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Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. (English) Zbl 1159.34020
The paper deals with the existence of positive solutions of the boundary value problem $$x^{(4)}-\lambda f(t,x)=\theta, \quad 0<t<1$$ in a Banach space $E$ under the boundary conditions $$x(0)=x(1)= \int_0^1g(s)x(s)\,ds \quad\text{and}\quad x''(0)=x''(1)= \int_0^1h(s)x''(s)\,ds.$$ The authors prove their results by using a cone in $E$ and applying the fixed point theory in cones for strict set contraction operators. An example is given to illustrate the result.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34G20Nonlinear ODE in abstract spaces
Full Text: DOI
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