Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations

Heikkilä, Seppo; Lakshmikantham, V.

doi:10.4324/9780203746493

Cited by 50 publications

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“…nonincreasing) sequence in E and x n → x * as n → ∞, then x n x * (resp. x n x * ) for all n ∈ N. A few details of a partially ordered normed linear space can be found in Dhage [6] and the conditions guaranteeing the regularity of E can be found in Guo and Lakshmikantham [7], Heikkilä and Lakshmikantham [8], Zeidler [9] and the references therein. A mapping…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

2017

Commun. Optim. Theory

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Abstract. In this paper, we prove the existence and uniqueness results for approximate solutions of a nonlinear periodic boundary value problem of first order nonlinear functional differential equations via construction of an algorithm. The main results rely on the Dhage iteration method embodied in a recent hybrid fixed point principle of Dhage (2014) in a partially ordered normed linear space. Examples are also furnished to illustrate the hypotheses and the abstract results of this paper.

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Section: Auxiliary Resultsmentioning

confidence: 99%

“…{−K} K = {θ }, where θ is a zero element of E. The details of cones and their properties may be found in Guo and Lakshmikantham [7], Heikkilä and Lakshmikantham [8] and references therein. We define an order relation ≤ in E by…”

Section: And Iii)mentioning

confidence: 99%

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

2017

Commun. Optim. Theory

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“…The Existence of mild solutions is developed by Heikkila and Lakshmikantham [1], Kamenski et al [2] and Hino and Murakami [3] for some semi-linear functional differential equations with the finite delay. Recently, there has been a significant development in functional evolution equations and inclusions; see Abbas and Benchohra [4], Baghli and Benchohra [5,6], Baliki and Benchohra [7,8], Benchohra, Henderson and Medjadj [9], Benchohra and Medjadj [10] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Successive approximations for functional evolution equations and inclusions

2017

J. Nonlinear Funct. Anal.

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“…In some cases, such indexation on the natural number are not su¢ cient to show that successive approximation from lower or upper solutions for a particular set of …xed points actually computes an extremal …xed point. See the example in Davey and Priestley [24], section 8.16 or Heikkila and Lakshmikantham [36], example 1.1.1. In such cases, one can still de…ne iterations on well-de…ned index sets that are subsets of chains.…”

Section: Computational Fixed Point Theorymentioning

confidence: 99%

“…For a more extensive discussion, see excellent surveys in Amann [4], Guo and Lakshmikantham [35], Heikkila and Lakshmikantham [36] and Jachymski [40].…”

Section: Fixed Point Theory In Ordered Spacesmentioning

confidence: 99%

Isotone Recursive Methods: The Case of Homogeneous Agents

Reffett

Handbook on Optimal Growth 1

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractOver the last decade, isotone recursive methods have provided uni…ed catalog of results on existence, characterization, and computation of Markovian Equilibrium Decision Processes (MEDPs) in in…-nite horizon economies where the second welfare theorem fails. Such economies include models with production nonconvexities, taxes, valued …at money, models with monopolistic competition, behavioral heterogeneity, and incomplete markets. In this paper, we survey this emerging class of methods. Our methods use a qualitative approach to economic equilibria …rst introduced in the work in operations research by Veinott and Topkis. As the methods emphasize the role of order, they are amenable for obtaining conditions for monotone comparison theorems on the space of economies. We are also able to describe monotone iterative procedures that provide the needed foundations for a theory of numerical solutions for MEDPs and stationary Markov equilibrium (SME). One interesting additional result of independent interest is we construct su¢ cient conditions for the existence of a new class of envelope theorems for nonconcave programming problems.Email addresses: Manjira.Datta@asu.edu, Kevin.Re¤ett@asu.edu. We are deeply indebted to Len Mirman and Olivier Morand for numerous lengthy discussions concerning many issues discussed in this survey. Many of the results presented in this paper were developed originally in some form during our joint work with Len and Olivier over the last …ve years. We dedicate this paper to Len Mirman on the occasion of his sixty-…fth birthday. Indeed, this paper would not have been written without Len's ongoing pioneering work on equilibrium growth under uncertainty. We also thank Elena Antoniadou, Hector Chade, John Coleman, Jeremy Greenwood, Seppo Heikkila, Ken Judd, Tom Krebs, Cuong Le Van, Robert Lucas, Jr., Jianjun Miao, Chris Shannon, John Stachurski, Yiannis Vailakis, Charles Van Marrewijk, Jean-Marie Viaene, Itzhak Zilcha, and especially Robert Becker and Manuel Santos for many helpful conversations over the past years. All mistakes remain our own.

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Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations

Cited by 50 publications

References 0 publications

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

Successive approximations for functional evolution equations and inclusions

Isotone Recursive Methods: The Case of Homogeneous Agents

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