| Title:
|
Positive solutions of a fourth-order differential equation with integral boundary conditions (English) |
| Author:
|
Padhi, Seshadev |
| Author:
|
Graef, John R. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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148 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
583-601 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the existence of positive solutions to the fourth-order two-point boundary value problem $$ \begin {cases} u^{\prime \prime \prime \prime }(t) + f(t,u(t))=0, & 0 < t < 1,\\ u^{\prime }(0) = u^\prime (1) = u^{\prime \prime }(0) =0, & u(0) = \alpha [u], \end {cases} $$ where $\alpha [u]=\int ^{1}_{0}u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq 0$ 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. (English) |
| Keyword:
|
boundary value problem |
| Keyword:
|
fixed point |
| Keyword:
|
positive solution |
| Keyword:
|
cone |
| Keyword:
|
existence \hbox {theorem} |
| MSC:
|
34B10 |
| MSC:
|
34B18 |
| DOI:
|
10.21136/MB.2022.0045-22 |
| . |
| Date available:
|
2023-11-23T12:40:07Z |
| Last updated:
|
2023-11-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151976 |
| . |
| Reference:
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[1] Anderson, D. R., Avery, R. I.: A fourth-order four-point right focal boundary value problem.Rocky Mt. J. Math. 36 (2006), 367-380. Zbl 1137.34008, MR 2234809, 10.1216/rmjm/1181069456 |
| Reference:
|
[2] Avery, R. I., Peterson, A. C.: Three positive fixed points of nonlinear operator on ordered Banach spaces.Comput. Math. Appl. 42 (2001), 313-322. Zbl 1005.47051, MR 1837993, 10.1016/S0898-1221(01)00156-0 |
| Reference:
|
[4] Benaicha, S., Haddouchi, F.: Positive solutions of a nonlinear fourth-order integral boundary value problem.An. Univ. Vest Timiş., Ser. Mat.-Inform. 54 (2016), 73-86. MR 3552473, 10.1515/awutm-2016-0005 |
| Reference:
|
[5] Graef, J. R., Qian, C., Yang, B.: A three point boundary value problem for nonlinear fourth order differential equations.J. Math. Anal. Appl. 287 (2003), 217-233. Zbl 1054.34038, MR 2010266, 10.1016/S0022-247X(03)00545-6 |
| Reference:
|
[7] 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. Zbl 1474.34170, MR 4251817 |
| Reference:
|
[8] Han, X., Gao, H., Xu, J.: Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter.Fixed Point Theory Appl. 2011 (2011), Article ID 604046, 11 pages. Zbl 1225.34032, MR 2764775, 10.1155/2011/604046 |
| Reference:
|
[9] Kang, P., Wei, Z., Xu, J.: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces.Appl. Math. Comput. 206 (2008), 245-256. Zbl 1169.34043, MR 2474970, 10.1016/j.amc.2008.09.010 |
| Reference:
|
[10] Krasnosel'skii, M. A.: Positive Solutions of Operator Equations.P. Noordhoff, Groningen (1964). Zbl 0121.10604, MR 0181881 |
| Reference:
|
[11] Li, Y.: Existence of positive solutions for the cantilever beam equations with fully nonlinear terms.Nonlinear Anal., Real World Appl. 27 (2016), 221-237. Zbl 1331.74095, MR 3400525, 10.1016/j.nonrwa.2015.07.016 |
| Reference:
|
[13] Ma, R.: Multiple positive solutions for a semipositone fourth-order boundary value problem.Hiroshima Math. J. 33 (2003), 217-227. Zbl 1048.34048, MR 1997695, 10.32917/hmj/1150997947 |
| Reference:
|
[14] 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. |
| Reference:
|
[15] Padhi, S., Graef, J. R., Pati, S.: Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions.Fract. Calc. Appl. Anal. 21 (2018), 716-745. Zbl 1406.34015, MR 3827151, 10.1515/fca-2018-0038 |
| Reference:
|
[16] Shen, W.: Positive solution for fourth-order second-point nonhomogeneous singular boundary value problems.Adv. Fixed Point Theory 5 (2015), 88-100. |
| Reference:
|
[17] Sun, Y., Zhu, C.: Existence of positive solutions for singular fourth-order three-point boundary value problems.Adv. Difference Equ. 2013 (2013), Article ID 51, 13 pages. Zbl 1380.34044, MR 3037642, 10.1186/1687-1847-2013-51 |
| Reference:
|
[18] Webb, J. R. L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions.Proc. R. Soc. Edinb., Sect. A, Math. 138 (2008), 427-446. Zbl 1167.34004, MR 2406699, 10.1017/S0308210506001041 |
| Reference:
|
[19] Wei, Y., Song, Q., Bai, Z.: Existence and iterative method for some fourth order nonlinear boundary value problems.Appl. Math. Lett. 87 (2019), 101-107. Zbl 1472.34042, MR 3848352, 10.1016/j.aml.2018.07.032 |
| Reference:
|
[20] Yan, D., Ma, R.: Global behavior of positive solutions for some semipositone fourth-order problems.Adv. Difference Equ. 2018 (2018), Article ID 443, 14 pages. Zbl 1448.34062, MR 3882988, 10.1186/s13662-018-1904-4 |
| Reference:
|
[21] Yang, B.: Positive solutions for a fourth order boundary value problem.Electron J. Qual. Theory Differ. Equ. 2005 (2005), Article ID 3, 17 pages. Zbl 1081.34025, MR 2121804, 10.14232/ejqtde.2005.1.3 |
| Reference:
|
[22] Yang, B.: Maximum principle for a fourth order boundary value problem.Differ. Equ. Appl. 9 (2017), 495-504. Zbl 1403.34024, MR 3737823, 10.7153/dea-2017-09-33 |
| Reference:
|
[23] Zhang, M., Wei, Z.: Existence of positive solutions for fourth-order $m$-point boundary value problem with variable parameters.Appl. Math. Comput. 190 (2007), 1417-1431. Zbl 1141.34018, MR 2339733, 10.1016/j.amc.2007.02.019 |
| Reference:
|
[25] Zhang, X., Liu, L.: Positive solutions of fourth-order multi-point boundary value problems with bending term.Appl. Math. Comput. 194 (2007), 321-332. Zbl 1193.34050, MR 2385904, 10.1016/j.amc.2007.04.028 |
| Reference:
|
[26] Zou, Y.: On the existence of of positive solutions for a fourth-order boundary value problem.J. Funct. Spaces 2017 (2017), Article ID 4946198, 5 pages. Zbl 1377.34031, MR 3690388, 10.1155/2017/4946198 |
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