Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi; John R. Graef

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 583-601
  • ISSN: 0862-7959

Abstract

top
We study the existence of positive solutions to the fourth-order two-point boundary value problem u ' ' ' ' ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ' ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = 0 , u ( 0 ) = α [ u ] , where α [ u ] = ∫ 0 1 u ( t ) d A ( t ) is a Riemann-Stieltjes integral with A ≥ 0 being a nondecreasing function of bounded variation and f ∈ 𝒞 ( [ 0 , 1 ] × ℝ + , ℝ + ) . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

How to cite

top

Padhi, Seshadev, and Graef, John R.. "Positive solutions of a fourth-order differential equation with integral boundary conditions." Mathematica Bohemica 148.4 (2023): 583-601. <http://eudml.org/doc/299359>.

@article{Padhi2023,
abstract = {We study the existence of positive solutions to the fourth-order two-point boundary value problem \[ \{\left\lbrace \begin\{array\}\{ll\} u^\{\prime \prime \prime \prime \}(t) + f(t,u(t))=0, & 0 < t < 1,\\ u^\{\prime \}(0) = u^\prime (1) = u^\{\prime \prime \}(0) =0, & u(0) = \alpha [u], \end\{array\}\right.\} \] where $\alpha [u]=\int ^\{1\}_\{0\}u(t)\{\rm d\}A(t)$ is a Riemann-Stieltjes integral with $A \ge 0$ being a nondecreasing function of bounded variation and $f \in \mathcal \{C\}([0,1] \times \mathbb \{R\}_\{+\}, \mathbb \{R\}_\{+\})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.},
author = {Padhi, Seshadev, Graef, John R.},
journal = {Mathematica Bohemica},
keywords = {boundary value problem; fixed point; positive solution; cone; existence theorem},
language = {eng},
number = {4},
pages = {583-601},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive solutions of a fourth-order differential equation with integral boundary conditions},
url = {http://eudml.org/doc/299359},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Padhi, Seshadev
AU - Graef, John R.
TI - Positive solutions of a fourth-order differential equation with integral boundary conditions
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 583
EP - 601
AB - We study the existence of positive solutions to the fourth-order two-point boundary value problem \[ {\left\lbrace \begin{array}{ll} u^{\prime \prime \prime \prime }(t) + f(t,u(t))=0, & 0 < t < 1,\\ u^{\prime }(0) = u^\prime (1) = u^{\prime \prime }(0) =0, & u(0) = \alpha [u], \end{array}\right.} \] where $\alpha [u]=\int ^{1}_{0}u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \ge 0$ being a nondecreasing function of bounded variation and $f \in \mathcal {C}([0,1] \times \mathbb {R}_{+}, \mathbb {R}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.
LA - eng
KW - boundary value problem; fixed point; positive solution; cone; existence theorem
UR - http://eudml.org/doc/299359
ER -

