Abstract Differential Equation with a Spectral Parameter in the Boundary Conditions

In the present paper, we first establish a coerciveness estimate with respect to both the space variable and the spectral parameter of a boundary-value problem for an operator differential equation with nonlocal boundary conditions containing the spectral parameter. Then, using the coerciveness estimate, we prove that the operator generated by this problem is a Fredholm, and, moreover is an isomorphism between appropriate spaces. Finally, the results obtained are applied to a class of non local problems for a quasi-elliptic partial differential equation.