Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces (Q955640)

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.