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Variational Approaches to Kirchhoff-Type Second-Order Impulsive Differential Equations on the Half-Line

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Abstract

In this work we continue the study of the multiplicity results for a Kirchhoff-type second-order impulsive differential equation on the half-line. In fact, using a consequence of the local minimum theorem due Bonanno we look into the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for the problem in a special case.

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References

  1. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. TMA 6, 1001–1022 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Autuori, G., Colasuonno, F., Pucci, P.: Blow up at infinity of solutions of polyharmonic Kirchhoff systems. Complex Var. Elliptic Equ. 57, 379–395 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Autuori, G., Colasuonno, F., Pucci, P.: Lifespan estimates for solutions of polyharmonic Kirchhoff systems. Math. Models Methods Appl. Sci. 22, 1150009 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Autuori, G., Colasuonno, F., Pucci, P.: On the existence of stationary solutions for higher-order \(p\)-Kirchhoff problems. Commun. Contemp. Math. 16(5), 1450002 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Averna, D., Bonanno, G.: A three critical point theorem and its applications to the ordinary Dirichlet problem. Topol. Methods Nonlinear Anal. 22, 93–103 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai, L., Dai, B.: Existence and multiplicity of solutions for impulsive boundary value problem with a parameter via critical point theory. Math. Comput. Model. 53, 1844–1855 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonanno, G.: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. TMA 75, 2992–3007 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonanno, G.: Multiple critical points theorems without the Palais–Smale condition. J. Math. Anal. Appl. 299, 600–614 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonanno, G.: Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal. 1, 205–220 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Bonanno, G., Barletta, G., O’Regan, D.: A variational approach to multiplicity results for boundary-value problems on the real line. Proc. R. Soc. Edinb. 145A, 13–29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bonanno, G., Chinnì, A.: Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent. J. Math. Anal. Appl. 418, 812–827 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bonanno, G., O’Regan, D.: A boundary value problem on the half-line via critical point methods. Dyn. Syst. Appl. 15, 395–408 (2006)

    MathSciNet  Google Scholar 

  15. Bonanno, G., O’Regan, D., Vetro, F.: Sequences of distinct solutions for boundary value problems on the real line. J. Nonlinear Convex Anal. 17, 365–375 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Bonanno, G., O’Regan, D., Vetro, F.: Triple solutions for quasilinear one-dimensional \(p\)-Laplacian elliptic equations in the whole space. Ann. Funct. Anal. 8, 248–258 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, H., He, Z., Li, J.: Multiplicity of solutions for impulsive differential equation on the half-line via variational methods. Bound. Value Probl. 2016(14), 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Chen, H., Sun, J.: An application of variational method to second-order impulsive differential equation on the half-line. Appl. Math. Comput. 217, 1863–1869 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Chipot, M., Lovat, B.: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Colasuonno, F., Pucci, P.: Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. TMA 74, 5962–5974 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dai, B., Zhang, D.: The existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Results Math. 63, 135–149 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gomes, J.M., Sanchez, L.: A variational approach to some boundary value problems in the half-line. Z. Angew. Math. Phys. 56, 192–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Graef, J.R., Heidarkhani, S., Kong, L.: A variational approach to a Kirchhoff-type problem involving two parameters. Results Math. 63, 877–889 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. He, X., Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Heidarkhani, S.: Infinitely many solutions for systems of \(n\) two-point Kirchhoff-type boundary value problems. Ann. Polon. Math. 107, 133–152 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Heidarkhani, S., Afrouzi, G.A., Moradi, S.: Existence results for a Kirchhoff-type second-order differential equation on the half-line with impulses. Asymptot. Anal. 105, 137–158 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heidarkhani, S., De Araujo, A.L.A., Afrouzi, G.A., Moradi, S.: Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions. Math. Nachr. (2017). https://doi.org/10.1002/mana.201600425

  28. Heidarkhani, S., Salari, A.: Nontrivial solutions for impulsive fractional differential systems through variational methods. Comput. Math. Appl. (2016). https://doi.org/10.1016/j.camwa.2016.04.016

    Google Scholar 

  29. Iffland, G.: Positive solutions of a problem EmdenFowler type with a type free boundary. SIAM J. Math. Anal. 18, 283–92 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kawano, N., Yanagida, E., Yotsutani, S.: Structure theorems for positive radial solutions to \(\Delta u + K(|x|)u^p = 0\) in \(\mathbb{R}^n\). Funkc. Ekvac. 36, 557–579 (1993)

    MATH  Google Scholar 

  31. Kaufmann, E.R., Kosmatov, N., Raffoul, Y.N.: A second-order boundary value problem with impulsive effects on an unbounded domain. Nonlinear Anal. TMA 69, 2924–2929 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kirchhoff, G.: Vorlesungen über Mathematische Physik Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  33. Kristály, A., Rădulescu, V., Varga, Cs: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, vol. 136. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  34. Li, J., Nieto, J.J.: Existence of positive solutions for multiple boundary value problems on the half-line with impulsive. Bound. Value Probl. Article ID 834158 (2009)

  35. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1977). North-Holland Mathematical Studies, vol. 30, pp. 284–346 (1978)

  36. Ma, R., Zhu, B.: Existence of positive solutions for a semipositone boundary value problem on the half-line. Comput. Math. Appl. 58, 1672–1686 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Motreanu, D., Rădulescu, V.: Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Applications. Kluwer, Dordrecht (2003)

    Book  MATH  Google Scholar 

  38. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence (1986)

  39. Ricceri, B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ricceri, B.: On an elliptic Kirchhoff-type problem depending on two parameters. J. Global Optim. 46, 543–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun, Y., Sun, Y., Debnath, L.: On the existence of positive solutions for singular boundary value problems on the half-line. Appl. Math. Lett. 22, 806–812 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tian, Y., Ge, W.: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509–528 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tian, Y., Ge, W.: Positive solutions for multi-point boundary value problem on the half-line. J. Math. Anal. Appl. 325, 1339–1349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, Q.R.: Oscillation and asymptotics for second-order half-linear differential equations. Appl. Math. Comput. 122, 253–266 (2001)

    MathSciNet  MATH  Google Scholar 

  45. Yan, B.: Multiple unbounded solutions of boundary value problems for second order differential equations on the half line. Nonlinear Anal. TMA 51, 1031–1044 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. III. Springer, New York (1985)

    Book  MATH  Google Scholar 

  47. Zhao, Y., Chen, H., Xu, C.: Existence of multiple solutions for three-point boundary-value problems on infinite intervals in Banach spaces. Electron. J. Differ. Equ. 2012(44), 1–11 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Zhao, Y., Wang, X., Liu, X.: New results for perturbed second-order impulsive differential equation on the half-line. Bound. Value Prob. 2014(246), 1–15 (2014)

    MathSciNet  Google Scholar 

  49. Zima, M.: On positive solution of boundary value problems on the half-line. J. Math. Anal. Appl. 259, 1270–136 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Economics, University of Messina, Via dei Verdi 75, Messina, Italy

    Giuseppe Caristi

  2. Department of Mathematics, Faculty of Sciences, Razi University, 67149, Kermanshah, Iran

    Shapour Heidarkhani

  3. Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

    Amjad Salari

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  1. Giuseppe Caristi
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  2. Shapour Heidarkhani
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  3. Amjad Salari
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Correspondence to Shapour Heidarkhani.

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Caristi, G., Heidarkhani, S. & Salari, A. Variational Approaches to Kirchhoff-Type Second-Order Impulsive Differential Equations on the Half-Line. Results Math 73, 44 (2018). https://doi.org/10.1007/s00025-018-0772-2

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