New PDF release: An Introduction to Minimax Theorems and Their Applications

By Maria do Rosário Grossinho, Stepan Agop Tersian

ISBN-10: 144194849X

ISBN-13: 9781441948496

ISBN-10: 1475733089

ISBN-13: 9781475733082

The ebook is meant to be an creation to serious aspect conception and its purposes to differential equations. even if the similar fabric are available in different books, the authors of this quantity have had the next objectives in brain:

  • to provide a survey of current minimax theorems,
  • to offer functions to elliptic differential equations in bounded domain names,
  • to contemplate the twin variational approach for issues of non-stop and discontinuous nonlinearities,
  • to provide a few parts of serious aspect thought for in the community Lipschitz functionals and provides functions to fourth-order differential equations with discontinuous nonlinearities,
  • to check homoclinic options of differential equations through the variational equipment.

The contents of the e-book encompass seven chapters, every one divided into numerous sections.
Audience: Graduate and post-graduate scholars in addition to experts within the fields of differential equations, variational equipment and optimization.

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Extra info for An Introduction to Minimax Theorems and Their Applications to Differential Equations

Example text

Proof. By (j) it follows that b ~ a > O. Suppose by contradiction that 0 and take 0 < E < ~. Let N1 EN be such that Kb = Minimization and Mountain-pass Theorems 38 inf xEoNl f (x) ~ b - c; and M > 0 be such that No c B M . 15, there exists a f-increasing homotopy 'fJ : [0,1] x X --+ X, such that 'fJ (1, fb-c n No) C fb+c' because No C BM and b> 2c;. ) : X --+ X is a homeomorphism, N2 = 'fJ (1, N l ) is open and aN2 = 'fJ (1, aNd· Moreover 0 E N2 and e r:J. N 2. Indeed, since b > 2c; we have f (0) ::; 0 < b - f (e) ::; 0 < b - 2c;, Then 0 and e belong to the set {x: If (x) - bl ~ 2c;} 2c;.

By (PS)c condition there exist If (x) - cl ~ EO =? fO, {3 > 0 such that 11/ (x)11 ~ {3. Otherwise, there exists a sequence (x j) j such that If (Xj) - cl ~ yand III (Xj)11 ~ y. By (P S) c condition it follows that c is a critical value, which contradicts to Kc = 0. Let 0 > 0 and f E (0, min (EO, (30/4)). 13 taking G = 0 and F = X . • Minimization and Mountain-pass Theorems 28 We introduce another (PS) type condition which generalizes (PSC)c condition. e. there are C> 0, R > 0 such that

A note on nonlinear eigenvalue problems for a class of differential equations. J. Diff. , 1971;9:536-548. [Ral] Rabinowitz P. The mountain-pass theorem: Theme and variations. : Springer-Verlag, 1982;237-269. [Ra2] Rabinowitz P. Minimax methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. , 1986. [Ram] Ramos, Miguel. Teoremas de Enlace na Teoria dos Pontos Criticos. Universidade de Lisboa, Faculdade de Ciencias, 1993. , Tsachev T. , The intrisic mountain pass principle.

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An Introduction to Minimax Theorems and Their Applications to Differential Equations by Maria do Rosário Grossinho, Stepan Agop Tersian


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