Mathematical Analysis and Applications: Selected Topics

By Hemen Dutta and Ravi P. Agarwal

An authoritative text that presents the current problems, theories, and applications of mathematical analysis research

Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors—a noted team of international researchers in the field— highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research.

This important text:

• Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc.
• Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided
• Offers references that help readers advance to further study

Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields.

• Language: English
• Publisher: Wiley
• Release date: Apr 11, 2018
• ISBN: 9781119414339

Book preview

Preface

This book is designed for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis in particular and in mathematics in general. The book aims to present theory, methods, and applications of the chosen topics under several chapters that have recent research importance and use. Emphasis is made to present the basic developments concerning each idea in full detail, and also contain the most recent advances made in the corresponding area of study. The text is presented in a self-contained manner, providing at least an idea of the proof of all results, and giving enough references to enable the interested reader to follow subsequent studies in a still developing field. There are 20 selected chapters in this book and they are organized as follows.

The chapter Spaces of Asymptotically Developable Functions and Applications presents the functional structure of the spaces of asymptotically developable functions in several complex variables. The authors also illustrate the notion of summability with some applications concerning singularly perturbed systems of ordinary differential equations and Pfaffian systems.

The chapter Duality for Gaussian Processes from Random Signed Measures proves a number of results for a general class of Gaussian processes. Two features are stressed, first the Gaussian processes are indexed by a general measure space; second, the authors adjust the associated reproducing kernel Hilbert spaces (RKHSs) to the measurable category. Among other things, this allows us to give a precise necessary and sufficient condition for equivalence of a pair of probability measures (in sample space), which determine the corresponding two Gaussian processes.

In the chapter Many-body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient, formulas are derived for solutions of many-body wave scattering problems by small impedance particles embedded in an inhomogeneous medium. The limiting case is considered when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotic of their solutions as a tends to zero. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied. Applications of this theory to creating materials with a desired refraction coefficient are given. A recipe is given for creating such materials by embedding into a given material many small impedance particles with prescribed boundary impedances.

The chapter Generalized Convex Functions and their Applications focuses on convex functions and their generalization. The definition of convex function along with some relevant properties of such functions is given first, followed by a discussion on a simple geometric property. Then the e-convex function is generalized and some of their properties are established. Moreover, a generalized s-convex function is presented in the second sense and the paper presents some new inequalities of generalized Hermite–Hadamard's type for the class of functions whose second local fractional derivatives of order α in absolute value at certain powers are generalized s-convex functions in the second sense. At the end, some applications to special means are also presented.

The chapter Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers presents an expository review and survey for analytic generalizations and properties of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, the Catalan function, the Catalan–Qi function, the q-Catalan–Qi numbers, and the Fuss–Catalan–Qi function.

The chapter Trace Inequalities of Jensen Type for Selfadjoint Operators in Hilbert Spaces: A Survey of Recent Results provides a survey of recent results for trace inequalities related to the celebrated Jensen's and Slater's inequalities and their reverses. Applications for various functions of interest such as power and logarithmic functions are also emphasized. Trace inequalities for bounded linear operators in complex Hilbert spaces play an important role in Physics, in general, and in Quantum Mechanics, in particular.

The chapter Spectral Synthesis and its Applications presents a survey about spectral analysis and spectral synthesis. The chapter recalls the most important classical results in the field and in some cases new proofs for them are given. It also presents the most recent results in discrete, nondiscrete, and spherical spectral synthesis together with some applications.

The chapter "Various Ulam–Hyers Stabilities of Euler–Lagrange–Jensen General (a, b; k = a + b)-Sextic Functional Equations" elucidates the historical development of well-known stabilities of various types of functional equations such as quintic, sextic, septic, and octic functional equations. It introduced a new generalized Euler–Lagrange–Jensen sextic functional equation, obtained its general solution and further investigated its various fundamental stabilities and instabilities by having employed the famous Hyers' direct method as well as the alternative fixed point method. The chapter is expected to be helpful to analyze the stability of various functional equations applied in the physical sciences.

The chapter A Note on the Split Common Fixed-Point Problem and its Variant Forms proposed new algorithms for solving the split common fixed point problem and its variant forms, and prove the convergence results of the proposed algorithms. The split common fixed point problems have found its applications in various branches of mathematics both pure and applied. It provides a unified structure to study a large number of nonlinear mappings.

The chapter Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a, b)-Sextic Functional Equations comprises the growth, importance and relevance of functional equations in other fields. Its fundamental and basic results of various stabilities are presented. The stability results of various rational and Euler–Lagrange–Jensen sextic functional equations are investigated. Application and geometrical interpretation of rational functional equation are also illustrated for the readers to study similar problems.

The chapter Attractor of the Generalized Contractive Iterated Function System deals with the problems to construct the fractal sets of the iterated function system for certain finite collection of F-contraction mappings defined on metric spaces as well as b-metric spaces. A new iterated function system called generalized F-contractive iterated function system is defined. Further, a method is presented to construct new fractals; where the resulting fractals are often self-similar but more general.

The chapter Regular and Rapid Variations and some Applications presents an overview of recent results on regular and rapid variations of functions and sequences and some their applications in selection principles theory, game theory, and asymptotic analysis of solutions of differential equations.

The chapter "n-Inner Products, n-Norms, and Angles Between Two Subspaces" discusses the concepts of n-inner products and n-norms for any natural number n, which are generalizations of the concepts of inner products and norms. It presents some geometric aspects of n-normed spaces and n-inner product spaces, especially regarding the notion of orthogonality and angles between two subspaces of such a space.

The chapter Proximal Fiber Bundles on Nerve Complexes introduces proximal fiber bundles of nerve complexes. Briefly, a nerve complex is a collection of filled triangles (2-simplexes) that have nonempty intersection. Nerve complexes are important in the study of shapes with a number of recent applications that include the classification of object shapes in digital images. The focus of this chapter is on fiber bundles defined by projections on a set of fibers that are nerve complexes into a base space such as the set of descriptions of nerve complexes. Two forms of fiber bundles are introduced, namely, spatial and descriptive, including a descriptive form BreMiller–Sloyer sheaf on a Vietoris–Rips complex. The introduction to nerve complexes includes a recent extension of nerve complexes that includes nerve spokes. A nerve spoke is a collection of filled triangles that always includes filled triangle in a nerve complex. A natural transition from the study of fibers that are nerve complexes is in the form of projections of pairs of fibers onto a local nervous system complex, which is a collection of nerve complexes that are glued together with spokes common to the nerve fibers. A number of results are given for fiber bundles viewed in the context of proximity spaces.

The chapter Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions deals with the approximation properties of the certain Baskakov–Szász operators. It estimates the results for these hybrid Baskakov–Szász type operators for exponential test functions. It also estimates a quantitative asymptotic formula for such operators. Mathematica software is used to estimate the results.

