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Let Y be a real Banach space and X* its conjugate Banach space. Let A be an unbounded monotone linear mapping from A"to X* and Na potential mapping from X* to X. In this paper we establish the existence of a solution of the equation u + ANu = v for a given v in X* using variational method. Our method consists in using a splitting of A via an auxiliary Hubert space and solving an equivalent equation in this auxiliary Hubert space. In §2, we prove the same result in the case when Y is a Hubert space using the natural splitting of A in terms of its square root. We do this to compare and contrast the proofs in the two cases. Introduction. Let Y be a real Banach space and Y* denote its conjugate Banach space. Let A be an unbounded monotone linear mapping from .Yto X* and A/a nonlinear mapping from X* to Y satisfying no monotone hypothesis. In this paper we study the solvability of the equation

The automobile industry in the United States is large, very complex and has great economic power. The major manufacturers are run as bureaucratic structures designed to operate efficiently at all levels. Those people and organizations that deal with the manufacturers have patterned their conduct to accommodate this model of economic efficiency. However, new models of automobiles must be designed several years before they are offered to the public and the demand for new automobiles fluctuates significantly. Bureaucratic operation in the service of economic efficiency, time-span and fluctuating demand are all critical factors which are reflected in many different kinds of exchange transactions found in this industry. In this paper we will consider some of the exchange relationships between the manufacturers, and their suppliers, their dealers and their customers in order to generalize about contract as it is found in this kind of large scale industry. This discussion will be limited to practices within the United States and the consequences of those practices under its legal system. We lack data for a full comparative analysis. Nonetheless what is said here is likely to have relevance in other nations. The automobile manufacturers are important transnational corporations and their practices are models for other large corporations. While the precise legal techniques used will differ, it is likely that the goal of risk avoidance and minimization will be pursued in all nations in which such organizations operate.

Let X be a real normed space with norm || ||, T the radial projection mapping defined by \( Tx = x,\quad {\text{if}}\left\| x \right\| \leqq 1,\quad {\text{and}}\quad Tx = x/\left\| x \right\|,\quad {\text{if}}\left\| x \right\| \geqq 1. \)

Let X be a real Banach space, X* its conjugate space, and let (w, u) denote the pairing between w in X * and u in X.

Ler X be a normed space. A mapping T: \( D \to X \) is called a contraction if \( \left\| {Tx - Ty} \right\| \le \left\| {x - y} \right\| \) for all x and y in D.