Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems. (English) Zbl 1139.34017
The authors are concerned with necessary and sufficient conditions for the existence of symmetric positive solutions of the fourth-order \(p\)-Laplacian differential equation
\[ (| u''| ^{p-1}u'')''=f(t,u,u',u''),\;\;t\in(0,1) \]
with the boundary condition
\[ u^{(2i)}(0)=u^{(2i)}(1)=\sum_{j=1}^{m}a_{ij}u^{(2i)}(t_j),\;\;i=0,1. \]
The proofs of the main results are based upon a fixed point theorem in cone, see [D. Guo and V. Lakshmikantham, Nonlinear problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5. Boston, MA: Academic Press (1988; Zbl 0661.47045)]. For related work see [R. I. Avery, J. Henderson, Appl. Math. Lett. 13, No. 3, 1–7 (2000; Zbl 0961.34014)].
\[ (| u''| ^{p-1}u'')''=f(t,u,u',u''),\;\;t\in(0,1) \]
with the boundary condition
\[ u^{(2i)}(0)=u^{(2i)}(1)=\sum_{j=1}^{m}a_{ij}u^{(2i)}(t_j),\;\;i=0,1. \]
The proofs of the main results are based upon a fixed point theorem in cone, see [D. Guo and V. Lakshmikantham, Nonlinear problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5. Boston, MA: Academic Press (1988; Zbl 0661.47045)]. For related work see [R. I. Avery, J. Henderson, Appl. Math. Lett. 13, No. 3, 1–7 (2000; Zbl 0961.34014)].
Reviewer: Ruyun Ma (Lanzhou)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
boundary value problems; existence; cone; multi-point boundary conditions; fixed point theoremReferences:
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