Nonlinear Functional Analysis and its Applications

1. Front Matter

   Pages i-xxii

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2. Introduction to the Subject

   1. Introduction to the Subject

      • Eberhard Zeidler

      Pages 1-11

3. Introductory Typical Examples

   1. Introductory Typical Examples

      • Eberhard Zeidler

      Pages 12-142

4. Two Fundamental Existence and Uniqueness Principles

   1. Front Matter

      Pages 143-143

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   2. Compactness and Extremal Principles

      • Eberhard Zeidler

      Pages 145-167

   3. Convexity and Extremal Principles

      • Eberhard Zeidler

      Pages 168-186

5. Extremal Problems without Side Conditions

   1. Front Matter

      Pages 187-187

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   2. Free Local Extrema of Differentiable Functionals and the Calculus of Variations

      • Eberhard Zeidler

      Pages 189-228

   3. Potential Operators

      • Eberhard Zeidler

      Pages 229-243

   4. Free Minima for Convex Functionals, Ritz Method and the Gradient Method

      • Eberhard Zeidler

      Pages 244-269

6. Extremal Problems with Smooth Side Conditions

   1. Front Matter

      Pages 271-272

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   2. Lagrange Multipliers and Eigenvalue Problems

      • Eberhard Zeidler

      Pages 273-312

   3. Ljusternik-Schnirelman Theory and the Existence of Several Eigenvectors

      • Eberhard Zeidler

      Pages 313-350

   4. Bifurcation for Potential Operators

      • Eberhard Zeidler

      Pages 351-360

7. Extremal Problems with General Side Conditions

   1. Front Matter

      Pages 361-361

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   2. Differentiable Functionals on Convex Sets

      • Eberhard Zeidler

      Pages 363-378

   3. Convex Functionals on Convex Sets and Convex Analysis

      • Eberhard Zeidler

      Pages 379-406

   4. General Lagrange Multipliers (Dubovickii-Miljutin Theory)

      • Eberhard Zeidler

      Pages 407-452

8. Saddle Points and Duality

   1. Front Matter

      Pages 453-455

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   2. General Duality Principle by Means of Lagrange Functions and Their Saddle Points

      • Eberhard Zeidler

      Pages 457-478

As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the Teubner-Texte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.

• Mathematica
• calculus
• functional analysis
• optimization

• Sektion Mathematik, Leipzig, German Democratic Republic

  Eberhard Zeidler

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