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Two-parameter nonresonance condition for the existence of fourth-order boundary value problems. (English) Zbl 1071.34016
Summary: We discuss the existence of the fourth-order boundary value problem \[ u^{(4)}= f(t,u,u''),\quad 0< t< 1,\quad u(0)= u(1)= u''(0)= u''(1)= 0, \] where \(f: [0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) is continuous, and partly solve the Del Pino and Manasevich’s conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We present a two-parameter nonresonance condition described by circle, too.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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