| Title:
|
On the solvability of some multi-point boundary value problems (English) |
| Author:
|
Gupta, Chaitan P. |
| Author:
|
Ntouyas, S. K. |
| Author:
|
Tsamatos, P. Ch. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
41 |
| Issue:
|
1 |
| Year:
|
1996 |
| Pages:
|
1-17 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f\colon [0,1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function satisfying Caratheodory’s conditions and let $e(t)\in L^{1}[0,1]$. Let $\xi _{i}, \tau _{j}\in (0,1)$, $ c_{i},a_{j}\in \mathbb{R}$, all of the $c_{i}$’s, (respectively, $a_{j}$’s) having the same sign, $i=1,2,\ldots ,m-2$, $j=1,2,\ldots ,n-2$, $0 < \xi _{1}<\xi _{2}<\ldots <\xi _{m-2}<1$, $0 < \tau _{1}<\tau _{2}<\ldots <\tau _{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems \begin{align*} x^{\prime\prime}(t)=f(t, x(t),x^{\prime}(t))+e(t),\qquad t\in (0,1)\tag{E} \\ x(0)=\sum\limits_{i=1}^{m-2} c_{i}x^{\prime}(\xi_{i}),\qquad x(1)=\sum\limits_{j=1}^{n-2} a_{j}x(\tau_{j}) \tag{BC$_{mn}$}\end{align*} and \begin{align*} x^{\prime\prime}(t)=f(t, x(t),x^{\prime}(t))+e(t),\qquad t\in (0,1)\tag {E}\\ x(0)=\sum\limits_{i=1}^{m-2} c_{i}x^{\prime}(\xi_{i}),\qquad x^{\prime}(1)=\sum\limits_{j=1}^{n-2} a_{j}x^{\prime}(\tau_{j}), \tag{BC$_{mn}$'} \end{align*} Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem. (English) |
| Keyword:
|
multi-point boundary value problems |
| Keyword:
|
four point boundary value problems |
| Keyword:
|
Leray-Schauder Continuation theorem |
| Keyword:
|
a priori bounds |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| idZBL:
|
Zbl 0858.34013 |
| idMR:
|
MR1365136 |
| DOI:
|
10.21136/AM.1996.134310 |
| . |
| Date available:
|
2009-09-22T17:49:55Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134310 |
| . |
| Reference:
|
[1] C. P. Gupta: Solvability of a three-point boundary value problem for a second order ordinary differential equation.Jour. Math. Anal. Appl. 168 (1992), 540–551. MR 1176010, 10.1016/0022-247X(92)90179-H |
| Reference:
|
[2] C. P. Gupta: A note on a second order three-point boundary value problem.Jour. Math. Anal. Appl. 186 (1994), 277–281. Zbl 0805.34017, MR 1290657, 10.1006/jmaa.1994.1299 |
| Reference:
|
[3] C. Gupta, S. Ntouyas and P. Tsamatos: On an $m$-point boundary value problem for second order ordinary differential equations.Nonlinear Analysis 23 (1994), 1427–1436. MR 1306681, 10.1016/0362-546X(94)90137-6 |
| Reference:
|
[4] C. Gupta, S. Ntouyas and P. Tsamatos: Existence results for $m$-point boundary value problems.Differential Equations and Dynamical Systems 2 (1994), 289–298. MR 1386275 |
| Reference:
|
[5] V. Il’in and E. Moiseev: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects.Differential Equations 23 (1987), 803–810. |
| Reference:
|
[6] V. Il’in and E. Moiseev: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.Differential Equations 23 (1987), 979–987. |
| Reference:
|
[7] S. A. Marano: A remark on a second order three-point boundary value problem.Jour. Math. Anal. Appl. 183 (1994), 518–522. Zbl 0801.34025, MR 1274852, 10.1006/jmaa.1994.1158 |
| Reference:
|
[8] J. Mawhin: Topological degree methods in nonlinear boundary value problems.“NSF-CBMS Regional Conference Series in Math.” No. 40, Amer. Math. Soc., Providence, RI, 1979. Zbl 0414.34025, MR 0525202 |
| . |
