najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-5-618-627najo-977Research ArticleMATHEMATICSМАТЕМАТИКАOn positive solutions of the homogeneous Hammerstein integral equationEshkabilovYu. Kh.

Tashkent

yusup62@mail.ruHaydarovF. H.

Tashkent

haydarov_imc@mail.ruNational University of UzbekistanUzbekistan20151508202565618627Copyright © Eshkabilov Y.K., Haydarov F.H., 20252025Eshkabilov Y.K., Haydarov F.H.Eshkabilov Y.K., Haydarov F.H.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/977

In this paper the existence and uniqueness of positive fixed points operator for a nonlinear integral operator are discussed. We prove the existence of a finite number of positive solutions for the Hammerstein type of integral equation. Obtained results are applied to the study of Gibbs measures for models on a Cayley tree. 

integral equation of Hammerstein typefixed point of operatorGibbs measureCayley treeReferencesM.A. Abdou, A.A. Badr. On a method for solving an integral equation in the displacement contact problem. J. Appl. Math Comput., 2002, 127, P. 65–78.M.A. Abdou, A.A. Badr. On a method for solving an integral equation in the displacement contact problem. J. Appl. Math Comput., 2002, 127, P. 65–78.R. Grimmer, J.H. Liu. Singular perturbations in viscoelasticity. Rocky Mountain J. Math., 1994, 24, P. 61–75.R. Grimmer, J.H. Liu. Singular perturbations in viscoelasticity. Rocky Mountain J. Math., 1994, 24, P. 61–75.J.B. Keller, W.E. Olmstead. Temperature of nonlinearly radiating semi-infinite solid. Quart. Appl. Math., 1972, 29, P. 559–566.J.B. Keller, W.E. Olmstead. Temperature of nonlinearly radiating semi-infinite solid. Quart. Appl. Math., 1972, 29, P. 559–566.M.A. Krasnoselskii, P.P. Zabrejko. Geometrical methods of Nonlinear Analysis. Springer-Verlag, 1984.M.A. Krasnoselskii, P.P. Zabrejko. Geometrical methods of Nonlinear Analysis. Springer-Verlag, 1984.W.E. Olmstead, R.A. Handelsman. Diffusion in a semi-infinite region with nonlinear surface dissipation. SIAM Rev., 1996, 18, P. 275–291.W.E. Olmstead, R.A. Handelsman. Diffusion in a semi-infinite region with nonlinear surface dissipation. SIAM Rev., 1996, 18, P. 275–291.M.A. Abdou. On the solution of linear and nonlinear integral equation. J. Appl. Math. Comput., 2003, 146, P. 857–871.M.A. Abdou. On the solution of linear and nonlinear integral equation. J. Appl. Math. Comput., 2003, 146, P. 857–871.M.A. Abdou, M.M. El-Borai, M.M. El-Kojok. Toeplitz matrix method and nonlinear integral equation of Hammerstein type. Journal of Computational and Applied Mathematics, 2009, 223, P. 765–776.M.A. Abdou, M.M. El-Borai, M.M. El-Kojok. Toeplitz matrix method and nonlinear integral equation of Hammerstein type. Journal of Computational and Applied Mathematics, 2009, 223, P. 765–776.M.A. Abdou, W.G. El-Sayed, E.I. Deebs. A solution of nonlinear integral equation. J. Appl. Math. Comput., 2005, 160, P. 1–14.M.A. Abdou, W.G. El-Sayed, E.I. Deebs. A solution of nonlinear integral equation. J. Appl. Math. Comput., 2005, 160, P. 1–14.A. Horvat-Marc. Positive solutions for nonlinear integral equations of Hammerstein type. Carpathian J. Math., 2008, 24 (2), P. 54–62.A. Horvat-Marc. Positive solutions for nonlinear integral equations of Hammerstein type. Carpathian J. Math., 2008, 24 (2), P. 54–62.J.A. Appell, E. de Pascale, P.P. Zabrejko. On the unique solvability of Hammerstein integral equations with non-symmetric kernels. Progress in Nonlinear Differential Equations and Their Applications, 2000, 40, P. 27–34.J.A. Appell, E. de Pascale, P.P. Zabrejko. On the unique solvability of Hammerstein integral equations with non-symmetric kernels. Progress in Nonlinear Differential Equations and Their Applications, 2000, 40, P. 27–34.J. Appell, A.S. Kalitvin. Existence results for integral equations: spectral methods VS. Fixed point theory, 2006, 7 (2), P. 219–234.J. Appell, A.S. Kalitvin. Existence