The chapter Well-Posed Minimization Problems via the Theory of Measures of Noncompactness presents an analysis of the minimization problems for functionals defined, lower bounded and lower semicontinuous on a closed subset of a metric space. The main focus is on the well-posedness of minimization problems from the viewpoint of the theory of measures of noncompactness. The minimization problems for several functionals defined on some Banach spaces are also investigated as well. Thus, the chapter clarifies the role of the theory of measures of noncompactness in the general approach to the well-posedness of minimization problems.

The chapter Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces discusses some important developments in the fixed point theory in metric spaces. Various advancements are explained in detail through useful and applicable results along with examples in generalized metric spaces.

The chapter The Basel Problem with an Extension presents some historical aspects to the famous Basel problem, which a number of brilliant mathematicians attempted, and which had remained unsolved for over 90 years. It was the genius Euler who provided a masterful solution and laid the foundations to the famous Riemann zeta function and the analysis of series. The chapter then investigates a related Euler sum and provides an explicit analytical representation, a closed form solution. The related Euler sum also represented in terms of logarithmic and hypergeometric functions. The integrals in question will be associated with the harmonic numbers of positive terms. A few examples of integrals provide an identity in terms of some special functions.

The chapter Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory focuses on the study of the coupled fixed point and coupled coincidence point problems for single- and multi-valued operators. The study of this chapter is based on appropriate fixed point theorems in two types of generalized metric spaces. Some applications are also given to illustrate part of the abstract results presented in this chapter.

The chapter The Corona Problem, Carleson Measures, and Applications reviews the developments and generalizations of the Corona problem, the results on Carleson measures themselves and some applications of Carleson measures, in several different settings, starting from the disc in ℂ (where the corona problem was originally set) arriving to the unit ball in ℂn, to bounded strongly pseudoconvex domains and even to domains in the quaternionic space. Both the corona problem and the Carleson measures still need investigation, as many open problems have not been solved yet. The open problems are also highlighted in this chapter. Carleson measures were introduced by Lennart Carleson in 1962 to solve an interpolation problem about bounded holomorphic functions called the corona problem.

The editors are grateful to the contributors for their timely contribution and co-operation throughout the editing process. The editors have benefited from the remarks and comments of several other experts on the topics covered in this book. The editors would also like to thank the book handling editors at Wiley and production staff members for their continuous support and help. Finally, the editors offer sincere thanks to all those who contributed directly or indirectly in completing this book project.

August 25, 2017

Michael Ruzhansky

London, UK

Hemen Dutta

Guwahati, India

Ravi P. Agarwal

Texas, USA

About the Editors

Michael Ruzhansky is a Professor at the Department of Mathematics, Imperial College London, UK. He has published over 100 research articles in leading international journals. He has also published 5 books and memoirs, and 9 edited volumes. His major research topics are related to pseudo-differential operators, harmonic analysis, functional analysis, partial differential equations, boundary value problems, and their applications. He is serving on the editorial board of many respected international journals and served as the President of the International Society of Analysis, Applications, and Computations (ISAAC) in the period 2009–2013.

Hemen Dutta is a Senior Assistant Professor of Mathematics at Gauhati University, India. He did his Master of Science (M.Sc.) in Mathematics, Post Graduate Diploma in Computer Application (PGDCA) and Ph.D. in Mathematics from Gauhati University, India. He received his M.Phil. in Mathematics from Madurai Kamaraj University, India. His research interest includes summability theory and functional analysis. He has to his credit more than 50 research papers and three books so far. He has delivered talks at foreign and national institutions. He has also organized a number of academic events. He is a member of several mathematical societies.

Ravi P. Agarwal is a Professor and the chair of the Department of Mathematics at Texas A&M University-Kingsville, USA. He has been actively involved in research as well as pedagogical activities for the last 45 years. Dr. Agarwal is the author or co-author of more than 1400 scientific papers and 40 research monographs. His major research interests include numerical analysis, differential and difference equations, inequalities, and fixed point theorems. Dr. Agarwal is the recipient of several notable honors and awards. He is on the editorial board of several journals in different capacities and also organized International Conferences.

List of Contributors

Mujahid Abbas

Department of Mathematics

Government College University

Katchery Road, Lahore 54000

Pakistan

and

Department of Mathematics

King Abdulaziz University

Jeddah 21589

Saudi Arabia

Józef Banaś

Department of Nonlinear Analysis

Rzeszów University of Technology

Aleja Powstańców Warszawy 8 35-959 Rzeszów

Poland

Dragan Djurčić

Faculty of Technical Sciences Department of Mathematics

University of Kragujevac

34000 Čačak

Serbia

Silvestru Sever Dragomir

Mathematics Department College of Engineering & Science

Victoria University

Melbourne 8001

Australia

and

DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences

School of Computer Science and Applied Mathematics

University of the Witwatersrand

Johannesburg 2000

South Africa

Jorge Mozo Fernández

Departamento de Álgebra, Análisis Matemático Geometría y Topología Facultad de Ciencias

Campus Miguel Delibes

Universidad de Valladolid

Paseo de Belén, 7, 47011 Valladolid

Spain

Hendra Gunawan

Department of Mathematics

Bandung Institute of Technology

Bandung 40132

Indonesia

Bai-Ni Guo

School of Mathematics and Informatics

Henan Polytechnic University

Jiaozuo, Henan, 454010

China

Vijay Gupta

Department of Mathematics

Netaji Subhas Institute of Technology

Dwarka, New Delhi 110078

India

Palle E.T. Jorgensen

Department of Mathematics

The University of Iowa

Iowa City, IA 52242-1419

USA

Adem Kiliçman

Department of Mathematics Faculty of Science Putra University of Malaysia

43400 Serdang, Selangor

Malaysia

Ljubiša D.R. Kočinac

Faculty of Sciences and Mathematics Department of Mathematics

University of Niš, 18000 Niš

Serbia

Somayya Komal

Theoretical and Computational Science (TaCS) Centre, Department of Mathematics, Faculty of Science

King Mongkut's University of Technology Thonburi

Thung Khru, Bangkok 10140

Thailand

Poom Kumam

Theoretical and Computational Science (TaCS) Centre, Department of Mathematics, Faculty of Science

