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Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary conditions

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Abstract

In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results.

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References

  1. Aftabizadeh, A.R.: Existence and uniqueness theorems for fourth order boundary value problems. J. Math. Anal. Appl. 116(2), 415–426 (1986). https://doi.org/10.1016/S0022-247X(86)80006-3

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves, E., Ma, T.F., Pelicer, M.L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlinear Anal. 71(9), 3834–3841 (2009). https://doi.org/10.1016/j.na.2009.02.051

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordred Banach spaces. SIAM Rev. 18(4), 620–709 (1976). https://doi.org/10.1137/1018114

    Article  MathSciNet  MATH  Google Scholar 

  4. Benaicha, S., Haddouchi, F.: Positive solutions of a nonlinear fourth-order integral boundary value problem. An. Univ. Vest Timiş. Ser. Mat.-Inform. 54(1), 73–86 (2016). https://doi.org/10.1515/awutm-2016-0005

    Article  MathSciNet  Google Scholar 

  5. Bouteraa, N., Benaicha, S., Djourdem, H., Benattia, M.E.: Positive solutions of nonlinear fourth-order two-point boundary value problem with a parameter. Rom. J. Math. Comput. Sci. 8(1), 17–30 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Cabada, A., Cid, J.A., Villamarin, B.M.: Computation of Green’s functions for boundary value problems with Mathematica. Appl. Math. Comput. 219(4), 1919–1936 (2012). https://doi.org/10.1016/j.amc.2012.08.035

    Article  MathSciNet  MATH  Google Scholar 

  7. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  Google Scholar 

  8. Graef, J.R., Kong, L., Kong, Q., Yang, B.: Positive solutions to a fourth order boundary value problem. Results Math. 59(1–2), 141–155 (2011). https://doi.org/10.1007/s00025-010-0068-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Guendouz, C., Haddouchi, F., Benaicha, S.: Existence of positive solutions for a nonlinear third-order integral boundary value problem. Ann. Acad. Rom. Sci. Ser. Math. Appl. 10(2), 314–328 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Haddouchi, F., Benaicha, S.: Positive solutions of a nonlinear three-point eigenvalue problem with integral boundary conditions. Rom. J. Math. Comput. Sci. 5(2), 202–2013 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Hao, X., Xu, N., Liu, L.: Existence and uniqueness of positive solutions for fourth-order \(m\)-point boundary value problems with two parameters. Rocky Mt. J. Math. 43(4), 1161–1180 (2013). https://doi.org/10.1216/RMJ-2013-43-4-1161

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, R., Zhai, C.: Positive solutions for a system of fourth-order differential equations with integral boundary conditions and two parameters. Nonlinear Anal. Model. Control. 23(3), 401–422 (2018). https://doi.org/10.15388/NA.2018.3.7

    Article  MathSciNet  MATH  Google Scholar 

  13. Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. P. Noordhoff, Groningen (1964)

    MATH  Google Scholar 

  14. Lan, K., Webb, J.R.L.: Positive solutions of semilinear differential equations with singularities. J. Differ. Equ. 148(2), 407–421 (1998). https://doi.org/10.1006/jdeq.1998.3475

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, S., Zhai, C.: New existence and uniqueness results for an elastic beam equation with nonlinear boundary conditions. Bound. Value Probl. (2015). https://doi.org/10.1186/s13661-015-0365-x

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, S., Zhang, X.: Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Comput. Math. Appl. 63(9), 1355–1360 (2012). https://doi.org/10.1016/j.camwa.2011.12.065

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Y., Weigho, Z., Chunfang, S.: Monotone and convex positive solutions for fourth-order multi-point boundary value problems. Bound. Value Probl. (2011). https://doi.org/10.1186/1687-2770-2011-21

    Article  MathSciNet  Google Scholar 

  18. Lv, X., Wang, L., Pei, M.: Monotone positive solution of a fourth-order BVP with integral boundary conditions. Bound. Value Probl. (2015). https://doi.org/10.1186/s13661-015-0441-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, J. P., Li, H. B.: Monotone positive solutions of nonlinear third-order BVP with integral boundary conditions. Bound. Value Probl. Art ID 874959 (2010). https://doi.org/10.1155/2010/874959

  20. Webb, J.R.L.: Solutions of nonlinear equations in cones and positive linear operators. J. Lond. Math. Soc. 82(2), 420–436 (2010). https://doi.org/10.1112/jlms/jdq037

    Article  MathSciNet  MATH  Google Scholar 

  21. Webb, J.R.L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth-order boundary value problems with local and nonlocal boundary conditions. Proc. R. Soc. Edinb. Sec. A 138(2), 427–446 (2008). https://doi.org/10.1017/S0308210506001041

    Article  MATH  Google Scholar 

  22. Yao, Q.: Positive solutions of nonlinear beam equations with time and space singularities. J. Math. Anal. Appl. 374(2), 681–692 (2011). https://doi.org/10.1016/j.jmaa.2010.08.056

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhai, C., Jiang, C.: Existence of nontrivial solutions for a nonlinear fourth-order boundary value problem via iterative method. J. Nonlinear Sci. Appl. 9(6), 4295–4304 (2016). https://doi.org/10.22436/jnsa.009.06.71

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhai, C., Jiang, C.: Existence and uniqueness of convex monotone positive solutions for boundary value problems of an elastic beam equation with a parameter. Electron. J. Qual. Theory Differ. Equ. 81, 1–11 (2015). https://doi.org/10.14232/ejqtde.2015.1.81

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhai, C., Jiang, C., Li, S.: Approximating monotone positive solutions of a nonlinear fourth-order boundary value problem via sum operator method. Mediterr. J. Math. 14(2), Paper No. 77 (2017). https://doi.org/10.1007/s00009-017-0844-7

  26. Zhai, C., Song, R., Han, Q.: The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem. Comput. Math. Appl. 62(6), 2639–2647 (2011). https://doi.org/10.1016/j.camwa.2011.08.003

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, X., Ge, W.: Positive solutions for a class of boundary value problems with integral boundary conditions. Comput. Math. Appl. 58(2), 203–215 (2009). https://doi.org/10.1016/j.camwa.2009.04.002

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for a careful reading of the paper and for his/her valuable comments.

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Authors and Affiliations

  1. Department of Physics, University of Sciences and Technology of Oran-MB, Oran, Algeria

    Faouzi Haddouchi

  2. Laboratory of Fundamental and Applied Mathematics of Oran, Department of Mathematics, University of Oran 1, Oran, Algeria

    Faouzi Haddouchi & Nourredine Houari

Authors
  1. Faouzi Haddouchi
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  2. Nourredine Houari
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Correspondence to Faouzi Haddouchi.

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Haddouchi, F., Houari, N. Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary conditions. J. Appl. Math. Comput. 66, 87–109 (2021). https://doi.org/10.1007/s12190-020-01426-4

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  • DOI: https://doi.org/10.1007/s12190-020-01426-4

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