Vol. 72, No. 1, 1977

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Monotonicity and alternative methods for nonlinear boundary value problems

R. Kent Nagle

Vol. 72 (1977), No. 1, 197–206

Let X be a Hilbert space, E a linear operator with finite dimensional null space, and N a nonlinear operator. In this paper we study the nonlinear equation

Ex = N x  x ∈ X.

Equations of this form arise in the study of boundary value problems for elliptic differential equations.

We use the alternative scheme of Bancroft, Hale, and Sweet and results from monotone operator theory with suitable monotonicity assumptions on E and N to reduce equation (1) to an alternative problem. We then use results from monotone operator theory to solve the alternative problem, hence prove the existence of solutions to equation (1). This extends to nonselfadjoint operators the results of Cesari and Kannan.

Mathematical Subject Classification
Primary: 47H15
Received: 30 November 1976
Published: 1 September 1977
R. Kent Nagle