Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Solvability Conditions for Some Systems of Nonlinear Non-Fredholm Elliptic Equations

Discontinuity, Nonlinearity, and Complexity 2(2) (2013) 159--165 | DOI:10.5890/DNC.2013.04.005

Vitali Vougalter

Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch 7701, South Africa

Download Full Text PDF

 

Abstract

We establish the existence and uniqueness of standing solitary wavelike solutions in H2 for certain systems of nonlocal nonlinear equations. These problems involve second order differential operatorswithout Fredholm property.

References

  1. [1]  Pelinovsky, D.E. and Yang, J. (2002), A normal form for nonlinear resonance of embedded solitons, The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences, 458( 2022), 1469- 1497.
  2. [2]  Cuccagna, S., Pelinovsky, D., and Vougalter, V. (2005), Spectra of positive and negative energies in the linearized NLS problem, Communications on Pure and Applied Mathematics, 58(1), 1-29.
  3. [3]  Vougalter, V. (2010), On threshold eigenvalues and resonances for the linearized NLS equation, Mathematical Modelling of Natural Phenomena, 5 (4), 448-469.
  4. [4]  Vougalter, V., Volpert, V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Analysis and Mathematical Physics, 2(4), 473-496.
  5. [5]  Volpert, V., Kazmierczak, B., Massot, M., and Peradzynski, Z. (2002), Solvability conditions for elliptic problems with non-Fredholm operators, Applicationes Mathematicae, 29 (2), 219-238.
  6. [6]  Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non Fredholm operators, Proceedings of the Edinburgh Mathematical Society, (2) 54(1), 249-271.
  7. [7]  Vougalter, V. and Volpert, V. (2010), On the solvability conditions for some non Fredholm operators, International Journal of Pure and Applied Mathematics, 60 (2), 169-191.
  8. [8]  Vougalter, V. and Volpert, V. (2012) On the solvability conditions for the diffusion equation with convection terms, Communications on Pure and Applied Analysis, 11 (1), 365-373.
  9. [9]  Vougalter, V. and Volpert, V. (2010), Solvability relations for some non Fredholm operators, International Electronic Journal of Pure and Applied Mathematics, 2(1), 75-83.
  10. [10]  Volpert, V. and Vougalter, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rendiconti dell'Istituto di Matematica dell'Universitàdi Trieste, 43 , 1-9.
  11. [11]  Vougalter, V., Volpert, V. (2012), Solvability conditions for a linearized Cahn-Hilliard equation of sixth order, Mathematical Modelling of Natural Phenomena, 7(2), 146-154.
  12. [12]  Ducrot, A., Marion, M., and Volpert, V. (2005), Systemes de réaction-diffusion sans propriété de Fredholm, Comptes Rendus Mathematique, 340(9), 659-664.
  13. [13]  Ducrot, A., Marion, M., and Volpert, V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances in Differential Equations, 13 (11-12), 1151-1192.
  14. [14]  Ducrot, A., Marion, M., and Volpert, V. (2009), Reaction- Diffusion Waves (with the Lewis Number Different From 1), Publibook, Paris.
  15. [15]  Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Documenta Mathematica, 16, 561-580.
  16. [16]  Volpert, V. and Vougalter, V. (2012), On the existence of stationary solutions for some systems of non-Fredholm integro-differential equations, Discontinuity, Nonlinearity and Complexity, 1(2), 85-98.