King Mongkut's University of Technology Thonburi

Thung Khru, Bangkok 10140

Thailand

Beri V. Senthil Kumar

Department of Mathematics

C. Abdul Hakeem College of Engg. and Tech.

Melvisharam 632 509, Tamil Nadu

India

Jelena V. Manojlović

Faculty of Sciences and Mathematics Department of Mathematics

University of Niš

18000 Niš

Serbia

L.B. Mohammed

Department of Mathematics Faculty of Science

Universiti Putra Malaysia

43400 Serdang, Selangor

Malaysia

Talat Nazir

Department of Mathematics

University of Jeddah

Jeddah 21589

Saudi Arabia

and

Department of Mathematics

COMSATS Institute of Information Technology

Abbottabad 22060

Pakistan

Narasimman Pasupathi

Department of Mathematics

Thiruvalluvar University College of Arts and Science

Tirupattur 635 901, Tamil Nadu

India

James F. Peters

Computational Intelligence Laboratory

University of Manitoba

WPG, MB, R3T 5V6

Canada

and

Department of Mathematics Faculty of Arts and Sciences

Adiyaman University

02040 Adiyaman

Turkey

Adrian Petruşel

Faculty of Mathematics and Computer Science

Babeş-Bolyai University

400084 Cluj-Napoca

Romania

Gabriela Petruşel

Faculty of Business

Babeş-Bolyai University

400084 Cluj-Napoca

Romania

Feng Qi

Institute of Mathematics

Henan Polytechnic University

Jiaozuo, Henan, 454010

China

and

College of Mathematics

Inner Mongolia University for Nationalities

Tongliao, Inner Mongolia, 028043

China

and

Department of Mathematics College of Science

Tianjin Polytechnic University

Tianjin, 300387

China

Alexander G. Ramm

Department of Mathematics

Kansas State University

Manhattan, KS 66506-2602

USA

John Michael Rassias

Pedagogical Department E.E., Section of Mathematics and Informatics

National and Capodistrian University of Athens

Athens 15342

Greece

Krishnan Ravi

Department of Mathematics

Sacred Heart College

Tirupattur 635 601, Tamil Nadu

India

Wedad Saleh

Department of Mathematics, Faculty of Science

Putra University of Malaysia

43400 Serdang, Selangor

Malaysia

Alberto Saracco

Dipartimento di Scienze Matematiche, Fisiche e Informatiche

Universitá degli Studi di Parma, 43124

Italy

Anthony Sofo

Victoria University

Melbourne City, Victoria 8001

Australia

László Székelyhidi

Institute of Mathematics

University of Debrecen, H-4010 Debrecen

Hungary

and

Department of Mathematics

University of Botswana

Gaborone

Botswana

Feng Tian

Department of Mathematics

Hampton University

Hampton, VA 23668

USA

Sergio Alejandro Carrillo Torres

Escuela de Ciencias Exactas e Ingeniería

Universidad Sergio Arboleda

Calle 74, 14-14, Bogotá

Colombia

Tomasz Zając

Department of Nonlinear Analysis

Rzeszów University of Technology

Aleja Powstańców Warszawy 8 35-959 Rzeszów

Poland

Chapter 1

Spaces of Asymptotically Developable Functions and Applications

Sergio Alejandro Carrillo Torres¹ and Jorge Mozo Fernández²

¹Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Calle 74, 14-14, Bogotá, Colombia

²Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Campus Miguel Delibes, Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain

2010 AMS Subject Classification Primary 34E05, 34E15

1.1 Introduction and Some Notations

This chapter is a short review of some results concerning asymptotic expansions in several complex variables, and summability. Different notions of asymptotic expansions have been developed in literature in last decades, trying to generalize classical results regarding asymptotics in one variable, dating back to H. Poincaré, and further developed by Wasow [1], Ramis [2], Écalle [3], Balser [4], Braaksma [5, 6], and many others. In several variables, as main contributions, let us mention those of Gérard and Sibuya [7] in the 1970s, Majima [8, 9] in the 1980s, and more recently the notion of monomial asymptotic expansions and monomial summability of Canalis-Durand et al. [10]. It is also worth to mention here that the notion of composite asymptotic expansions, developed by Fruchard and Schäfke [11], has been very useful in the treatment of singularly perturbed linear differential equations.

We shall review mainly the notion of strong asymptotic expansions of Majima and monomial asymptotic expansions of Canalis-Durand, Mozo, and Schäfke, and clarifying some relations between them. These notions will be applied to several problems concerning summability of solutions of systems of ordinary differential equations ODEs and Pfaffian systems. Concerning Pfaffian systems, we will state some recent progresses of S. Carrillo, see [12] for complete details.

This chapter does not intend to be complete at all, the objective is only to present, in the opinion of the authors, some of the more relevant contributions concerning this wide theory. For the new results presented here, relevant precise references are given in the text. Such a survey, as we know, does not exist in the literature, so we think that it may be useful as a starting point for researchers in the area.

Some of the main notations used throughout the text will be the following:

c01-math-001 will denote the set of natural numbers including 0, and c01-math-002 .

c01-math-003 , c01-math-004 , c01-math-005 ,…(boldface) will denote vectors: c01-math-006 , and so on.

If c01-math-007 , c01-math-008 , c01-math-009 means that c01-math-010 , c01-math-011 .

If c01-math-012 , c01-math-013 , c01-math-014 means that c01-math-015 , c01-math-016 .

c01-math-017 .

If c01-math-018 is an open set in c01-math-019 , c01-math-020 is the set of holomorphic functions on c01-math-021 . Similarly, c01-math-022 will denote the set of c01-math-023 functions on c01-math-024 , identifying c01-math-025 .

c01-math-026 is the ring of formal power series in the variables c01-math-027 .

c01-math-028 is the ring of convergent power series (at the origin) in the variables c01-math-029 .

If c01-math-030 and c01-math-031 (or c01-math-032 ), c01-math-033 will denote the sector of radius c01-math-034 and opening between the rays c01-math-035 and c01-math-036 , that is, the set

equation

Note that we are not restricted to the case c01-math-037 . If c01-math-038 , the sectors are to be understood in the Riemann surface of c01-math-039 , that is, the universal covering of c01-math-040 .

If c01-math-041 and c01-math-042 , c01-math-043 will denote the polysector

equation

In general, multiindex notation will be freely used throughout the text. So, c01-math-044 will denote c01-math-045 , c01-math-046 is c01-math-047 , and so on.

1.2 Strong Asymptotic Expansions

The notion of asymptotics in one variable was introduced by Poincaré, trying to give a meaning to the notion of a sum for divergent series, that had been controversed and widely study since the times of Euler and Abel and was developed by different authors during the twentieth century, as Birkhoff, Wasow, Hukuhara, and Sibuya. Some good expositions with emphasis in the history, of the theory of divergent series, are due to Ramis [13, 14]. An important improvement was done at the end of the 1970s by Ramis with the introduction of the theory of Gevrey summability, generalizing Borel summability. The objective was to give a meaning to the formal power series appearing as solutions of systems of ODEs with irregular singular points. In other words, to define a sum (or several sums) for these series. It turns out that not every solution of a system of ODEs with irregular singular points is summable in this sense, but nevertheless it was shown in the next years that solutions of these systems are multisummable, that is, choosing a direction in the complex plane that avoids a finite number of directions, the formal series solutions can be uniquely decomposed as a sum of formal series, and a process of c01-math-048 -summability (with different values of c01-math-049 ) can be applied to each of these summands in order to obtain a true holomorphic solution of the system.

Different essays were done in order to generalize this notion to several variables. Asymptotics and summability with parameters were used by different authors, from Wasow, Hukuhara, and others, but this was not a true complete notion of summability in several variables, that could be used, for instance, to study systems of partial differential equations. The first notion that clearly generalized that of Poincaré was due to H. Majima, who in 1983 presented what he called strong asymptotic expansions, and applied in 1984 to the study of integrable connections.

In this section, we shall recall the notion of strong asymptotic expansions. In the Gevrey case, his work was generalized by Haraoka [15]. Further developments of this notion were established by Zurro [16], Hernández [17], Galindo and Sanz [18] and the second author, among others.