results for integral equations: spectral methods VS. Fixed point theory, 2006, 7 (2), P. 219–234.D. Bugajewski. On BV-Solutions of some nonlinear integral equations. Integral Equations Operator Theory, 2003, 46, P. 387–398.D. Bugajewski. On BV-Solutions of some nonlinear integral equations. Integral Equations Operator Theory, 2003, 46, P. 387–398.P.S. Milojevic. Solvability and the number of solutions of Hammerstein equations. Electronic J. Diff. Equat., 2004, 54, P. 1–25.P.S. Milojevic. Solvability and the number of solutions of Hammerstein equations. Electronic J. Diff. Equat., 2004, 54, P. 1–25.H.K. Pathak, G.S. Jeong, Y.J. Cho. Recent fixed point theorems and their applications. Proceedings of nonlinear functional analysis and applications, 1997, 2, P. 89–114.H.K. Pathak, G.S. Jeong, Y.J. Cho. Recent fixed point theorems and their applications. Proceedings of nonlinear functional analysis and applications, 1997, 2, P. 89–114.P. Kumlin. A note on fixed point theory. Functional Analysis. Mathematics, Chalmers ℘ GU, 2003/2004, TMA 401/MAN 670.P. Kumlin. A note on fixed point theory. Functional Analysis. Mathematics, Chalmers ℘ GU, 2003/2004, TMA 401/MAN 670.W.V. Petryshyn, P.M. Fitzpatrik. New existence theorems for nonlinear integral equations of Hammerstein type. Trams. Amer. Math. Soc., 1970, 160, P. 39–63.W.V. Petryshyn, P.M. Fitzpatrik. New existence theorems for nonlinear integral equations of Hammerstein type. Trams. Amer. Math. Soc., 1970, 160, P. 39–63.F.E. Browder. Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to Nonlinear Functional Analysis (ed. C.H. Zarantonello, Academic Press), 1971, P. 99– 154.F.E. Browder. Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to Nonlinear Functional Analysis (ed. C.H. Zarantonello, Academic Press), 1971, P. 99– 154.H. Brezis, F.E. Browder. Nonlinear integral equations and systems of Hammerstein type. Advances in Math, 1975, 18, P. 115–147.H. Brezis, F.E. Browder. Nonlinear integral equations and systems of Hammerstein type. Advances in Math, 1975, 18, P. 115–147.E.Yu. Elenskaya. The existence fixed point of monoton left-continuous operators in space with right cone. Izvestiya vuzov. Matematika, 2011, 10, P. 40–47 [in Russian].E.Yu. Elenskaya. The existence fixed point of monoton left-continuous operators in space with right cone. Izvestiya vuzov. Matematika, 2011, 10, P. 40–47 [in Russian].Yu.Kh. Eshkabilov, F.H. Haydarov, U.A. Rozikov. Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree. Math. Phys. Anal. Geom., 2013, 16 (1), P. 1–17.Yu.Kh. Eshkabilov, F.H. Haydarov, U.A. Rozikov. Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree. Math. Phys. Anal. Geom., 2013, 16 (1), P. 1–17.Yu.Kh. Eshkabilov, U.A. Rozikov. On models with uncountable set of spin values on a Cayley tree: Integral equations. Math. Phys. Anal. Geom., 2010, 13, P. 275–286.Yu.Kh. Eshkabilov, U.A. Rozikov. On models with uncountable set of spin values on a Cayley tree: Integral equations. Math. Phys. Anal. Geom., 2010, 13, P. 275–286.U.A. Rozikov. Gibbs measures on a Cayley trees. World Sci. Pub, Singapore, 2013.U.A. Rozikov. Gibbs measures on a Cayley trees. World Sci. Pub, Singapore, 2013.H.O. Georgii. Gibbs measures and phase transitions. Second edition, de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 2011.H.O. Georgii. Gibbs measures and phase transitions. Second edition, de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 2011.C. Preston. Gibbs states on countable sets. Cambridge University Press, London, 1974.C. Preston. Gibbs states on countable sets. Cambridge University Press, London, 1974.I.V. Proskuryakov. Problems in Linear Algebra. MIR Publishers Moscow, 1978 (English translation).I.V. Proskuryakov. Problems in Linear Algebra. MIR Publishers Moscow, 1978 (English translation).

The authors declare that there are no conflicts of interest present.