Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness

Martina Pavlačková; Valentina Taddei

Archivum Mathematicum (2023)

  • Volume: 059, Issue: 1, page 99-107
  • ISSN: 0044-8753

Abstract

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We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.

How to cite

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Pavlačková, Martina, and Taddei, Valentina. "Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness." Archivum Mathematicum 059.1 (2023): 99-107. <http://eudml.org/doc/298975>.

@article{Pavlačková2023,
abstract = {We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.},
author = {Pavlačková, Martina, Taddei, Valentina},
journal = {Archivum Mathematicum},
keywords = {second-order differential inclusion; nonlocal conditions; Banach spaces; cosine family; approximation solvability method; mild solution},
language = {eng},
number = {1},
pages = {99-107},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness},
url = {http://eudml.org/doc/298975},
volume = {059},
year = {2023},
}

TY - JOUR
AU - Pavlačková, Martina
AU - Taddei, Valentina
TI - Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 059
IS - 1
SP - 99
EP - 107
AB - We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.
LA - eng
KW - second-order differential inclusion; nonlocal conditions; Banach spaces; cosine family; approximation solvability method; mild solution
UR - http://eudml.org/doc/298975
ER -

References

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  1. Andres, J., Malaguti, L., Pavlačková, M., On second-order boundary value problems in Banach spaces: a bound sets approach, Topol. Methods Nonlinear Anal. 37 (2) (2011), 303–341. (2011) MR2849825
  2. Andres, J., Malaguti, L., Pavlačková, M., A Scorza-Dragoni approach to second-order boundary value problems in abstract spaces, Appl. Math. Inf. Sci. 6 (2) (2012), 29–44. (2012) MR2914078
  3. Balachandran, K., Park, J.Y., Existence of solutions of second order nonlinear differential equations with nonlocal conditions in Banach spaces, Indian J. Pure Appl. Math. 32 (12) (2001), 1883–1891. (2001) MR1879632
  4. Benchohra, M., Nieto, J.J., Rezoug, N., 10.1515/dema-2017-0029, Demonstr. Math. 50 (2017), 309–319. (2017) MR3742568DOI10.1515/dema-2017-0029
  5. Benchohra, M., Rezoug, N., Samet, B., Zhou, Y., 10.3390/math7121134, Mathematics 7 (12) (2019), 20 pp., art. no. 1134. (2019) DOI10.3390/math7121134
  6. Benedetti, I., Loi, N.V., Malaguti, L., Obukhovskii, V., An approximation solvability method for nonlocal differential problems in Hilbert spaces, Commun. Contemp. Math. 19 (2) (2017), 34 pp. (2017) MR3611657
  7. Benedetti, I., Loi, N.V., Malaguti, L., Taddei, V., 10.1016/j.jde.2016.10.019, J. Differential Equations 262 (2007), 1499–1523. (2007) MR3582201DOI10.1016/j.jde.2016.10.019
  8. Benedetti, I., Loi, N.V., Taddei, V., 10.3934/dcds.2017128, Discrete Contin. Dyn. Syst. Ser. A 37 (6) (2017), 2977–2998. (2017) MR3622071DOI10.3934/dcds.2017128
  9. Benedetti, I., Malaguti, L., Taddei, V., Nonlocal semilinear evolution equations without strong compactness: theory and applications, Rend. Istit. Mat. Univ. Trieste 44 (2012), 371–388. (2012) MR3047115
  10. Bochner, S., Taylor, A.E., 10.2307/1968472, Ann. Math. 39 (1938), 913–944. (1938) DOI10.2307/1968472
  11. Byszewski, L., Lakshmikantham, V., 10.1080/00036819008839989, Appl. Anal. 40 (1) (1991), 11–19. (1991) DOI10.1080/00036819008839989
  12. Cardinali, T., Gentili, S., 10.24193/subbmath.2017.0008, Stud. Univ. Babeş-Bolyai Math. 62 (1) (2017), 101–117. (2017) MR3627069DOI10.24193/subbmath.2017.0008
  13. Cernea, A., A note on the solutions of a second-order evolution in non separable Banach spaces, Comment. Math. Univ. Carolin. 58 (3) (2017), 307–314. (2017) MR3708775
  14. Henríquez, H.R., Poblete, V., Pozo, J., 10.1016/j.jmaa.2013.10.086, J. Math. Anal. Appl. 412 (2014), 1064–1083. (2014) MR3147269DOI10.1016/j.jmaa.2013.10.086
  15. Hernández, E.M., Henríquez, H.R., 10.4064/ap83-2-6, Ann. Polon. Math. 83 (2) (2004), 149–170. (2004) MR2111405DOI10.4064/ap83-2-6
  16. Hernández, E.M., Henríquez, H.R., 10.1619/fesi.52.113, Funkcial. Ekvac. 52 (1) (2009), 113–137. (2009) MR2538282DOI10.1619/fesi.52.113
  17. Kakutani, S.A., 10.1215/S0012-7094-41-00838-4, Duke Math. J. 8 (1941), 451–459. (1941) DOI10.1215/S0012-7094-41-00838-4
  18. Kamenskii, M., Obukhovskii, V., Zecca, P., Condensing multivalued maps and semilinear differential inclusions in Banach spaces, W. de Gruyter, Berlin, 2001. (2001) MR1831201
  19. Malaguti, L., Perrotta, S., Taddei, V., 10.3934/eect.2021053, Evol. Equ. Control Theory 11 (5) (2022), 1533–1564. (2022) MR4475865DOI10.3934/eect.2021053
  20. Pavlačková, M., Taddei, V., 10.3390/math10040672, Mathematics 10 (672) (2022). (2022) DOI10.3390/math10040672
  21. Tidke, H.L., Dhakne, M.B., 10.1007/s12591-009-0024-8, Differ. Equ. Dyn. Syst. 17 (4) (2009), 331–342. (2009) MR2610901DOI10.1007/s12591-009-0024-8
  22. Travis, C.C., Webb, G.F., 10.1007/BF01902205, Acta Math. Acad. Sci. Hungar. 32 (1–2) (1978), 75–96. (1978) DOI10.1007/BF01902205
  23. Wu, J., Theory and application of partial functional differential equations, Springer-Verlag, New York, 1996. (1996) 
  24. Xiao, J.-Z., Zhu, X.-H., Cheng, R., 10.1016/j.camwa.2012.02.015, Comput. Math. Appl. 64 (2) (2012), 147–160. (2012) MR2928211DOI10.1016/j.camwa.2012.02.015

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