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Existence and nonexistence of positive solutions for a class of $n$th-order three-point boundary value problems in Banach spaces. (English) Zbl 1176.34030
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.

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 34G20 Nonlinear ODE in abstract spaces 34B10 Nonlocal and multipoint boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
Full Text:
References:
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