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Positive Solutions for a Nonlocal Resonant Problem of First Order

  • Mirosława Zima
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We study a first order differential system subject to a nonlocal condition. Our goal in this paper is to establish conditions sufficient for the existence of positive solutions when the considered problem is at resonance. The key tool in our approach is Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. We conclude the paper with several examples illustrating the main result.

Keywords

Positive solution Cone Resonant problem Nonlocal condition 

Mathematics Subject Classification (2010)

Primary 34B18; Secondary 34B10 

Notes

Acknowledgements

This work was completed with the partial support of the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów. The author wishes to express her thanks to the referees for careful reading of the manuscript and constructive comments.

References

  1. 1.
    D.R. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions. J. Math. Anal. Appl. 408, 318–323 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    O. Bolojan, R. Precup, Implicit first order differential systems with nonlocal conditions. Electron. J. Qual. Theor. Differ. Equ. 2014(69), 1–13 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    O. Bolojan-Nica, G. Infante, R. Precup, Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal. 94, 231–242 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Boucherif, First-order differential inclusions with nonlocal initial conditions. Appl. Math. Lett. 15, 409–414 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C.T. Cremins, A fixed point index and existence theorems for semilinear equations in cones. Nonlinear Anal. 46, 789–806 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Franco, G. Infante, M. Zima, Second order nonlocal boundary value problems at resonance. Math. Nachr. 284, 875–884 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Franco, J.J. Nieto, D. O’Regan, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions. Appl. Math. Comput. 153, 793–802 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    R.E. Gaines, J. Santanilla, A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 12, 669–678 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J.R. Graef, S. Padhi, S. Pati, Periodic solutions of some models with strong Allee effects. Nonlinear Anal. Real World Appl. 13, 569–581 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    G. Infante, M. Zima, Positive solutions of multi-point boundary value probelms at resonance. Nonlinear Anal. 69, 2458–2465 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    T. Jankowski, Boundary value problems for first order differential equations of mixed type. Nonlinear Anal. 64, 1984–1997 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    W. Jiang, C. Yang, The existence of positive solutions for multi-point boundary value problem at resonance on the half-line. Bound. Value Probl. 2016, 13 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 68, 2158–2171 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    N. Kosmatov, A singular non-local problem at resonance. J. Math. Anal. Appl. 394, 425–431 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    V. Lakshmikantham, S. Leela, Existence and monotone method for periodic solutions of first order differential equations. J. Math. Anal. Appl. 91, 237–243 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. Liu, Existence and uniqueness of solutions to first-order multipoint boundary value problems. Appl. Math. Lett. 17, 1307–1316 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y. Liu, Multiple solutions of periodic boundary value problems for first order differential equations. Comput. Math. Appl. 54, 1–8 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    B. Liu, Z. Zhao, A note on multi-point boundary value problems. Nonlinear Anal. 67, 2680–2689 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces. J. Differ. Equ. 12, 610–636 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O. Nica, Nonlocal initial value problems for first order differential systems. Fixed Point Theory 13, 603–612 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    J.J. Nieto, R. Rodríguez-López, Greens function for first-order multipoint boundary value problems and applications to the existence of solutions with constant sign. J. Math. Anal. Appl. 388, 952–963 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    D. O’Regan, M. Zima, Leggett-Williams norm-type theorems for coincidences. Arch. Math. 87, 233–244 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    W.V. Petryshyn, On the solvability of x ∈ Tx + λFx in quasinormal cones with T and F k-set contractive. Nonlinear Anal. 5, 585–591 (1981)Google Scholar
  24. 24.
    R. Precup, D. Trif, Multiple positive solutions of non-local initial value problems for first order differential systems. Nonlinear Anal. 75, 5961–5970 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. Santanilla, Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 105, 357–371 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    J. Santanilla, Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 126, 397–408 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    C.C. Tisdell, Existence of solutions to first-order periodic boundary value problems. J. Math. Anal. Appl. 323, 1325–1332 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    M. Zima, P. Drygaś, Existence of positive solutions for a kind of periodic boundary value problem at resonance. Bound. Value Probl. 2013, 19 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural Sciences, Department of Functional AnalysisUniversity of RzeszówRzeszówPoland

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