Given c01-math-050 , and c01-math-051 , denote c01-math-052 the polysector c01-math-053 .

Definition 1.1

A total family of coefficients in c01-math-054 is a family of holomorphic functions

equation

Given such a family, let us define, for c01-math-055 , the c01-math-056 -approximant of c01-math-057 as

equation

where c01-math-058 denotes c01-math-059 , c01-math-060 .

Definition 1.2

Let c01-math-061 be a polysector in c01-math-062 , c01-math-063 , and c01-math-064 a total family of coefficients in c01-math-065 . We will say that c01-math-066 admits c01-math-067 as a strong asymptotic expansion in c01-math-068 if for every c01-math-069 , and every strict subpolysector c01-math-070 of c01-math-071 (see Remark 1.1), there are constants c01-math-072 such that, if c01-math-073 ,

equation

If c01-math-074 , the asymptotic expansion is called of c01-math-075 -Gevrey type if there exists constants c01-math-076 and c01-math-077 , depending on the subpolysector c01-math-078 , such that c01-math-079 can be chosen as

equation

Remark 1.1

In this remark, and throughout the text, we will say that c01-math-080 is a strict subpolysector of c01-math-081 if it is bounded, and moreover,

equation

We will denote this situation as c01-math-082 .

Let us denote c01-math-083 (resp. c01-math-084 ) the set of functions on c01-math-085 admitting a strong asymptotic expansion (resp. of c01-math-086 -Gevrey type). They are differential c01-math-087 -algebras. In particular, if c01-math-088 and c01-math-089 admits a family c01-math-090 as a strong asymptotic expansion, the derivative c01-math-091 admits c01-math-092 , where

equation

as strong asymptotic expansion. The unicity of the asymptotic expansion can be deduced from the following fact: take, for instance, c01-math-093 , c01-math-094 . Then, we have that

equation

in a proper subpolysector c01-math-095 . Taking limits when c01-math-096 tends to 0, we have

equation

Stability under derivation allows us to conclude. Due to this unicity, the total family of coefficients of strong asymptotic expansion for a function c01-math-097 is denoted as c01-math-098 . Denoting c01-math-099 , the formal power series

equation

is the formal power series of asymptotic expansion of c01-math-100 , denoted as c01-math-101 . In the particular case, this series, when expanded with respect to any variable, has coefficients holomorphic in a common disk around the origin in the other variables, then it determines the family c01-math-102 , and we will say that c01-math-103 has the series c01-math-104 as strong asymptotic expansion at the origin.

Definition 1.3

A total family of coefficients c01-math-105 in c01-math-106 is called consistent if each c01-math-107 and moreover the family

equation

equals c01-math-108 . If c01-math-109 , the family c01-math-110 is a consistent one.

The following characterization turns out to be very useful in order to work with strong asymptotic expansions in polysectors:

Theorem 1.1

Let c01-math-111 be a polysector in c01-math-112 and c01-math-113 . The following conditions are equivalent:

1. c01-math-114 .

2. If c01-math-115 , every derivative c01-math-116 is bounded in c01-math-117 .

3. If c01-math-118 , the restriction c01-math-119 can be extended to the whole space c01-math-120 as a c01-math-121 -function (considering c01-math-122 ).

Proof

(1) c01-math-123 (2) is evident, as strongly asymptotically developable functions are bounded in subpolysectors and the c01-math-124 -algebra c01-math-125 is stable by derivation.

(2) c01-math-126 (3). The subpolysector c01-math-127 is 1-regular in the sense of H. Whitney: for every c01-math-128 , there exists a neighborhood c01-math-129 of c01-math-130 , and a constant c01-math-131 such that, if c01-math-132 , a rectifiable curve c01-math-133 exists in c01-math-134 joining c01-math-135 and c01-math-136 and such that its length c01-math-137 satisfies a bound

equation

Given c01-math-138 , let us take a sequence c01-math-139 in c01-math-140 converging to c01-math-141 . If c01-math-142 , c01-math-143 , a curve c01-math-144 exists joining c01-math-145 and c01-math-146 , and such that c01-math-147 . Then

equation

where c01-math-148 is a bound for the first derivatives of c01-math-149 . The sequence c01-math-150 is a Cauchy sequence, so c01-math-151 can be extended to c01-math-152 . The same argument allows us to extend to c01-math-153 all the derivatives of c01-math-154 .

As c01-math-155 is 1-regular, these extensions define a c01-math-156 -function in the sense of Whitney, and the result follows.

(3) c01-math-157 (1). Consider a c01-math-158 extension c01-math-159 of c01-math-160 , and define

equation

on c01-math-161 , functions that patch together giving a holomorphic function in c01-math-162 . These functions define a total family of coefficients c01-math-163 . Taylor integral formula allows us to show that, on c01-math-164 ,

equation

where c01-math-165 is a bound of c01-math-166 on c01-math-167 .

In classical asymptotics in one variable, Borel–Ritt theorem is of great utility: It says that every formal power series is the asymptotic expansion of some function in an arbitrarily chosen sector. There is an analogue for strong asymptotic expansions, as follows:

Theorem 1.2 (Borel-Ritt)

Given a consistent family c01-math-168 of coefficients on a polysector c01-math-169 , there exists a function c01-math-170 such that c01-math-171 .

Sketch of proof

In [8], Majima proves a weaker result. More precisely, he shows that given a formal power series c01-math-172 , there exists a function c01-math-173 such that c01-math-174 . He follows the same idea as in the classical proof in one variable: from the expansion

equation

he constructs a function

equation

defining c01-math-175 and c01-math-176 appropriately in order to guarantee that the previous expression defines a holomorphic function in the polysector, and the bounds of the definition of the strong asymptotic expansion are verified. A modification of this proof is presented in [9] to show the general case stated here. He employs induction on c01-math-177 , assuming in each step that part of the functions c01-math-178 are zero.

Let us mention that another proof may be done as follows: a consistent family c01-math-179 verifies regularity conditions (in fact, it is formally holomorphic), and so, defines a c01-math-180 -function in the sense of Whitney (see [19] for precise definitions and details). So, there exists c01-math-181 , a c01-math-182 function, defined in a neighborhood of c01-math-183 in c01-math-184 , such that the restriction of its derivatives coincide with the functions c01-math-185 . Truncated Laplace transform of c01-math-186 defines an element of c01-math-187 and this allows us to conclude.

Strong asymptotic expansions may be defined from one variable asymptotic expansions using functional analysis techniques. For, let us consider the space c01-math-188 . For every c01-math-189 and c01-math-190 , define

equation

This number exists, by Theorem 1.1, and defines a family of seminorms c01-math-191 , that provides c01-math-192 with a Frechet space structure. If c01-math-193 is a Frechet space and c01-math-194 a polysector, the space c01-math-195 of strongly asymptotically developable functions with values in c01-math-196 may be defined. It turns out that there are natural isomorphisms c01-math-197 [17], and this allows us to make recurrence on the number of variables.

