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Multiple solutions and eigenvalues for third-order right focal boundary value problems. (English) Zbl 1003.34021

The authors consider the third-order nonlinear ordinary differential equation \[ x'''(t)= f(t,x(t)),\quad t_1\leq t\leq t_3, \] and the associated eigenvalue problem \[ x'''(t)= \lambda a(t) f(x(t)) \] both with the same boundary conditions \(x(t_1)= x'(t_2)= x''(t_3)= 0\). Making various assumptions on \(f\), \(a\) and \(\lambda\), they establish intervals of the parameter \(\lambda\) that yield the existence of a positive solution to the eigenvalue problem. By placing certain restrictions on the nonlinearity, they prove the existence of at least one or at least two, or at least three, or infinitely many positive solutions to the boundary value problem.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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