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Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods

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Abstract

The purpose of this paper is to use a three critical point theorem due to Ricceri to obtain the existence of at least three solutions for the following Sturm–Liouville boundary value problem with impulses

$$\begin{cases}(\phi_{p}(x'(t)))'=(a(t)\phi_{p}(x)+\lambda f(t,x)+\mu h(x))g(x'(t)),\quad \mbox{a.e. }t\in[0,1],\\\Delta G(x'(t_{i}))=I_{i}(x(t_{i})),\quad i=1,2,\ldots,k,\\\alpha_{1}x(0)-\alpha_{2}x'(0)=0,\\\beta_{1}x(1)+\beta_{2}x'(1)=0,\end{cases}$$

where p>1, φ p (x)=|x|p−2 x, λ, μ are positive parameters, \(G(x)=\int_{0}^{x}\frac{(p-1)|s|^{p-2}}{g(s)}\,ds\) . The interest is that the nonlinear term includes x′. We exhibit the existence of at least three solutions and h(x) can be an arbitrary C 1 functional with compact derivative. As an application, an example is given to illustrate the results.

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References

  1. Agarwal, R.P., Perera, K., O’Regan, D.: Multiple positive solutions of singular problems by variational methods. Proc. Am. Math. Soc. 134(3), 817–824 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ricceri, B.: On a three critical points theorem. Arch. Math. 75, 220–226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonanno, G.: A minimax inequality and its applications to ordinary differential equations. J. Math. Anal. Appl. 270, 210–229 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Du, Z., Lin, X., Tistell, C.C.: A multiplicity result for p-Laplacian boundary value problems via critical points theorem. Appl. Math. Comput. (2008). doi:10.1016/j.amc.2008.07.011

    Google Scholar 

  5. Afrouzi, G.A., Heidarkhani, S.: Three solutions for a quasilinear boundary value problem. Nonlinear Anal. 69, 3330–3336 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ricceri, B.: A three critical points theorem revisited. Nonlinear Anal. (2008). doi:10.1016/j.na.2008.04.010

    MathSciNet  Google Scholar 

  7. George, P.K., Nandakumaran, A.K., Arapostathis, A.: A note on contrability of impulsive systems. J. Math. Anal. Appl. 241, 276–283 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Akhmetov, M.U., Zafer, A.: Controllability of the Vallée-Poussin problem for impulsive differential systems. J. Optim. Theory Appl. 102, 263–276 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Nenov, S.: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. TMA 36, 881–890 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Georescu, P., Morosanu, G.: Pest regulation by means of impulsive controls. Appl. Math. Compact. 190, 790–803 (2007)

    Article  Google Scholar 

  11. Liu, X., Willms, A.R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277–299 (1996)

    Article  MATH  Google Scholar 

  12. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    MATH  Google Scholar 

  13. Samilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  14. Neto, I.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal. RWA. doi:10.1016/j.nonrwa.2007.10.022

  15. Mawhin, J., Willen, M.: Critical Point Theorem and Hamiltonian Systems. Appl. Math. Sci., vol. 74. Springer, New York (1989)

    Google Scholar 

  16. Simon, J.: Regularite de la Solution d’Une Equation Non Lineaire dans R n. Lecture Notes in Mathematics, vol. 665, pp. 205–227. Springer, New York (1978)

    Google Scholar 

  17. Ricceri, B.: Existence of three solutions for a class of elliptic eigenvalue problem. Math. Comput. Model. 32, 1485–1494 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Part 2 B: Nonlinear Monotone Operators. Springer, New York (1990)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Li Zhang & Weigao Ge

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  1. Li Zhang
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  2. Weigao Ge
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Correspondence to Li Zhang.

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Supported by the National Natural Science Foundation of China (No. 10671012) and the Doctoral Program Foundation of the Education Ministry of China (20050007011).

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Zhang, L., Ge, W. Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods. Acta Appl Math 110, 1237–1248 (2010). https://doi.org/10.1007/s10440-009-9504-7

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  • DOI: https://doi.org/10.1007/s10440-009-9504-7

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