Returning to Theorem 1.1, let us denote c01-math-198 , that is, the space of holomorphic functions in c01-math-199 that are c01-math-200 in the sense of Whitney in the compact set c01-math-201 . Due to the regularity of c01-math-202 , this implies that they can be extended as a c01-math-203 -function to the whole space c01-math-204 . So, we have that c01-math-205 . The space c01-math-206 , as a subspace of c01-math-207 , is a nuclear space [20, 21], and hence, c01-math-208 is also nuclear. As c01-math-209 is dense in c01-math-210 , it can be shown that in fact, c01-math-211 , where c01-math-212 denotes the topological tensor product, as defined by A. Grothendieck. Precise details may be found in [22].

Let us comment briefly further properties of strong asymptotic expansions.

1. Consider a multidirection c01-math-213 on a polysector c01-math-214 ; c01-math-215 , where c01-math-216 . Assume that c01-math-217 is a holomorphic and bounded function having a strong asymptotic expansion on c01-math-218 : the bounds are verified when restricting to this multidirection (which in fact defines a c01-math-219 -dimensional real space). Then c01-math-220 , that is, the asymptotic expansion exists in the whole polysector. This result is shown in [23], and generalizes a result for the one variable case stated by Fruchard and Zhang [24].

2. From the sheaf of Whitney c01-math-221 -functions, Honda and Prelli construct in [25] the sheaf of strongly asymptotically developable functions by applying a functor of specialization. This functional setting appears to be rather interesting for future applications, and it deserves further development.

Most of the main results presented in this section have been stated in the context of the so-called Poincaré asymptotics. In the Gevrey case, they are still valid, with more or less straightforward modifications. In the literature on the subject you can find complete statements and proofs. Let us, nevertheless, mention some interesting issues concerning the Gevrey case.

For, recall that in one variable, Watson's lemma says that if c01-math-222 has a series c01-math-223 as c01-math-224 -Gevrey asymptotic expansion, and the opening of the sector c01-math-225 is strictly greater than c01-math-226 , then c01-math-227 is unique, and therefore is called the sum of c01-math-228 in c01-math-229 .

In the context of strong asymptotic expansions, a similar result is the following one:

Theorem 1.3

Let c01-math-230 be a polysector, c01-math-231 having a total family of coefficients c01-math-232 as c01-math-233 -Gevrey strong asymptotic expansion. Then, if for some c01-math-234 , the opening of c01-math-235 is greater than c01-math-236 , c01-math-237 is unique.

In order to give a proof of this result it is enough to use the fact that c01-math-238 , and techniques of topological vector spaces. In [26], the reader can find precise details in a rather more general setting.

1.3 Monomial Asymptotic Expansions

Several existing examples in the literature may lead us to a different notion of asymptotic expansions in several complex variables, more precisely, asymptotics expansions with respect to a monomial. Let us state here, as an example, one of these situations concerning resonant holomorphic foliations.

Martinet and Ramis [27] studied the analytic classification of resonant holomorphic foliations in dimension two, that is, foliations generated by a 1-form

equation

where c01-math-239 , c01-math-240 . Such a 1-form turns out to be formally equivalent to a certain formal normal form. In order to study the analytical equivalence, they introduce the space of formal power series

equation

Elements c01-math-241 define a formally holomorphic map, in the sense of Łojasiewicz. Considering the map c01-math-242 defined by c01-math-243 , define

equation

In fact, c01-math-244 is a formal space. But, if considering c01-math-245 -Gevrey series in c01-math-246 , they turn out to be different. Moreover, from c01-math-247 -Gevrey asymptotic expansions in c01-math-248 , with coefficients in the variables c01-math-249 and c01-math-250 , functions in two variables may be defined as representing a kind of asymptotic expansions in c01-math-251 Martinet and Ramis define also a notion of summability with respect to this monomial, but with the warning that La notion de resommabilité à plusieurs variables reste peu claire [ c01-math-252 ] [[27], Section 4.3]. The notion of monomial summability we will present here will clarify these definitions.

This example, and another ones that can be found in the literature (see, for instance, Chapter X in [28]) lead us to state a notion of asymptotic expansion and summability that depends on a monomial. This is what we will do briefly in the sequel.

Definition 1.4

Consider a monomial c01-math-253 , c01-math-254 , c01-math-255 . A monomial sector in c01-math-256 is a set

equation

By analogy, the number c01-math-257 is called the opening of the monomial sector.

Given a monomial c01-math-258 , denote c01-math-259 the set of analytic functions in a neighborhood of the origin c01-math-260 , such that its power series expansion at the origin is

equation

where c01-math-261 if c01-math-262 and c01-math-263 . c01-math-264 will denote the set c01-math-265 , with obvious identifications.

Definition 1.5

Let c01-math-266 be a holomorphic function defined over a monomial sector c01-math-267 , and c01-math-268 . We will say that c01-math-269 has c01-math-270 as monomial asymptotic expansion at the origin, or c01-math-271 -asymptotic expansion, if the following conditions are satisfied:

1. There exists c01-math-272 such that

equation

where c01-math-273 .

2. For every proper subsector c01-math-274 (i.e., c01-math-275 , with c01-math-276 , c01-math-277 ), and every c01-math-278 , there exists a constant c01-math-279 such that

equation

over c01-math-280 .

The asymptotic expansion is called of c01-math-281 -Gevrey type if, moreover:

1. c01-math-282 is of c01-math-283 -Gevrey type in c01-math-284 , that is, there exists c01-math-285 such that

equation

for some constants c01-math-286 and c01-math-287 (here, c01-math-288 denotes the supremum norm on c01-math-289 ).

2. c01-math-290 can be chosen as c01-math-291 , where c01-math-292 and c01-math-293 depend only on c01-math-294 , but they do not depend on c01-math-295 .

The definition of monomial asymptotic expansion is equivalent to the following one: There exists a family of holomorphic functions defined in a fixed neighborhood of c01-math-296 , c01-math-297 , such that, given a monomial subsector c01-math-298 and c01-math-299 ,

equation

on c01-math-300 , for appropriate constants c01-math-301 .

Let us observe that if a formal series c01-math-302 is the c01-math-303 -Gevrey monomial asymptotic expansion of a function c01-math-304 , writing

equation

the coefficients of c01-math-305 satisfy bounds

equation

Equivalently, c01-math-306 is a c01-math-307 -Gevrey series in c01-math-308 (with coefficients holomorphic in a common neighborhood of c01-math-309 in c01-math-310 ), and c01-math-311 -Gevrey series in c01-math-312 (with coefficients holomorphic in a common neighbourhood of c01-math-313 in c01-math-314 ).

The main tools to study monomial asymptotics and summability are the following operators, c01-math-315 and c01-math-316 , which allow us to reduce monomial asymptotic expansions to asymptotic expansions in one variable. Let us define first the operator c01-math-317 , acting on the space of formal power series.

