Existence of positive solutions for a fourth-order three-point BVP with sign-changing Green’s function. (Chinese. English summary) Zbl 1363.34069
Summary: We apply an iterative method to study the solutions for a fourth-order three-point boundary value problem, and obtain the existence of positive solutions of the problem
\[
u^{(4)}(t)=f(t, u(t)),\;t\in [0,1],
\]
\[ u'(0)=u''(\eta)=u'''(0)=u(1)=0, \] where \(f:[0,1]\times[0, +\infty)\to[0, +\infty)\) is continuous, \(\eta\in[\sqrt{3}/3,1]\) is a constant. In the case of sign-changing Green’s function, the existence of a positive solution of this problem can be obtained, and the solution is monotonically decreasing. The existence of a positive solution of this problem is no longer restricted to the case of a positive Green’s function.
\[ u'(0)=u''(\eta)=u'''(0)=u(1)=0, \] where \(f:[0,1]\times[0, +\infty)\to[0, +\infty)\) is continuous, \(\eta\in[\sqrt{3}/3,1]\) is a constant. In the case of sign-changing Green’s function, the existence of a positive solution of this problem can be obtained, and the solution is monotonically decreasing. The existence of a positive solution of this problem is no longer restricted to the case of a positive Green’s function.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
