Multiple solutions for a class BVPs for second order ODEs via an extension of Leray-Schauder boundary condition

Abstract

We study a class of boundary value problems for second order ODEs. A new topological approach is developed and applied to prove the existence of at least two nonnegative classical solutions. The interesting points
of our results are that derivative operator is time depending and the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. The arguments are based upon recent fixed point index theory in cones of Banach spaces for a k-set contraction perturbed by an expansive operator. An example is given to illustrate the obtained result.