For, given a formal power series c01-math-318 , consider the filtration by the powers of c01-math-319 , and let us write uniquely

equation

where c01-math-320 , the set of series such that c01-math-321 does not divide any of its terms. Define

equation

Note that c01-math-322 . A similar operator can be constructed in the analytic setting: consider c01-math-323 , c01-math-324 being a c01-math-325 -sector. Suppose first that c01-math-326 . The function c01-math-327 is defined if c01-math-328 , c01-math-329 , and so, it admits a Laurent expansion

equation

If c01-math-330 , the term c01-math-331 can be rewritten, taking into account that c01-math-332 , as

equation

Define c01-math-333 , and consider

equation

It is defined on c01-math-334 , for appropriate c01-math-335 small enough, and it verifies c01-math-336 .

For a general monomial c01-math-337 , c01-math-338 can be decomposed as

equation

where c01-math-339 , c01-math-340 . This decomposition is obtained explicitly solving a Vandermonde type linear system. Applying the previous operator c01-math-341 , we obtain a holomorphic function c01-math-342 such that:

1. c01-math-343 .

2. Fixing c01-math-344 , the Taylor expansion at the origin is an element of c01-math-345 .

This function c01-math-346 is uniquely determined. It allows us to transfer asymptotic properties of c01-math-347 to asymptotic properties of c01-math-348 , in the new variable c01-math-349 . More precisely:

Theorem 1.4

Let c01-math-350 be a c01-math-351 -sector, c01-math-352 . Assume that c01-math-353 , where c01-math-354 , and all the functions c01-math-355 have a common disk of convergence. Then, the following conditions are satisfied:

1. c01-math-356 has c01-math-357 as c01-math-358 -asymptotic expansion over c01-math-359 .

2. c01-math-360 has c01-math-361 as c01-math-362 -asymptotic expansion.

Moreover, if we assume that the asymptotic expansions are of c01-math-363 -Gevrey type, then the previous equivalences are still valid.

This transfer between monomial and classical summability allows us to define properly summability with respect to a monomial.

Definition 1.6

Let c01-math-364 , c01-math-365 , and c01-math-366 be given. Let c01-math-367 be a c01-math-368 -sector. We will say that c01-math-369 is c01-math-370 summable in c01-math-371 if c01-math-372 , and there exists c01-math-373 having c01-math-374 as c01-math-375 -Gevrey asymptotic expansion in c01-math-376 . An adaptation of Watson's lemma allows us to prove the unicity of such c01-math-377 . In fact, functions having the null series as c01-math-378 -Gevrey monomial asymptotic expansion with respect to c01-math-379 are exactly the functions that are exponentially small, that is, in each proper monomial subsector, they satisfy bounds

equation

c01-math-380 is called c01-math-381 summable in a direction c01-math-382 if there exists a c01-math-383 -sector c01-math-384 , bisected by c01-math-385 , of opening greater than c01-math-386 , such that c01-math-387 in c01-math-388 -summable in c01-math-389 .

We will say shortly that c01-math-390 is c01-math-391 -summable if it is c01-math-392 -summable in every direction with finitely many exceptions modulo c01-math-393 . The space of c01-math-394 summable series will be denoted by c01-math-395 .

In one variable, it is crucial to establish the result about the incompatibility of summability in different levels (i.e., with respect to different values of c01-math-396 ). More precisely, J.-P. Ramis shows that if a formal power series c01-math-397 is both c01-math-398 - and c01-math-399 -summable, with c01-math-400 , then it is convergent (this result is known as Ramis' Tauberian theorem). In order to make a precise statement of a similar result for monomial summability, we must first notice that, if c01-math-401 , there is a natural equality c01-math-402 . So, in order to compare two algebras c01-math-403 and c01-math-404 , we must take into account this fact and assume that there does not exist c01-math-405 such that c01-math-406 . A particular case is the following:

Proposition 1.1

If c01-math-407 , then c01-math-408 .

This is a consequence of the properties of the operator c01-math-409 : If c01-math-410 , then c01-math-411 is both c01-math-412 - and c01-math-413 -summable as a series in c01-math-414 with coefficients in c01-math-415 . Applying Ramis' Tauberian theorem, c01-math-416 is convergent, so it is c01-math-417 .

Consider now the general case:

Theorem 1.5

Consider the differential algebras c01-math-418 and c01-math-419 . Then, if the three numbers c01-math-420 , c01-math-421 , and c01-math-422 are not equal, c01-math-423 .

Proof

In [12], two different proofs of this interesting result are shown. We will follow schematically the first of them.

If c01-math-424 , it is the previous proposition. Assume c01-math-425 .

If c01-math-426 , it is enough to observe that c01-math-427 is a c01-math-428 -summable series in c01-math-429 , and moreover, it is c01-math-430 - Gevrey, so it is convergent. Analogously, if c01-math-431 .

Suppose now that c01-math-432 . Positive numbers c01-math-433 , c01-math-434 , c01-math-435 , c01-math-436 exist such that

equation

Fixing c01-math-437 , the formal series in c01-math-438 ,

c01-math-439

is c01-math-440 - and c01-math-441 -summable, so, if c01-math-442 , it is convergent. From this, it can be deduced that c01-math-443 converges, using complex analysis standard arguments. If c01-math-444 , consider c01-math-445 , c01-math-446 with c01-math-447 and observe that c01-math-448 , c01-math-449 .

Remark 1.2

As has been said before, another proof using properties of blow-ups is given in [12].

Let us finish this section linking monomial asymptotic expansions with strong asymptotic expansions. For, assume c01-math-450 , and let us consider c01-math-451 , c01-math-452 a function admitting a monomial asymptotic expansion with respect to c01-math-453 . Let c01-math-454 , c01-math-455 be two sectors in c01-math-456 , such that c01-math-457 . If c01-math-458 , c01-math-459 , there exists c01-math-460 a c01-math-461 -sector strictly contained in c01-math-462 , and such that c01-math-463 .

If c01-math-464 is the monomial asymptotic expansion of c01-math-465 in c01-math-466 , c01-math-467 as asymptotic expansion in the variable c01-math-468 in c01-math-469 , c01-math-470 being the sector c01-math-471 . Restrict c01-math-472 to c01-math-473 , c01-math-474 , c01-math-475 ; it can be extended as a c01-math-476 -function to the whole space c01-math-477 . As c01-math-478 , the restriction of c01-math-479 to c01-math-480 admits a c01-math-481 extension to c01-math-482 . As a consequence, c01-math-483 has a strong asymptotic expansion in c01-math-484 . We have shown:

Theorem 1.6

Monomial asymptotic expansion implies strong asymptotic expansion. More precisely, if a holomorphic function defined over a monomial sector c01-math-485 has a monomial asymptotic expansion, then it has a strong asymptotic expansion when restricted to every polysector included in c01-math-486 .

