| Title:
|
Existence results for a class of high order differential equation associated with integral boundary conditions at resonance (English) |
| Author:
|
Nhan, Le Cong |
| Author:
|
Hoang, Do Huy |
| Author:
|
Truong, Le Xuan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
53 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
111-130 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By using Mawhin’s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form $$x^{(n)} =f(t,x,x^{\prime },\dots ,x^{(n-1)})\,, \quad t \in [0, 1]\,,$$ associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator. (English) |
| Keyword:
|
coincidence degree |
| Keyword:
|
high order differential equation |
| Keyword:
|
resonance |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| idZBL:
|
Zbl 06770056 |
| idMR:
|
MR3672785 |
| DOI:
|
10.5817/AM2017-2-111 |
| . |
| Date available:
|
2017-06-09T07:52:55Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146798 |
| . |
| Reference:
|
[1] Du, Z., Lin, X., Ge, W.: Some higher-order multi-point boundary value problem at resonance.J. Comput. Appl. Math. 177 (2005), 55–65. Zbl 1059.34010, MR 2118659, 10.1016/j.cam.2004.08.003 |
| Reference:
|
[2] Feng, W., Webb, J.R.L.: Solvability of m-point boundary value problems with nonlinear growth.J. Math. Anal. Appl. 212 (1997), 467–480. Zbl 0883.34020, MR 1464891, 10.1006/jmaa.1997.5520 |
| Reference:
|
[3] Feng, W., Webb, J.R.L.: Solvability of three-point boundary value problems at resonance.Nonlinear Anal. 30 (6) (1997), 3227–3238. Zbl 0891.34019, MR 1603039, 10.1016/S0362-546X(96)00118-6 |
| Reference:
|
[4] Franco, D., Infante, G., Zima, M.: Second order nonlocal boundary value problems at resonance.Math. Nachr. 284 (2011), 875–884. Zbl 1273.34029, MR 2815958, 10.1002/mana.200810841 |
| Reference:
|
[5] Gaines, R.E., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations.Lecture Notes in Math., vol. 568, Springer Verlag Berlin, 1977. Zbl 0339.47031, MR 0637067, 10.1007/BFb0089537 |
| Reference:
|
[6] Gao, Y., Pei, M.: Solvability for two classes of higher-order multi-point boundary value problems at resonance.Boundary Value Problems (2008), Art. ID 723828, 14 pp. Zbl 1149.34008, MR 2392915 |
| Reference:
|
[7] Gupta, C.P.: Existence theorems for a second order m-point boundary value problem at resonance.Internat. J. Math. Math. Sci. 18 (4) (1995), 705–710. Zbl 0839.34027, MR 1347059, 10.1155/S0161171295000901 |
| Reference:
|
[8] Il’in, V.A., Moiseev, E.I.: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator.J. Differential Equations 23 (1987), 803–810. MR 0903975 |
| Reference:
|
[9] Il’in, V.A., Moiseev, E.I.: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.J. Differential Equations 23 (1987), 979–978. Zbl 0668.34024, MR 0909590 |
| Reference:
|
[10] Infante, G., Zima, M.: Positive solutions of multi-point boundary value problems at resonance.Nonlinear Anal. 69 (2008), 622–633. Zbl 1203.34041, MR 2446343, 10.1016/j.na.2007.08.024 |
| Reference:
|
[11] Kosmatov, N.: A multi-point boundary value problem with two critical conditions.Nonlinear Anal. 65 (2006), 622–633. Zbl 1121.34023, MR 2231078, 10.1016/j.na.2005.09.042 |
| Reference:
|
[12] Kosmatov, N.: A singular non-local problem at resonance.J. Math. Anal. Appl. 394 (2012), 425–431. Zbl 1260.34037, MR 2926233, 10.1016/j.jmaa.2012.04.069 |
| Reference:
|
[13] Lin, X., Du, Z., Ge, W.: Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations.Comput. Math. Appl. 49 (2005), 1–11. Zbl 1081.34017, MR 2123180, 10.1016/j.camwa.2005.01.001 |
| Reference:
|
[14] Lu, S., Ge, W.: On the existence of m-point boundary value problem at resonance for higher order differential equation.J. Math. Anal. Appl. 287 (2003), 522–539. Zbl 1046.34029, MR 2024338, 10.1016/S0022-247X(03)00567-5 |
| Reference:
|
[15] Ma, R.: Existence results of a m-point boundary value problem at resonance.J. Math. Anal. Appl. 294 (2004), 147–157. Zbl 1070.34028, MR 2059796, 10.1016/j.jmaa.2004.02.005 |
| Reference:
|
[16] Mawhin, J.: Topological degree methods in nonlinear boundary value problems.Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, R.I., 1979. Zbl 0414.34025, MR 0525202 |
| Reference:
|
[17] Mawhin, J.: Functional analysis and nonlinear boundary value problems: the legacy of Andrzej Lasota.Ann. Math. Sil. 27 (2013), 7–38. Zbl 1326.34013, MR 3157115 |
| Reference:
|
[18] O’Regan, D.: Boundary value problems for second and higher order differential equations.Proc. Amer. Math. Soc. 113 (1991), 761–775. Zbl 0742.34023, MR 1069295, 10.1090/S0002-9939-1991-1069295-2 |
| Reference:
|
[19] Przeradzki, B.: Three methods for the study of semilinear equations at resonance.Colloq. Math. (1993), 110–129. Zbl 0828.47054, MR 1242650 |
| Reference:
|
[20] Wong, F.H.: An application of Schander’s fixed point theorem with respect to higher order BVPs.Proc. Amer. Math. Soc. 126 (1998), 2389–2397. MR 1476399, 10.1090/S0002-9939-98-04709-1 |
| Reference:
|
[21] Zhang, X., Feng, M., Ge, W.: Existence result of second-order differential equations with integral boundary conditions at resonance.J. Math. Anal. Appl. 353 (2009), 311–319. Zbl 1180.34016, MR 2508869, 10.1016/j.jmaa.2008.11.082 |
| . |