References

top
  1. Anderson, D. R., Avery, R. I., 10.1216/rmjm/1181069456, Rocky Mt. J. Math. 36 (2006), 367-380. (2006) Zbl1137.34008MR2234809DOI10.1216/rmjm/1181069456
  2. Avery, R. I., Peterson, A. C., 10.1016/S0898-1221(01)00156-0, Comput. Math. Appl. 42 (2001), 313-322. (2001) Zbl1005.47051MR1837993DOI10.1016/S0898-1221(01)00156-0
  3. Benaicha, S., Haddouchi, F., 10.1515/awutm-2016-0005, An. Univ. Vest Timiş., Ser. Mat.-Inform. 54 (2016), 73-86. (2016) MR3552473DOI10.1515/awutm-2016-0005
  4. Graef, J. R., Qian, C., Yang, B., 10.1016/S0022-247X(03)00545-6, J. Math. Anal. Appl. 287 (2003), 217-233. (2003) Zbl1054.34038MR2010266DOI10.1016/S0022-247X(03)00545-6
  5. Haddouchi, F., Guendouz, C., Benaicha, S., Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem, Mat. Vesn. 73 (2021), 25-36. (2021) Zbl1474.34170MR4251817
  6. Han, X., Gao, H., Xu, J., 10.1155/2011/604046, Fixed Point Theory Appl. 2011 (2011), Article ID 604046, 11 pages. (2011) Zbl1225.34032MR2764775DOI10.1155/2011/604046
  7. Kang, P., Wei, Z., Xu, J., 10.1016/j.amc.2008.09.010, Appl. Math. Comput. 206 (2008), 245-256. (2008) Zbl1169.34043MR2474970DOI10.1016/j.amc.2008.09.010
  8. Krasnosel'skii, M. A., Positive Solutions of Operator Equations, P. Noordhoff, Groningen (1964). (1964) Zbl0121.10604MR0181881
  9. Li, Y., 10.1016/j.nonrwa.2015.07.016, Nonlinear Anal., Real World Appl. 27 (2016), 221-237. (2016) Zbl1331.74095MR3400525DOI10.1016/j.nonrwa.2015.07.016
  10. Ma, R., 10.32917/hmj/1150997947, Hiroshima Math. J. 33 (2003), 217-227. (2003) Zbl1048.34048MR1997695DOI10.32917/hmj/1150997947
  11. Padhi, S., Bhuvanagiri, S., Monotone iterative method for solutions of a cantilever beam equation with one free end, Adv. Nonlinear Var. Inequal. 23 (2020), 15-22. (2020) 
  12. Padhi, S., Graef, J. R., Pati, S., 10.1515/fca-2018-0038, Fract. Calc. Appl. Anal. 21 (2018), 716-745. (2018) Zbl1406.34015MR3827151DOI10.1515/fca-2018-0038
  13. Shen, W., Positive solution for fourth-order second-point nonhomogeneous singular boundary value problems, Adv. Fixed Point Theory 5 (2015), 88-100. (2015) 
  14. Sun, Y., Zhu, C., 10.1186/1687-1847-2013-51, Adv. Difference Equ. 2013 (2013), Article ID 51, 13 pages. (2013) Zbl1380.34044MR3037642DOI10.1186/1687-1847-2013-51
  15. Webb, J. R. L., Infante, G., Franco, D., 10.1017/S0308210506001041, Proc. R. Soc. Edinb., Sect. A, Math. 138 (2008), 427-446. (2008) Zbl1167.34004MR2406699DOI10.1017/S0308210506001041
  16. Wei, Y., Song, Q., Bai, Z., 10.1016/j.aml.2018.07.032, Appl. Math. Lett. 87 (2019), 101-107. (2019) Zbl1472.34042MR3848352DOI10.1016/j.aml.2018.07.032
  17. Yan, D., Ma, R., 10.1186/s13662-018-1904-4, Adv. Difference Equ. 2018 (2018), Article ID 443, 14 pages. (2018) Zbl1448.34062MR3882988DOI10.1186/s13662-018-1904-4
  18. Yang, B., 10.14232/ejqtde.2005.1.3, Electron J. Qual. Theory Differ. Equ. 2005 (2005), Article ID 3, 17 pages. (2005) Zbl1081.34025MR2121804DOI10.14232/ejqtde.2005.1.3
  19. Yang, B., 10.7153/dea-2017-09-33, Differ. Equ. Appl. 9 (2017), 495-504. (2017) Zbl1403.34024MR3737823DOI10.7153/dea-2017-09-33
  20. Zhang, M., Wei, Z., 10.1016/j.amc.2007.02.019, Appl. Math. Comput. 190 (2007), 1417-1431. (2007) Zbl1141.34018MR2339733DOI10.1016/j.amc.2007.02.019
  21. Zhang, X., Liu, L., 10.1016/j.amc.2007.04.028, Appl. Math. Comput. 194 (2007), 321-332. (2007) Zbl1193.34050MR2385904DOI10.1016/j.amc.2007.04.028
  22. Zou, Y., 10.1155/2017/4946198, J. Funct. Spaces 2017 (2017), Article ID 4946198, 5 pages. (2017) Zbl1377.34031MR3690388DOI10.1155/2017/4946198

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.