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Zbl 1176.34030
Zhang, Xuemei; Feng, Meiqiang; Ge, Weigao
Existence and nonexistence of positive solutions for a class of $n$th-order three-point boundary value problems in Banach spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 2, A, 584-597 (2009). ISSN 0362-546X

The authors study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear three-point boundary value problem for a differential equation of order $n$ in an ordered Banach space. The boundary value problem has the form $$x^{(n)}(t)+ f(t, x(t), x'(t),\dots, x^{(n-2)}(t))= \theta,\quad t\in [0,1],$$ $$x^{(i)}(0)= \theta,\quad i= 0,1,\dots, n-2,$$ $$x^{(n-2)}(1)= \rho x^{(n-2)}(\eta),$$ where $\rho,\eta\in (0,1)$, and $\theta$ is the zero element of the Banach space. The proofs employ fixed-point theory in cones.
[Sergei A. Brykalov (Ekaterinburg)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE
34G20 Nonlinear ODE in abstract spaces
34B10 Multipoint boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: three-point nonlinear boundary value problems in Banach spaces; positive solutions

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