By Antonio Cañada, Salvador Villegas
This publication highlights the present nation of Lyapunov-type inequalities via a close research. aimed at researchers and scholars operating in differential equations and people drawn to the purposes of balance concept and resonant platforms, the booklet starts off with an outline Lyapunov’s unique effects and strikes ahead to incorporate favourite effects bought long ago ten years. special proofs and an emphasis on simple principles are supplied for various boundary stipulations for traditional differential equations, together with Neumann, Dirichlet, periodic, and antiperiodic stipulations. Novel result of better eigenvalues, platforms of equations, partial differential equations in addition to variational ways are provided. To this appreciate, a brand new and unified variational perspective is brought for the remedy of such difficulties and a scientific dialogue of other kinds of boundary stipulations is featured.
Various difficulties make the examine of Lyapunov-type inequalities of curiosity to these in natural and utilized arithmetic. Originating with the research of the soundness homes of the Hill equation, different questions arose for example in platforms at resonance, crystallography, isoperimetric difficulties, Rayleigh style quotients and oscillation and periods of disconjugacy and it result in the research of Lyapunov-type inequalities for differential equations. This classical zone of
mathematics remains to be of significant curiosity and is still a resource of inspiration.
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Additional resources for A Variational Approach to Lyapunov Type Inequalities: From ODEs to PDEs
Moreover, we assume one of the following conditions: a. Z ˝ ˛ 0; ˛ 6Á 0 b. t. 123) has a unique solution. 23. 123) has been previously employed by different authors. The interested reader may consult [13, 25, 27, 29] for the case where the nonlinearity f is restricted in one direction. 44 2 A Variational Characterization of the Best Lyapunov Constants References 1. : On a Lyapunov criterion of stability. Am. J. Math. 71, 67–70 (1949) 2. : Analyse Fonctionnelle. Masson, Paris (1983) 3. : Lyapunov type inequalities and Neumann boundary value problems at resonance.
A Neumann problem at resonance with the nonlinearity restricted in one direction. Nonlinear Anal. 51, 739–747 (2002) 14. : Nonlinear eigenvalue problems and Fredholm alternative. , Takáˇc, P. ) Nonlinear Differential Equations. Research Notes in Mathematics Series, vol. 404, pp. 1–46. Chapman and Hall/CRC, London (1999) 15. : On an inequality of Lyapunov for disfocality. J. Math. Anal. Appl. 146, 495–500 (1990) 16. : Ordinary Differential Equations. Wiley, New York (1964) 17. : Existence and uniqueness of periodic solutions for Duffing equations across many points of resonance.
Proof. 0; d/ Therefore, problem PM(0,d) has only the trivial solution. x0 ; L/ a similar reasoning is valid and we obtain that problem PM(d,L) has only the trivial solution. L x0 /2 in the interval Œx0 ; L, we deduce that either problem PM(0,x0 ) or problem PM(x0 ,L) has only the trivial solution. 115) has only the trivial solution and therefore we have the desired conclusion. 2, we obtain the classical result related to the so-called nonuniform nonresonance conditions with respect to the first positive 2 eigenvalue L2 [24–26].
A Variational Approach to Lyapunov Type Inequalities: From ODEs to PDEs by Antonio Cañada, Salvador Villegas