Three solutions for three-point boundary value problems. (English) Zbl 1076.34011

Summary: We study the existence of at least three solutions for the three-point nonlinear boundary value problems \[ u''(t)+ a(t) f(u)= 0,\quad 0< t< 1;\qquad u(0)= 0= u(1)-\gamma u(\eta), \] with \(\eta\in (0,1)\), \(\gamma\in [0,1)\), \(a\in C([0, 1],(0,\infty))\) and \(f\in C(\mathbb{R},\mathbb{R})\). Without any monotonicity assumptions on the nonlinear term \(f\), by using the increasing operator theory and approximation process, we prove that the three-point boundary value problems has at least three solutions.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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