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Multiple unbounded positive solutions for three-point BVPs with sign-changing nonlinearities on the positive half-line. (English) Zbl 1195.34042
The authors consider the following second-order nonlinear three-point boundary value problem on the positive half-line
\[ -x''+cx'+\lambda x=f(t,x(t), x'(t)), \quad t\in (0,\infty), \]
\[ x(0)-\alpha x(\eta)=a_0, \quad \lim_{t\to \infty}\frac{x'(t)}{re^{rt}}=b_0, \]
where \(a_0, b_0\) are nonnegative real numbers, \(\alpha\geq 0,\) \(\eta>0,\) \(c,\lambda\) are real positive constants, \(r\in (0,c),\) and \(f: (0,\infty)\times [0,\infty)\times {\mathbb R}\to {\mathbb R}\) is a Carathéodory function which may change sign. Under general polynomial growth conditions on \(f\) the existence of nontrivial single and multiple unbounded positive solutions is proved via fixed point theorems in a cone in a special weighted Banach space.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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