular function spaces 74. The purpose of this paper is to study the existence and uniqueness of common fixed point results in partially ordered modular function spaces. Iterative approximation of fixed point of multivalued. Fixed point theorems for contraction mappings in modular. Banach operator pairs and common fixed points in modular function spaces. A patatype fixed point theorem in modular spaces with.
Modular function spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces such as. The existence of fixed points of singlevalued mappings in modular function spaces has been studied by many authors. Kkm and ky fan theorems in modular function spaces fixed. Pdf asymptotically nonexpansive mappings in modular. A korovkin theorem in multivariate modular function spaces. Introduction the theory of modular spaces was initiated by nakano 14 in 1950 in connection with the theory of order spaces and redened and generalized by musielak and orlicz in 1959. In this paper, we initiate the study of approximating. Sep 23, 2011 in modular function spaces, we introduce knasterkuratowskimazurkiewicz mappings in short kkmmappings and prove an analogue to ky fan s fixed point theorem. Metric spaces are seen as a nonlinear version of vector spaces endowed with a norm. Common fixed points in partially ordered modular function spaces. Best proximity points in modular function spaces pdf. This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like lebesgue, orlicz, musielakorlicz, lorentz.
Moduli spaces often carry natural geometric and topological structures as well. This chapter introduces the general notions related to modular function spaces. Pdf banach operator pairs and common fixed points in. Apr 24, 2019 journal of function spaces table of contents journal of function spaces publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. Some fixed point theorems in modular function spaces. The moduli space is, therefore, the positive real numbers. Pdf onelocal retract and common fixed point in modular. We now prove a much more precise result about the dimension of spaces of mod ular forms. We have tried to generalize the idea of best proximity points in modular function spaces and to present an application for integral operators in orlicz function spaces, endowed with an orlicz function modular. We may identify the noneuclidean triangle formed by 0. On relative weighted summability in modular function spaces. An example of an orlicz function which is uc is 17, 18. A modular form for g of weight k is a function on h satisfying the above functional equation for all matrices in g, that is holomorphic on h and at all cusps of g. Corresponding to each there is a solution of, which determines a conformal class and the corresponding field of elliptic functions.
Fixed points of multivalued maps in modular function spaces. Pdf a convexity property in modular function spaces. Oct 11, 2012 the purpose of this paper is to give an outline of the recent results in fixed point theory for asymptotic pointwise contractive and nonexpansive mappings, and semigroups of such mappings, defined on some subsets of modular function spaces. Abstract let be a modular function satisfying a 2type condition and l the cor responding modular space. We prove the existence of fixed points of monotonecontraction mappings in modular function spaces. In modular function spaces, we introduce knasterkuratowskimazurkiewicz mappings in short kkmmappings and prove an analogue to ky fan s fixed point theorem. Kkm and ky fan theorems in modular function spaces. It is easier to construct functions that transform in a certain way under the action of. Fixed point theory in modular function spaces mohamed a. Asymptotically regular mappings in modular function spaces. The main result in this paper states that if c is a bounded and a.
This is the modular version of the ran and reurings fixed point theorem. As an application, we study the existence of solutions of integral equations in modular function spaces. The new feature is that the elements of the vector spaces are functions, and the spaces are in nite dimensional. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are close. The approximation of fixed points in such spaces via convergence of an iterative process for singlevalued mappings has also been attempted very recently by dehaish and kozlowski fixed point theory appl. Asymptotically nonexpansive mappings in modular function spaces. Best proximity points in modular function spaces pdf paperity. Recently, khan and abbas initiated the study of approximating fixed points of multivalued nonlinear mappings in modular function spaces. In particular, we know that the uc property of the modular in orlicz spaces is satisfied if and only if the orlicz function is uc 15, 16.
Advancements in fixed point theory in modular function spaces. In the present paper, we give a modular version of the korovkin theorem in multivariate modular function spaces. Nonlinear semigroups in modular function spaces mohamed a. Continued fractions and modular functions 7 it is straightforward to compute the. The objective of this paper is to initiate the study of common fixed point results in partially ordered modular function spaces. It is our purpose in this study to continue this recent trend in the study of fixed point theory of multivalued nonlinear mappings in modular function spaces. We also discus the extension of these results to the case of pointwise monotonecontraction mappings in modular function spaces.