1.4 Monomial Summability for Singularly Perturbed Differential Equations

Let us apply the notion of monomial summability to the study of the so-called doubly singular systems of ODEs, summarizing here the main results of [10, 29]. For, let us consider a differential system as follows:

1.1 equation

where c01-math-488 is a holomorphic function c01-math-489 defined in a neighborhood of c01-math-490 , c01-math-491 . Assume that this equation has invertible linear part, that is, c01-math-492 is an invertible matrix. Such a system has a unique formal solution

equation

Implicit function theorem allows us to assume that c01-math-493 , that is, c01-math-494 . Assume, for simplicity, that c01-math-495 , and write

equation

Plugging in the series c01-math-496 in the equation leads to a recurrence equation as

equation

and the c01-math-497 turn out to be unique, holomorphic functions in a neighborhood of the origin. A majorant series argument shows that c01-math-498 is 1-Gevrey in c01-math-499 .

A similar argument, using Nagumo norms, shows that c01-math-500 is also 1-Gevrey as a formal power series in c01-math-501 . Precise details of these computations may be found in [29] and [10]. See also [30] for details concerning Nagumo norms and for a precise and detailed study of systems analogous to (1.1) but with c01-math-502 .

Consider now the linear case, that is,

1.2 equation

This system, from the point of view of summability with respect to c01-math-504 and c01-math-505 separately, has been studied in [29]. Again, assume c01-math-506 , and apply Borel transform c01-math-507 to (1.2), with respect to c01-math-508 . If c01-math-509 , we obtain an equation

equation

where c01-math-510 denotes the Borel transform of c01-math-511 , and c01-math-512 the Borel transform of c01-math-513 . As c01-math-514 is invertible, this equation may be solved for c01-math-515 small enough, and the solution extends in every direction that avoids the eigenvalues of c01-math-516 . If c01-math-517 are the directions of these eigenvalues, then we obtain the following theorem.

Theorem 1.7

c01-math-518 is 1-summable in c01-math-519 in direction c01-math-520 , for every c01-math-521 such that

equation

Using the fact that summability implies unicity if the opening of the sectors are wide enough, it is also shown in the aforementioned paper that c01-math-522 is 1-summable in c01-math-523 , in every direction c01-math-524 such that

equation

Moreover we have the following lemma:

Lemma 1.1

If a formal power series in two variables,

equation

is 1-Gevrey on c01-math-525 , and 1-Gevrey on c01-math-526 , and c01-math-527 , then it is also a c01-math-528 -Gevrey series in both variables, that is, there exists positive constants c01-math-529 and c01-math-530 , such that

equation

This result follows easily from the inequality

equation

So, the problem of the summability in both variables c01-math-531 and c01-math-532 can be stated. In this context, in [29] it is shown that c01-math-533 is c01-math-534 -summable. The sum is here defined via a weighted Borel–Laplace transform: from the formal solution

equation

construct

equation

This turns out to be convergent, and a weighted-Laplace transform may be applied, in the form

equation

from which the sum is obtained, for more details, see [29].

The nonlinear case is studied in [10], using monomial summability. More precisely, it is shown the following result:

Theorem 1.8

Consider the system of differential equations

equation

with c01-math-535 , c01-math-536 an invertible matrix. Then, the unique formal power series solution is c01-math-537 -summable.

The proof is rather technical, and uses Banach fixed point theorem in an appropriate space obtained after applying the operator c01-math-538 to the system of differential equations.

Monomial summability may be related with Borel–Laplace transforms, with weighted variables, in order to construct the sum. Using this, it can be shown that Theorem 1.8 implies easily that the only formal power series solution in summable with respect to either the variable c01-math-539 , or with respect to the variable c01-math-540 , obtaining similar results to those previously obtained in the linear case. In fact, in [31], the following result is shown:

Theorem 1.9

Let c01-math-541 be a c01-math-542 -Gevrey series in c01-math-543 (Definition 1.5). Then c01-math-544 is c01-math-545 -summable in direction c01-math-546 if and only if for some c01-math-547 , c01-math-548 is c01-math-549 -summable in direction c01-math-550 .

So, both approaches to asymptotic expansions in monomial sectors coincide.

We will not explore this in these pages, precise details are given in [12, 31].

1.5 Pfaffian Systems

One of the main applications for asymptotic expansions in several variables concerns doubly singular Pfaffian systems, with normal crossings, that is, systems of differential equations of the form

1.3 equation

1.4 equation

where c01-math-553 , c01-math-554 , c01-math-555 holomorphic functions defined in a neighborhood of the origin in c01-math-556 , and c01-math-557 , c01-math-558 , c01-math-559 , c01-math-560 . The system is said to satisfy the complete integrability condition, or, shortly, to be completely integrable, if the equality

equation

holds, for every c01-math-561 . These systems have been studied by Gérard and Sibuya [7] and by Majima [9]. Let us summarize here some of the main results obtained by these authors, concerning mainly the asymptotic behavior of the solutions.

For instance, Gérard and Sibuya studied Pfaffian systems of the simplified form (with respect to the previous one):

1.5 equation

1.6 equation

with the hypothesis of the invertibility of the linear parts of both parts of the system, that is, of the matrices

equation

and assuming complete integrability. It can be shown by indeterminate coefficients that there is a unique formal power series c01-math-564 solution of (1.5) and (1.6). Write this solution as

equation

This solution, by classical results on holomorphic ordinary differential equations (ODEs), turns out to be c01-math-565 -summable in c01-math-566 , and c01-math-567 -summable in c01-math-568 , considering summability with respect to one variable, parameterized by the other one. This second condition means that there exists a finite number of directions

equation

such that if c01-math-569 , there exists c01-math-570 , where c01-math-571 is a disk,

c01-math-572

, c01-math-573 , with

equation

Write

equation

with c01-math-574 . We have bounds

equation

where c01-math-575 denotes a circle of radius c01-math-576 around the origin. So, each function c01-math-577 turns out to be c01-math-578 -summable, with sum c01-math-579 . As it is also a convergent series, then c01-math-580 in fact converges in a neighborhood of c01-math-581 . So, the family of functions c01-math-582 glue together defining a function that coincides with c01-math-583 around the origin, so c01-math-584 glue in c01-math-585 , holomorphic in c01-math-586 , c01-math-587 , and solution of the equation. Then c01-math-588 converges. This is an adaptation of the proof given by Sibuya [32].

In the framework of strong asymptotic expansions, Pfaffian systems have been studied by Majima [9]. He considers systems (1.3) and (1.4) assuming that at least one of the matrices c01-math-589 , c01-math-590 is invertible, and proving several results. Under this assumption, and under complete integrability condition, he manages to show that the only formal solution of this system has a strong asymptotic expansion. In fact, this can be deduced from Majima's results about the existence of strongly asymptotically developable function solutions of systems of ODEs, and the hypothesis of complete integrability. More precisely, the following result is shown:

Proposition 1.2

Let us consider the Pfaffian system

1.7 equation

1.8 equation

Suppose that the system is completely integrable. Assume that c01-math-593 is a consistent total family of coefficients of strong asymptotic expansion, defining formally a solution of the Pfaffian system. Let c01-math-594 be a strictly proper sector with respect to c01-math-595 (see Remark 1.3). Then, there exists c01-math-596 solution of the system, having the family c01-math-597 as the strong asymptotic expansion.

