| Title:
|
On solvability of nonlinear boundary value problems for the equation $(x'+g(t,x,x'))'=f(t,x,x')$ with one-sided growth restrictions on $f$ (English) |
| Author:
|
Staněk, Svatoslav |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
38 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
129-148 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider boundary value problems for second order differential equations of the form $(x^{\prime }+g(t,x,x^{\prime }))^{\prime }=f(t,x,x^{\prime })$ with the boundary conditions $r(x(0),x^{\prime }(0),x(T)) + \varphi (x)=0$, $w(x(0),x(T),x^{\prime }(T))+ \psi (x)=0$, where $g,r,w$ are continuous functions, $f$ satisfies the local Carathéodory conditions and $\varphi , \psi $ are continuous and nondecreasing functionals. Existence results are proved by the method of lower and upper functions and applying the degree theory for $\alpha $-condensing operators. (English) |
| Keyword:
|
nonlinear boundary value problem |
| Keyword:
|
existence |
| Keyword:
|
lower and upper functions |
| Keyword:
|
$\alpha $-condensing operator |
| Keyword:
|
Borsuk antipodal theorem |
| Keyword:
|
Leray-Schauder degree |
| Keyword:
|
homotopy |
| MSC:
|
34B15 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1087.34007 |
| idMR:
|
MR1909594 |
| . |
| Date available:
|
2008-06-06T22:30:11Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107827 |
| . |
| Reference:
|
[1] Cabada A., Pouso R. L.: Existence results for the problem $(\Phi (u^{\prime }))^{\prime }=f(t,u,u^{\prime })$ with nonlinear boundary conditions.Nonlinear Anal. 35 (1999), 221–231. MR 1643240 |
| Reference:
|
[2] Deimling K.: Nonlinear Functional Analysis.Springer-Verlag, Berlin Heidelberg, 1985. Zbl 0559.47040, MR 0787404 |
| Reference:
|
[3] Kiguradze I. T.: Boundary Value Problems for Systems of Ordinary Differential Equations.In “Current Problems in Mathematics: Newest Results", Vol. 30, 3–103, Moscow 1987 (in Russian); English transl.: J. Soviet Math. 43 (1988), 2340–2417. Zbl 0782.34025, MR 0925829 |
| Reference:
|
[4] Kiguradze I. T., Lezhava N. R.: Some nonlinear two-point boundary value problems.Differentsial’nyie Uravneniya 10 (1974), 2147–2161 (in Russian); English transl.: Differ. Equations 10 (1974), 1660–1671. MR 0364733 |
| Reference:
|
[5] Kiguradze I. T., Lezhava N. R.: On a nonlinear boundary value problem.Funct. theor. Methods in Differ. Equat., Pitman Publ. London 1976, 259–276. Zbl 0346.34008, MR 0499409 |
| Reference:
|
[6] Staněk S.: Two-point functional boundary value problems without growth restrictions.Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 37 (1998), 123–142. Zbl 0963.34061, MR 1690481 |
| Reference:
|
[7] Thompson H. B.: Second order ordinary differential equations with fully nonlinear two point boundary conditions.Pacific. J. Math. 172 (1996), 255–277. Zbl 0862.34015, MR 1379297 |
| Reference:
|
[8] Thompson H. B.: Second order ordinary differential equations with fully nonlinear two point boundary conditions II.Pacific. J. Math. 172 (1996), 279–297. Zbl 0862.34015, MR 1379297 |
| Reference:
|
[9] Wang M. X., Cabada A., Nieto J. J.: Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions.Ann. Polon. Math. 58 (1993), 221–235. Zbl 0789.34027, MR 1244394 |
| . |
