Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
$\phi $-Laplacian; impulses; lower and upper functions; periodic boundary value problem
$\phi $-Laplacian; impulses; lower and upper functions; periodic boundary value problem
Summary:
The paper deals with the impulsive boundary value problem \[ \frac{d}{dt}[\phi (y^{\prime }(t))] = f(t, y(t), y^{\prime }(t)), \quad y(0) = y(T),\quad y^{\prime }(0) = y^{\prime }(T), y(t_{i}+) = J_{i}(y(t_{i})), \quad y^{\prime }(t_{i}+) = M_{i}(y^{\prime }(t_{i})),\quad i = 1, \ldots m. \] The method of lower and upper solutions is directly applied to obtain the results for this problems whose right-hand sides either fulfil conditions of the sign type or satisfy one-sided growth conditions.
References:
[1] Cabada A., Pouso L. R.: Existence results for the problem $(\phi (u^{\prime }))^{\prime } = f(t, u, u^{\prime })$ with nonlinear boundary conditions. Nonlinear Analysis 35 (1999), 221–231. MR 1643240
[2] Cabada A., Pouso L. R.: Existence result for the problem $(\phi (u^{\prime }))^{\prime } = f(t, u, u^{\prime })$ with periodic and Neumann boundary conditions. Nonlinear Anal. T.M.A 30 (1997), 1733–1742. MR 1490088
[3] O’Regan D.: Some general principles and results for $(\phi (u^{\prime }))^{\prime } = qf(t, u, u^{\prime })$, $0 < t < 1$. SIAM J. Math. Anal. 24 (1993), 648–668. MR 1215430
[4] Manásevich R., Mawhin J.: Periodic solutions for nonlinear systems with p-Laplacian like operators. J. Differential Equations 145 (1998), 367–393. MR 1621038
[5] Bainov D., Simeonov P.: Impulsive differential equations: periodic solutions, applications. : Pitman Monographs and Surveys in Pure and Applied Mathematics 66, Longman Scientific and Technical, Essex, England. 1993. MR 1266625
[6] Cabada A., Nieto J. J., Franco D., Trofimchuk S. I.: A generalization of the monotone method for second order periodic boundary value problem with impulses at fixed points. Dyn. Contin. Discrete Impulsive Syst. 7 (2000), 145–158. MR 1744974 | Zbl 0953.34020
[7] Yujun Dong: Periodic solutions for second order impulsive differential systems. Nonlinear Anal. T.M.A 27 (1996), 811–820. MR 1402168
[8] Erbe L. H., Xinzhi Liu: Existence results for boundary value problems of second order impulsive differential equations. J. Math. Anal. Appl. 149 (1990), 56–59. MR 1054793
[9] Shouchuan Hu, Laksmikantham V.: Periodic boundary value problems for second order impulsive differential equations. Nonlinear Anal. T.M.A 13 (1989), 75–85. MR 0973370
[10] Liz E., Nieto J. J.: Periodic solutions of discontinuous impulsive differential systems. J. Math. Anal. Appl. 161 (1991), 388–394. MR 1132115 | Zbl 0753.34027
[11] Liz E., Nieto J. J.: The monotone iterative technique for periodic boundary value problems of second order impulsive differential equations. Comment. Math. Univ. Carolinae 34 (1993), 405–411. MR 1243071 | Zbl 0780.34006
[12] Rachůnková I., Tomeček J.: Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions. J. Math. Anal. Appl. 292 (2004), 525–539. MR 2047629 | Zbl 1058.34031
[13] Rachůnková I., Tvrdý M.: Impulsive Periodic Boudary Value Problem and Topological Degree. Funct. Differ. Equ. 9 (2002), 471–498. MR 1971622
[14] Zhitao Zhang: Existence of solutions for second order impulsive differential equations. Appl. Math., Ser. B (eng. Ed.) 12 (1997), 307–320. MR 1482919
Search
Partner of
