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Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. (English) Zbl 1158.34336

Summary: This paper investigates the existence and multiplicity of positive solutions for the class of nonlinear boundary-value problems
\[ x''+f(t,x)=\theta,\quad 0<t<1, \]
subject to one of the following integral boundary conditions:
\[ x(0)=\int^1_0g(t)x(t)\,dt,\quad x(1)=\theta, \]
or
\[ x(0)=\theta,\quad x(1)=\int^1_0g(t)x(t)\,dt. \]
The arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators. The nonexistence of a positive solution is also studied.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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