File Name: monotone operators in banach space and nonlinear partial differential equations .zip
In this chapter we present the basic theory of maximal monotone operators in reflexive Banach spaces along with its relationship and implications in convex analysis and existence theory of nonlinear elliptic boundary value problems. However, the latter field is not treated exhaustively but only from the perspective of its implications to nonlinear dynamics in Banach spaces. Unable to display preview.
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We deal with the existence and uniqueness of positive solutions to a class of nonlinear parabolic partial differential equations, by using some fixed point theorems for mixed monotone operators with perturbation. In this paper, we consider a class of nonlinear parabolic partial differential equations of the form where is a bounded smooth domain in and ; we denote. Problems related to nonlinear parabolic equations arise in many mathematical models of applied science, such as nuclear science, chemical reactions, heat transfer, population dynamics, and biological sciences, and have attracted a great deal of attention in the literature; see [ 1 — 6 ] and the references therein. In recent years, there are many results about existence, uniqueness, blowing-up, global existence, critical exponent, and other properties of the solution; see [ 4 , 5 , 7 — 12 ], among others. Some of the authors who investigated parabolic equations were using the method of upper and lower solutions; see [ 11 ], for example. Different from the works mentioned above, in the present paper, we will utilize some fixed point theorems for mixed monotone operators with perturbation to study the existence and uniqueness of positive solutions to the nonlinear parabolic partial differential equation 1. With this context in mind, the outline of this paper is as follows.
His research concerned partial differential equations and biological. This book is designed for students who have had no previous knowledge of the theory of heat conduction nor indeed of the general theory of partial differential equations. Books on partial differential equations and boundary value problems generally. This module extends the knowledge and skills that students have gained concerning matrices and systems of linear equations. It discusses how to translate physical problems into mathematics and covers such topics as differential equations , dynamics,. A similar discussion of the notions of hyperbolic, parabolic and elliptic partial differential equations which pertain to the case of two independent variables also appears. There is particular emphasis on the special functions which arise as solutions of differential equations which occur frequently in physics.
This book is concerned with basic results on Cauchy problems associated with nonlinear monotone operators in Banach spaces with applications to partial differential equations of evolutive type. This is a monograph about the most significant results obtained in this area in last decades but is also written as a graduate textbook on modern methods in partial differential equations with main emphasis on applications to fundamental mathematical models of mathematical physics, fluid dynamics and mechanics. Skip to main content Skip to table of contents. Advertisement Hide.
The existence and uniqueness results of entropy solutions are established. Full text available only in PDF format. Andreianov, M.
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Metrics details. Without assuming the existence of upper-lower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct two iterative schemes to approximate it. As applications, we research a nonlinear fractional differential equation with multi-point fractional boundary conditions. By using the obtained fixed point theorems of sum-type operator, we get the sufficient conditions which guarantee the existence and uniqueness of positive solutions. At last, a specific example is provided to illustrate our result. With a significant development and extensive applications in various differential and integral equations, nonlinear operators theory has been an active area of research in nonlinear functional analysis. Over the past several decades, much attention has been paid to various fixed point theorems for the single nonlinear operator, and a lot of important results have been obtained, see for example [ 1 — 10 ].