Mathematica Bohemica, Vol. 148, No. 4, pp. 583-601, 2023
Positive solutions of a fourth-order differential equation with integral boundary conditions
Seshadev Padhi, John R. Graef
Received March 28, 2022. Published online December 15, 2022.
Abstract: 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 $\alpha[u]=\int^1_0u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq0$ 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.
Keywords: boundary value problem; fixed point; positive solution; cone; existence theorem
References: [1] D. R. Anderson, R. I. Avery: A fourth-order four-point right focal boundary value problem. Rocky Mt. J. Math. 36 (2006), 367-380. DOI 10.1216/rmjm/1181069456 | MR 2234809 | Zbl 1137.34008
[2] R. I. Avery, A. C. Peterson: Three positive fixed points of nonlinear operator on ordered Banach spaces. Comput. Math. Appl. 42 (2001), 313-322. DOI 10.1016/S0898-1221(01)00156-0 | MR 1837993 | Zbl 1005.47051
[4] S. Benaicha, F. Haddouchi: Positive solutions of a nonlinear fourth-order integral boundary value problem. An. Univ. Vest Timiş., Ser. Mat.-Inform. 54 (2016), 73-86. DOI 10.1515/awutm-2016-0005 | MR 3552473
[5] J. R. Graef, C. Qian, B. Yang: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287 (2003), 217-233. DOI 10.1016/S0022-247X(03)00545-6 | MR 2010266 | Zbl 1054.34038
[7] F. Haddouchi, C. Guendouz, S. Benaicha: Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem. Mat. Vesn. 73 (2021), 25-36. MR 4251817 | Zbl 1474.34170
[8] X. Han, H. Gao, J. Xu: 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. DOI 10.1155/2011/604046 | MR 2764775 | Zbl 1225.34032
[9] P. Kang, Z. Wei, J. Xu: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Appl. Math. Comput. 206 (2008), 245-256. DOI 10.1016/j.amc.2008.09.010 | MR 2474970 | Zbl 1169.34043
[10] M. A. Krasnosel'skii: Positive Solutions of Operator Equations. P. Noordhoff, Groningen (1964). MR 0181881 | Zbl 0121.10604
[11] Y. Li: Existence of positive solutions for the cantilever beam equations with fully nonlinear terms. Nonlinear Anal., Real World Appl. 27 (2016), 221-237. DOI 10.1016/j.nonrwa.2015.07.016 | MR 3400525 | Zbl 1331.74095
[13] R. Ma: Multiple positive solutions for a semipositone fourth-order boundary value problem. Hiroshima Math. J. 33 (2003), 217-227. DOI 10.32917/hmj/1150997947 | MR 1997695 | Zbl 1048.34048
[14] S. Padhi, S. Bhuvanagiri: Monotone iterative method for solutions of a cantilever beam equation with one free end. Adv. Nonlinear Var. Inequal. 23 (2020), 15-22.
[15] S. Padhi, J. R. Graef, S. Pati: Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions. Fract. Calc. Appl. Anal. 21 (2018), 716-745. DOI 10.1515/fca-2018-0038 | MR 3827151 | Zbl 1406.34015
[16] W. Shen: Positive solution for fourth-order second-point nonhomogeneous singular boundary value problems. Adv. Fixed Point Theory 5 (2015), 88-100.
[17] Y. Sun, C. Zhu: Existence of positive solutions for singular fourth-order three-point boundary value problems. Adv. Difference Equ. 2013 (2013), Article ID 51, 13 pages. DOI 10.1186/1687-1847-2013-51 | MR 3037642 | Zbl 1380.34044
[18] J. R. L. Webb, G. Infante, D. Franco: 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. DOI 10.1017/S0308210506001041 | MR 2406699 | Zbl 1167.34004
[19] Y. Wei, Q. Song, Z. Bai: Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 87 (2019), 101-107. DOI 10.1016/j.aml.2018.07.032 | MR 3848352 | Zbl 1472.34042
[20] D. Yan, R. Ma: Global behavior of positive solutions for some semipositone fourth-order problems. Adv. Difference Equ. 2018 (2018), Article ID 443, 14 pages. DOI 10.1186/s13662-018-1904-4 | MR 3882988 | Zbl 1448.34062
[21] B. Yang: Positive solutions for a fourth order boundary value problem. Electron J. Qual. Theory Differ. Equ. 2005 (2005), Article ID 3, 17 pages. DOI 10.14232/ejqtde.2005.1.3 | MR 2121804 | Zbl 1081.34025
[22] B. Yang: Maximum principle for a fourth order boundary value problem. Differ. Equ. Appl. 9 (2017), 495-504. DOI 10.7153/dea-2017-09-33 | MR 3737823 | Zbl 1403.34024
[23] M. Zhang, Z. Wei: Existence of positive solutions for fourth-order $m$-point boundary value problem with variable parameters. Appl. Math. Comput. 190 (2007), 1417-1431. DOI 10.1016/j.amc.2007.02.019 | MR 2339733 | Zbl 1141.34018
[25] X. Zhang, L. Liu: Positive solutions of fourth-order multi-point boundary value problems with bending term. Appl. Math. Comput. 194 (2007), 321-332. DOI 10.1016/j.amc.2007.04.028 | MR 2385904 | Zbl 1193.34050
[26] Y. Zou: On the existence of of positive solutions for a fourth-order boundary value problem. J. Funct. Spaces 2017 (2017), Article ID 4946198, 5 pages. DOI 10.1155/2017/4946198 | MR 3690388 | Zbl 1377.34031
Affiliations: Seshadev Padhi, Department of Mathematics, Birla Institute of Technology, Mesra, Ranchi, Jharkhand-835215, India, e-mail: spadhi@bitmesra.ac.in; John R. Graef (corresponding author), Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA, e-mail: john-graef@utc.edu
