Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.