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On the Solutions of a Singular Nonlinear Periodic Boundary Value Problem

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Abstract

In this paper, we consider the following singular nonlinear problem

$$\left\{ \begin{gathered} \tfrac{1}{A}(Au')' - qu = - f( \cdot ,u){\text{ a}}{\text{.e on (0,1),}} \hfill \\ u(0) = u(1),{\text{ }}Au'(0) = Au'(1), \hfill \\ \end{gathered} \right.$$

where A is a positive continuous function on (0,1), q is a nonnegative measurable function on [0,1] and f is a nonnegative regular function on (0,1)×(0,∞).

We suppose that ∫ 10 dt/A(t)<∞ and 0<∫ 10 A(t)q(t) dt<∞. Then we prove the existence and the uniqueness of a positive solution of this problem (P).

Our approach is based on the use of the Green's function and the Schauder's fixed point theorem.

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References

  1. Agarwall, R. P. and O'Regan, D.: ‘Nonlinear superlinear and nonsingular second order boundary value problems’, Differential Equations 143(1998), 60-95.

    Google Scholar 

  2. Bosiud, L. E. and O'Regan, D.: ‘Positive solutions for a class of nonlinear singular boundary value problems at reasonance’, J. Math. Anal. Appl. 168(1992), 413-424.

    Google Scholar 

  3. Cabada, A. and Sanchez, L.: ‘A positive operator approach of the Neumann problem of a second order ordinary differential equation’, J. Math. Anal. Appl. 204(1996), 774-785.

    Google Scholar 

  4. Dalmasso, R.: ‘On a singular nonlinear elliptic problems of second and fourth orders’, Bull. Sci. Math. Série 2 116(1992), 95-110.

    Google Scholar 

  5. Kannan, R. and O'Regan, D.: ‘A note on singular boundary value problems with solutions in weighted spaces’, Nonlinear Anal. 37(1999), 791-796.

    Google Scholar 

  6. Maâgli, H. and Masmoudi, S.: ‘Sur les solutions d'un opérateur différentiel singulier semilinéaire’, Potential Anal. 10(1999), 289-304.

    Google Scholar 

  7. O'Regan, D.: ‘Singular Dirichlet boundary value problem. Superlinear and nonreasonant case’, Nonlinear Anal. 29(1997), 221-245.

    Google Scholar 

  8. Saranen, J. and Seikkala, S.: ‘Some remarks on nonlinear second-order differential equations with periodic boundary conditions’, J. Math. Anal. Appl. 139(1989), 465-476.

    Google Scholar 

  9. Wang, J.: ‘On positive solutions of singular nonlinear two-point boundary problems’, J. Differential Equations 107(1994), 163-174.

    Google Scholar 

  10. Wang, J.: ‘Solvability of singular nonlinear two-point boundary problems’, Nonlinear Anal. 24(1995), 555-561.

    Google Scholar 

  11. Wang, J. and Jiang, D.: ‘A singular nonlinear second order periodic boundary value problem’, Tôhoku Math. J. 50(1998), 203-210.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Tunis, Campus universitaire, 1060, Tunis, Tunisia

    Habib Maagli

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  1. Habib Maagli
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Maagli, H. On the Solutions of a Singular Nonlinear Periodic Boundary Value Problem. Potential Analysis 14, 437–447 (2001). https://doi.org/10.1023/A:1011254307524

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