 
Nonlinear extension of Brezzi's theoremThe following lemma provides equivalent formulations for the LBB condition in an abstract setting.
Lemma 7
Define the linear operator
and its adjoint operator
in abstract spaces , by
Proof. The lemma is proved, e.g., in [3, Lemma 4.2].
Then, the following formulations of the LBB condition , , are equivalent. such that
The operator is an isomorphism, and the estimate
The operator is an isomorphism, and the estimate
Next, we present an extension of Brezzi's theorem to a class of nonlinear problems.
THEOREM 5
Suppose a mixed variational problem in abstract setting
Proof.
>From (A4) and (A5) it follows that the nonlinear problem
Find such that where the bivariate form is assumed to be linear with respect to its second argument only, and is a bilinear form. Assume that the following conditions are satisfied (A1) , (A2) with (A3) the LBB (LadyzenskayaBabuškaBrezzi) condition, i.e. such that
(A4) the strong  monotonicity of , i.e. with
(A5) and the Lipschitz continuity of , i.e., with Then there exists a unique solution . Find with has a unique solution (see, e.g., [19, Theorem 24.2.]). Then we define by Since we get , and with Lemma 7 () it follows that there exists a unique with Finally, we conclude from , (73), (74), that the pair fulfills (67), (68). Now, we apply Theorem 5 to the variational formulation (VF2).
THEOREM 6
Suppose that
and that the material function fulfills
(50) and (M1), (M2), (M4) and (M5).
Then (55), (56) have a unique solution.
Proof.
The assumptions (A1), (A2) and (A3) are verified in the
proof of Theorem 2. Further, (A4) and
(A5) follow from the Lemmata 5 and 6.
The results of Section 3.4 can be carried over to the nonlinear case without modification.
THEOREM 7
Assume that is bounded, Lipschitzcontinuous and simplyconnected.
Assume further that
is divergencefree,
i.e.,
Proof. Coincides with that of Theorem 4.
Indeed, the only term in (35)
involving is not changed during the proof of Theorem 4.
Then, the unique solution of fulfills Thus, the variational formulation of the nonlinear problem is adequate to the physical problem. Again, we have such that we can omit the term and solve the coercive problem: (VF2) Find such that
 
