On a new class of nonlinear wave equations

In this paper we prove the existence and uniqueness of regular solutions for the Cauchy problem for the evolution equation $\text{u″ + A}{}^2\text{u + (α + M(¦A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u¦}{}^2\text{) Au = 0}$, suggested by the study of beams and plates. We represent by A a linear operator of a Hilbert space H with norm ∥, α is a real number, and M(λ) > 0 a real function, for λ ? 0.