| Title:
|
Existence results and iterative method for fully third order nonlinear integral boundary value problems (English) |
| Author:
|
Dang, Quang A |
| Author:
|
Dang, Quang Long |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
66 |
| Issue:
|
5 |
| Year:
|
2021 |
| Pages:
|
657-672 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the boundary value problem \begin {gather} u'''(t)=f(t,u(t),u'(t),u''(t)), \quad 0<t<1, \nonumber \\ u(0)=u'(0)=0, \quad u(1)= \int _0^1 g(s)u(s) \mathrm{d} s,\nonumber \end {gather} where $f\colon [0, 1] \times \mathbb {R}^3 \rightarrow \mathbb {R}^+$, $g\colon [0, 1] \rightarrow \mathbb {R}^+$ are continuous functions. The case when $f=f(u(t))$ was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method. (English) |
| Keyword:
|
fully third order nonlinear differential equation |
| Keyword:
|
integral boundary condition |
| Keyword:
|
positive solution |
| Keyword:
|
iterative method |
| MSC:
|
34B15 |
| MSC:
|
34B27 |
| idZBL:
|
07396172 |
| idMR:
|
MR4299879 |
| DOI:
|
10.21136/AM.2021.0040-20 |
| . |
| Date available:
|
2021-08-18T08:28:33Z |
| Last updated:
|
2023-11-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149076 |
| . |
| Reference:
|
[1] 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:
|
[2] Benchohra, M., Nieto, J. J., Ouahab, A.: Second-order boundary value problem with integral boundary conditions.Bound. Value Probl. 2011 (2011), Article ID 260309, 9 pages. Zbl 1208.34015, MR 2734188, 10.1155/2011/260309 |
| Reference:
|
[3] Boucherif, A.: Positive solutions of second order differential equations with integral boundary conditions.Discrete and Continuous Dynamical Systems 2007, Suppl AIMS, Springfield (2007), 155-159. Zbl 1163.34315, MR 2409209, 10.3934/proc.2007.2007.155 |
| Reference:
|
[4] Boucherif, A.: Second-order boundary value problems with integral boundary conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 364-371. Zbl 1169.34310, MR 2468243, 10.1016/j.na.2007.12.007 |
| Reference:
|
[5] Boucherif, A., Bouguima, S. M., Al-Malki, N., Benbouziane, Z.: Third order differential equations with integral boundary conditions.Nonlinear Anal., Theory Methods Appl., Ser. A, e-Suppl. 71 (2009), e1736--e1743. Zbl 1238.34031, MR 2671952, 10.1016/j.na.2009.02.055 |
| Reference:
|
[6] Dang, Q. A: Mixed boundary-domain operator in approximate solution of biharmonic type equation.Vietnam J. Math. 26 (1998), 243-252. Zbl 0939.35061, MR 1684351 |
| Reference:
|
[7] Dang, Q. A, Dang, Q. L.: A simple efficient method for solving sixth-order nonlinear boundary value problems.Comput. Appl. Math. 37 (2018), 16-26. Zbl 1438.65152, MR 3896686, 10.1007/s40314-018-0643-1 |
| Reference:
|
[8] Dang, Q. A, Dang, Q. L., Ngo, T. K. Q.: A novel efficient method for nonlinear boundary value problems.Numer. Algorithms 76 (2017), 427-439. Zbl 1378.65149, MR 3704876, 10.1007/s11075-017-0264-6 |
| Reference:
|
[9] Dang, Q. A, Ngo, T. K. Q.: Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term.Nonlinear Anal., Real World Appl. 36 (2017), 56-68. Zbl 1362.34036, MR 3621230, 10.1016/j.nonrwa.2017.01.001 |
| Reference:
|
[10] Dang, Q. A, Ngo, T. K. Q.: New fixed point approach for a fully nonlinear fourth order boundary value problem.Bol. Soc. Parana Mat. (3) 36 (2018), 209-223. Zbl 1424.34083, MR 3744750, 10.5269/bspm.v36i4.33584 |
| Reference:
|
[11] Denche, M., Kourta, A.: Boundary value problem for second-order differential operators with integral conditions.Appl. Anal. 84 (2005), 1247-1266. Zbl 1084.47036, MR 2178770, 10.1080/00036810500287255 |
| Reference:
|
[12] Guendouz, C., Haddouchi, F., Benaicha, S.: Existence of positive solutions for a nonlinear third-order integral boundary value problem.Ann. Acad. Rom. Sci., Math. Appl. 10 (2018), 314-328. Zbl 1438.34095, MR 3933051 |
| Reference:
|
[13] Guo, Y., Liu, Y., Liang, Y.: Positive solutions for the third-order boundary value problems with the second derivatives.Bound. Value Probl. 2012 (2012), Articles ID 34, 9 pages. Zbl 1279.34040, MR 2944543, 10.1186/1687-2770-2012-34 |
| Reference:
|
[14] Guo, Y., Yang, F.: Positive solutions for third-order boundary-value problems with the integral boundary conditions and dependence on the first-order derivatives.J. Appl. Math. 2013 (2013), Article ID 721909, 6 pages. Zbl 1271.34028, MR 3074314, 10.1155/2013/721909 |
| Reference:
|
[15] Jankowski, T.: Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 73 (2010), 1289-1299. Zbl 1200.34072, MR 2661226, 10.1016/j.na.2010.04.055 |
| Reference:
|
[16] Lepin, A. Y., Lepin, L. A.: On a boundary value problem with integral boundary conditions.Differ. Equ. 51 (2015), 1666-1668 translation from Differ. Uravn. 51 2015 1686-1688. Zbl 1337.34026, MR 3453243, 10.1134/S0012266115120149 |
| Reference:
|
[17] Li, H., Wang, L., Pei, M.: Solvability of a fourth-order boundary value problem with integral boundary conditions.J. Appl. Math. 2013 (2013), Article ID 782363, 7 pages. Zbl 1270.34033, MR 3035180, 10.1155/2013/782363 |
| Reference:
|
[18] Lv, X., Wang, L., Pei, M.: Monotone positive solution of a fourth-order BVP with integral boundary conditions.Bound. Value Probl. 2015 (2015), Article ID 172, 12 pages. Zbl 1341.34033, MR 3400645, 10.1186/s13661-015-0441-2 |
| Reference:
|
[19] Sun, J.-P., Li, H.-B.: Monotone positive solution of nonlinear third-order BVP with integral boundary conditions.Bound. Value Probl. 2010 (2010), Article ID 874959, 11 pages. Zbl 1208.34017, MR 2739196, doi.org/10.1155/2010/874959 |
| Reference:
|
[20] Zhang, X., Ge, W.: Positive solutions for a class of boundary-value problems with integral boundary conditions.Comput. Math. Appl. 58 (2009), 203-215. Zbl 1189.34035, MR 2535787, 10.1016/j.camwa.2009.04.002 |
| . |
