Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Fixed Point Theorems for Mixed Monotone Vector Operators with Application to Systems of Nonlinear Boundary Value Problems

  • Sadrati, Abdellatif (MSISI Laboratory, AM2CSI Group, Department of Mathematics, FST, University Moulay Ismal of Meknes) ;
  • Aouragh, My Driss (MSISI Laboratory, AM2CSI Group, Department of Mathematics, FST, University Moulay Ismal of Meknes)
  • Received : 2020.04.11
  • Accepted : 2020.11.23
  • Published : 2021.09.30
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Abstract

In this paper, we present and prove new existence and uniqueness fixed point theorems for vector operators having a mixed monotone property in partially ordered product Banach spaces. Our results extend and improve existing works on τ-φ-concave operators in the scalar case. As an application, we study the existence and uniqueness of positive solutions for systems of nonlinear Neumann boundary value problems.

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References

  1. R. P. Agarwal, M. Meehan and D. O'Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics 141, Cambridge University Press, 2001.
  2. A. Bensedik and M. Bouchekif, Symmetry and uniqueness of positive solutions for a Neumann boundary value problem, Appl. Math. Lett., 20(2007), 419-426. https://doi.org/10.1016/j.aml.2006.05.008
  3. V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74(2011), 4889-4897. https://doi.org/10.1016/j.na.2011.03.032
  4. X. Du and Z. Zhao, On fixed point theorems of mixed monotone operators, Fixed Point Theory Appl., (2011), Art. ID 563136, 8 pp.
  5. C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168(1992), 540-551. https://doi.org/10.1016/0022-247x(92)90179-h
  6. C. P. Gupta and S. I. Trofimchuk, A sharper condition for solvability of a three-point second order boundary value problem, J. Math. Anal. Appl., 205(1997), 586-597. https://doi.org/10.1006/jmaa.1997.5252
  7. J. Henderson and H. Y. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208(1997), 252-259. https://doi.org/10.1006/jmaa.1997.5334
  8. J. Hernndez, J. Kartson and P. L. Simon, Multiplicity for semilinear elliptic equations involving singular nonlinearity, Nonlinear Anal., 65(2006), 265-283. https://doi.org/10.1016/j.na.2004.11.024
  9. J. Hernndez, F. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137(2007), 41-62. https://doi.org/10.1017/S030821050500065X
  10. V. A. Ilin and E. L. Moiseev, A nonlocal boundary value problem of the first kind for the SturmLiouville operator in differential and difference interpretations, Differentsial'nye Uravneniya, 23(7)(1987), 11981207.
  11. V. A. Ilin and E. L. Moiseev, A nonlocal boundary value problem of the second kind for the SturmLiouville operator, J. Differentsial'nye Uravneniya, 23(8)(1987), 14221431.
  12. D. Jiang and H. Liu, Existence of positive solutions to second order Neumann boundary value problem, J. Math. Res. Exposition, 20(2000), 360-364.
  13. I. Y. Karaca, Nonlinear triple-point problems with change of sign, Comput. Math. Appl., 55(4)(2008), 691-703. https://doi.org/10.1016/j.camwa.2007.04.032
  14. H. B. Keller, Some positive problems suggested by nonlinear heat generation, Bifurcation Theory and Nonlinear Eigenvalue Problems, 217-255, Benjamin, Elmsford, NY, 1969.
  15. M. A. Krasnoselskii, Positive solutions of operators equations, Noordhooff, Groningen, 1964.
  16. H. J. Kuiper, On positive solutions of nonlinear elliptic eigenvalue problems, Rend. Circ. Mat. Palermo (2), 20(1979), 113-138. https://doi.org/10.1007/BF02844166
  17. B. Liu, Positive solutions of a nonlinear three-point boundary value problem, Appl. Math. Comput., 132(2002), 11-28. https://doi.org/10.1016/S0096-3003(02)00341-7
  18. R. Ma, Positive solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equations, (1999), No. 34, 8 pp.
  19. R. Ma, Multiplicity of positive solutions for second-order three-point boundary value problems, Comput. Math. Appl., 40(2000), 193-204. https://doi.org/10.1016/S0898-1221(00)00153-X
  20. H. Magli and M. Zribi, Existence and estimates of solutions for singular nonlinear elliptic problems, J. Math. Anal. Appl., 263(2001), 522-542. https://doi.org/10.1006/jmaa.2001.7628
  21. S. A. Marano, A remark on a second order three-point boundary value problem, J. Math. Anal. Appl., 183(1994), 518-522. https://doi.org/10.1006/jmaa.1994.1158
  22. J. Sun and W. Li, Multiple positive solutions to second-order Neumann boundary value problems, Appl. Math. Comput., 146(2003), 187-194. https://doi.org/10.1016/S0096-3003(02)00535-0
  23. J. Sun, W. Li and S. Cheng, Three positive solutions for second-order Neumann boundary value problems, Appl. Math. Lett., 17(2004), 1079-1084. https://doi.org/10.1016/j.aml.2004.07.012
  24. Y. P. Sun and Y. Sun, Positive solutions for singular semi-positone Neumann boundary value problems, Electron. J. Differential Equations, (2004), No. 133, 8 pp.
  25. C. B. Zhai and X. M. Cao, Fixed point theorems for τ - φ-concave operators and applications, Comput. Math. Appl., 59(2010), 532-538. https://doi.org/10.1016/j.camwa.2009.06.016
  26. C. Zhai and L. Zhang, New fixed point theorems for mixed monotone operators and local existenceuniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl., 382(2011), 594-614. https://doi.org/10.1016/j.jmaa.2011.04.066
  27. J. Zhang and C. B. Zhai, Existence and uniqueness results for perturbed Neumann boundary value problems, Bound. Value Probl., (2010), Art. ID 494210, 10 pp.