Remark 1.3

The condition of c01-math-598 being a strictly proper sector with respect to c01-math-599 means the following: c01-math-600 is contained in c01-math-601 , where

equation

such that the interval c01-math-602 (considered in c01-math-603 ) does not contain completely a negative interval for c01-math-604 , that is, an interval where c01-math-605 is exponentially flat, c01-math-606 being an eigenvalue of c01-math-607 , and the same with c01-math-608 , c01-math-609 , c01-math-610 an eigenvalue of c01-math-611 .

The family c01-math-612 that appears in the statement can be obtained from a formal power series c01-math-613 solution of the system, so, this would mean that such a formal power series is summable in appropriate polysectors. The existence of c01-math-614 follows from the invertibility of c01-math-615 , c01-math-616 , and from the complete integrability of the system. These are very technical results, that we will not be developed further here, whose complete proofs are given in [9]. They can be generalized, assuming that at least one of the matrices c01-math-617 and c01-math-618 is invertible.

The conditions on the polysectors (they are strictly proper with respect to c01-math-619 ) allows us to think that monomial asymptotics could be used. Moreover, if c01-math-620 , two different levels of summability seem to appear, which would imply convergence. But complete integrability condition implies very serious restrictions on the Pfaffian system (1.7), (1.8). More precisely, we have the following proposition.

Proposition 1.3 [33]

Let us consider the Pfaffian system (1.7), (1.8), and denote c01-math-621 , c01-math-622 . Then:

1.

If c01-math-623 or c01-math-624 , then c01-math-625 is nilpotent.

2.

If c01-math-626 or c01-math-627 , then c01-math-628 is nilpotent.

3.

If c01-math-629 , c01-math-630 . Then, for every eigenvalue c01-math-631 of c01-math-632 , there exists an eigenvalue c01-math-633 of c01-math-634 such that c01-math-635 .

So, we must be careful when imposing integrability condition in these singularly perturbed Pfaffian systems. It may happen that this integrability condition forbids the system to have invertible matrices of the linear part at the origin. At least, this happens for a great number of values c01-math-636 . Nevertheless, in the absence of complete integrability condition, even if the existence of a formal solution cannot be guaranteed, if it exists, we can provide useful information about it regarding summability. More precisely we have:

Theorem 1.10

Consider the Pfaffian system (1.7) and (1.8), with previous notations.

1. Suppose that a formal solution exists. If c01-math-637 and c01-math-638 are invertible, and c01-math-639 , then the solution is convergent.

2. If the system is completely integrable and c01-math-640 is invertible, there is a unique formal solution, c01-math-641 -summable.

3. If the system is completely integrable and c01-math-642 is invertible, there is a unique formal solution, c01-math-643 -summable.

References

1 Wasow, W. (1965) Asymptotic Expansions for Ordinary Differential Equations, John Wiley & Sons, Inc.

2 Ramis, J.-P. (1980) Les séries c01-math-644 -sommables et leurs applications Complex analysis, microlocal calculus and relativistic quantum theory (Proceedings of International Colloquium, Centre of Physics, Les Houches, 1979), Lecture Notes in Physics, vol. 126, Springer-Verlag, Berlin, New York, pp. 178–199.

3 Écalle, J. (1981–1985) Les fonctions résurgentes. Tome I, II, III, Publications Mathématiques d'Orsay. Université de Paris-Sud, Département de Mathématique, Orsay.

4 Balser, W. (1994) From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics, No. 1582, Springer-Verlag, Berlin.

5 Braaksma, B.L.J. (1992) Multisummability of formal power series solutions of nonlinear meromorphic differential equations. Ann. Inst. Fourier Grenoble, 42, 517–540.

6 Balser, W., Braaksma, B.L.J., Ramis, J.-P., and Sibuya, Y. (1991) Multisummability of formal power series solutions of linear ordinary differential equations. Asympt. Anal., 5, 27–45.

7 Gérard, R. and Sibuya, Y. (1979) Étude de certains systèmes de Pfaff avec singularités, Lecture Notes in Mathematics, vol. 172, Springer-Verlag, Berlin, pp. 131–288.

8 Majima, H. (1983) Analogues of Cartan's decomposition theorem in asymptotic analysis. Funk. Ekvac., 26, 131–154.

9 Majima, H. (1984) Asymptotic Analysis for Integrable Connections with Irregular Singular Points, Lecture Notes in Mathematics vol. 1075, Springer-Verlag, Berlin.

10 Canalis-Durand, M., Mozo-Fernández, J., and Schäfke, R. (2007) Monomial summability and doubly singular differential equations. J. Differ. Equ., 233 (2), 485–511.

11 Fruchard, A. and Schäfke, R. (2013) Composite Asymptotic Expansions, Lecture Notes in Mathematics, vol. 2066, Springer-Verlag.

12 Carrillo, S. (2016) Monomial summability through Borel-Laplace transforms. Applications to sngularly perturbed differential equations and Pfaffian systems. PhD thesis. University of Valladolid.

13 Ramis, J.-P. (1993) Séries divergentes et théories asymptotiques. Bull. Soc. Math. France, 121, coll. Panoramas et Syntheses.

14 Ramis, J.-P. (2006) Les séries divergentes. Pour la Sci., 350, 132–139.

15 Haraoka, Y. (1989) Theorems of Sibuya-Malgrange type for Gevrey functions of several variables. Funk. Ekvac., 32, 365–388.

16 Zurro, M.A. (1997) A new Taylor type formula and c01-math-645 extensions for asymptotically developable functions. Stud. Math., 123 (2), 151–163.

17 Hernández, J.A. (1994) Desarrollos asintóticos en polisectores. Problemas de existencia y unicidad (Asymptotic expansions in polysectors. Existence and uniqueness problems). PhD dissertation. Universidad de Valladolid, Spain.

18 Galindo, F. and Sanz, J. (1999) On strongly asymptotically developable functions and the Borel-Ritt theorem. Stud. Math., 133, 231–248.

19 Malgrange, B. (1967) Ideals of Differentiable Functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, vii+106 pp.

20 Grothendieck, A. (1955) Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc., 16.

21 Douady, R. (1974) Produits tensoriels topologiques et espaces nucléaires. Astérisque, 16, I.01I.25.

22 Mozo-Fernández, J. (1999) Topological tensor products and asymptotic developments. Ann. Fac. Sci. Toulouse 6 Srie t., 8 (2), 281–295.

23 Lastra, A., Mozo-Fernández, J., and Sanz, J. (2012) Strong asymptotic expansions in a multidirection. Funk. Ekvac., 55 (2), 317–345.

24 Fruchard, A. and Zhang, C. (1999) Remarques sur les développements asymptotiques. Ann. Fac. Sci. Toulouse Math. (6), 8 (1), 91–115.

25 Honda, N. and Prelli, L. (2013) Multi-specialization and multi-asymptotic expansions. Adv. Math., 232, 432–498.

26 Sanz, J. (2003) Linear continuous extension operators for Gevrey classes on polysectors. Glasgow Math. J., 45, 199–216.

27 Martinet, J. and Ramis, J.-P. (1983) Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann. Sci. É.N.S. 4e. sér., t, 16 (4), 571–621.

28