The results discussed in this chapter is used throughout the rest of the book. Table of contents page 8 journal of function spaces. In this paper we prove that a modular function space satisfies a property that implies uniformopial and uniformkadecklee proper ties with respect to convergence almost everywhere. Chapter 2 function spaces many di erential equations of physics are relations involving linear di erential operators. This chapter also presents the reader with an exhaustive list of examples that will frequently reoccur in later parts of the book. For an extensive study of fixed point theory in modular function spaces and the geometry of modular function spaces we refer to 8 and for modular space in 19,20. In particular, the map f 7fg is an isomorphism of m. Uniformly lipschitzian mappings in modular function spaces. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. Again, modular forms that vanish at all cusps are called cusp forms for g. Fixed point theorems for contraction mappings in modular metric spaces chirasak mongkolkeha wutiphol sintunavarat poom kumam in this article, we study and prove the new existence theorems of fixed points for contraction mappings in modular metric spaces. Modular function spaces are natural generalizations of both.
Some fixed point theorems in modular function spaces endowed. To construct a modular function, we have to construct a meromorphic function on h that is invariant under the action of. Pdf on monotone mappings in modular function spaces. Mar 25, 2017 the purpose of the present article is to extend the notion of relative convergence on modular function spaces using the weighted double \\alpha,\beta \density. Onelocal retract and common fixed point in modular function spaces.
Asymptotically nonexpansive mappings in modular function. Metric modular spaces are extensions of metric spaces, metric linear spaces, and classical modular linear spaces. On monotone contraction mappings in modular function spaces. These operators, like matrices, are linear maps acting on vector spaces. Pdf kkm and ky fan theorems in modular function spaces.
On monotone asymptotic pointwise nonexpansive mappings in. The purpose of this paper is to give an outline of the recent results in fixed point theory for asymptotic pointwise contractive and nonexpansive mappings, and semigroups of such mappings, defined on some subsets of modular function spaces. We give the definitions of statistically relatively modularly \\mathcal p\summability and its strong form for double sequences of functions. The modular uniform convexity in orlicz function spaces was initiated in the work of khamsi et al. Pdf asymptotically regular mappings in modular function. Musielak, orlicz spaces and modular spaces, lecture notes in mathematics, vol. The new feature is that the elements of the vector spaces are functions, and the spaces are in. Pdf nonlinear semigroups in modular function spaces. We have found sufficient conditions for the existence and uniqueness of best proximity points for cyclic maps in modular function spaces. The purpose of this paper is to study the existence of. Let be a modular function space where is a nonzero regular modular function see 6, 11, 12 for more. In most cases, particularly in applications to integral operators.
The topics covered include the classification of modulars, metrizability of modular spaces, modular transforms and duality between modular spaces, metric and modular topologies. As we said then, the extent to which we can express modular forms in terms of. Pdf asymptotically regular mappings in modular function spaces. Journal of function spaces table of contents journal of function spaces publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines. In the context of modular function spaces with 2property, khamsi 30 discussed the existence of nonlinear semigroups in musielakorlicz spaces and considered some applications to differential equations. Modos is a customizable modular shelving system of which the elements can be used to create a tabouret or standing desk too. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like lebesgue, orlicz, musielakorlicz, lorentz, orliczlorentz, calderonlozanovskii spaces, and others. T, where t is a onexpansive mapping in the modular sense. The cvector spaces of modular and cusp forms of weight k are denoted m k g and s k g, respectively. Modular function spaces are natural generalizations of both function and sequence variants of many important, from applications. Indeed throughout this book we have seen that a modular function space is a vector space endowed with a modular function. Table of contents page 8 journal of function spaces hindawi.
The purpose of this chapter is to give an introduction into the theory of modular function spaces and into the fixed point theory for nonlinear mappings, and semigroups of such mappings, defined. Larger sized elements allow you to create life size desks, tables, chairs and other items all without the use of tools. May 12, 2009 the purpose of this paper is to study the existence of fixed points for contractivetype and nonexpansivetype multivalued maps in the setting of modular function spaces. Mar 25, 2015 metric spaces are seen as a nonlinear version of vector spaces endowed with a norm.
The modular furniture system toolless and almost limitless furniture creation. These results extend several similar results proved in metric and banach spaces. Metric modular spaces vyacheslav v chistyakov springer. Fixed point theory in modular function spaces books pics.
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