### A study of variational inequalities for set-valued mappings.

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A condition weaker than the insatiability condition is given.

We have given several proofs on the existence of the price equilibrium --- via variational inequality --- via degree theory and via Brouwer's theorems.

The aim of this paper is to establish a random coincidence degree theory. This degree theory possesses all the usual properties of the deterministic degree theory such as existence of solutions, excision and Borsuk’s odd mapping theorem. Our degree theory provides a method for proving the existence of random solutions of the equation $Lx\in N(\omega ,x)$ where $L:\mathit{\text{dom}}\phantom{\rule{1.3pt}{0ex}}L\subset X\to Z$ is a linear Fredholm mapping of index zero and $N:\Omega \times \overline{G}\to {2}^{Z}$ is a noncompact Carathéodory mapping. Applications to random differential inclusions are also considered.

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