Periodic BVP with φ -Laplacian and impulses

Vladimír Polášek

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2005)

  • Volume: 44, Issue: 1, page 131-150
  • ISSN: 0231-9721

Abstract

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The paper deals with the impulsive boundary value problem d d t [ φ ( y ' ( t ) ) ] = f ( t , y ( t ) , y ' ( t ) ) , y ( 0 ) = y ( T ) , y ' ( 0 ) = y ' ( T ) , y ( t i + ) = J i ( y ( t i ) ) , y ' ( t i + ) = M i ( y ' ( t i ) ) , i = 1 , ... 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.

How to cite

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Polášek, Vladimír. "Periodic BVP with $\phi $-Laplacian and impulses." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 44.1 (2005): 131-150. <http://eudml.org/doc/32445>.

@article{Polášek2005,
abstract = {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.},
author = {Polášek, Vladimír},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {$\phi $-Laplacian; impulses; lower and upper functions; periodic boundary value problem; -Laplacian; impulses; lower and upper functions; periodic boundary value problem},
language = {eng},
number = {1},
pages = {131-150},
publisher = {Palacký University Olomouc},
title = {Periodic BVP with $\phi $-Laplacian and impulses},
url = {http://eudml.org/doc/32445},
volume = {44},
year = {2005},
}

TY - JOUR
AU - Polášek, Vladimír
TI - Periodic BVP with $\phi $-Laplacian and impulses
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2005
PB - Palacký University Olomouc
VL - 44
IS - 1
SP - 131
EP - 150
AB - 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.
LA - eng
KW - $\phi $-Laplacian; impulses; lower and upper functions; periodic boundary value problem; -Laplacian; impulses; lower and upper functions; periodic boundary value problem
UR - http://eudml.org/doc/32445
ER -

References

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  1. Cabada A., Pouso L. R., Existence results for the problem ( φ ( u ' ) ) ' = f ( t , u , u ' ) with nonlinear boundary conditions, Nonlinear Analysis 35 (1999), 221–231. (1999) MR1643240
  2. Cabada A., Pouso L. R., Existence result for the problem ( φ ( u ' ) ) ' = f ( t , u , u ' ) with periodic and Neumann boundary conditions, Nonlinear Anal. T.M.A 30 (1997), 1733–1742. (1997) MR1490088
  3. O’Regan D., Some general principles and results for ( φ ( u ' ) ) ' = q f ( t , u , u ' ) , 0 < t < 1 , SIAM J. Math. Anal. 24 (1993), 648–668. (1993) MR1215430
  4. Manásevich R., Mawhin J., Periodic solutions for nonlinear systems with p-Laplacian like operators, J. Differential Equations 145 (1998), 367–393. (1998) MR1621038
  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. (1993) MR1266625
  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. Zbl0953.34020MR1744974
  7. Yujun Dong, Periodic solutions for second order impulsive differential systems, Nonlinear Anal. T.M.A 27 (1996), 811–820. (1996) MR1402168
  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. (1990) MR1054793
  9. Shouchuan Hu, Laksmikantham V., Periodic boundary value problems for second order impulsive differential equations, Nonlinear Anal. T.M.A 13 (1989), 75–85. (1989) MR0973370
  10. Liz E., Nieto J. J., Periodic solutions of discontinuous impulsive differential systems, J.  Math. Anal. Appl. 161 (1991), 388–394. (1991) Zbl0753.34027MR1132115
  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. (1993) Zbl0780.34006MR1243071
  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. Zbl1058.34031MR2047629
  13. Rachůnková I., Tvrdý M., Impulsive Periodic Boudary Value Problem and Topological Degree, Funct. Differ. Equ. 9 (2002), 471–498. MR1971622
  14. Zhitao Zhang, Existence of solutions for second order impulsive differential equations, Appl. Math., Ser. B (eng. Ed.) 12 (1997), 307–320. (1997) MR1482919

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