Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition. (English) Zbl 1140.34009

The paper deals with the second-order three-point boundary value problem \[ y''(t)+f(t,y)=0,\quad 0\leq t\leq 1, \tag{1} \]
\[ y(0)=0,\quad y(1)-\alpha y(\eta)=0, \tag{2} \] where \(f\in C([0,1]\times\mathbb R,\mathbb R)\) and \(0<\eta< 1\), \(0<\alpha <1\). The purpose of this paper is to prove some multiplicity results for solutions of problem (1), (2) under conditions of non-well-ordered upper and lower solutions. The authors prove the existence of at least two, three or six solutions of the problem. The proofs are based on the fixed point index of some increasing operator with respect to some closed convex sets, which are translations of some special cones. The main idea is associating a fixed point index with pairs of lower and upper solutions. In the proofs the athors employ monotonicity conditions on the nonlinear term \(f\) of (1).


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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