In this article, we develop some general theorems governing the existence of solutions of a class of nonlinear problems of elastostatics of continuous media. We arrive at these theorems in a deductive way; i.e., we develop some general existence theorems applicable to abstract equations defined on reflexive Banach spaces, and we then describe sufficient conditions under which certain problems in nonlinear elasticity fall within the framework of the general theorems. To simplify the analysis, we restrict ourselves to the problem of place in nonlinear elastostatics; i.e., we consider the class of nonlinear boundary-value problems in which the displacements are prescribed on the boundary. Our theory does not require the existence of a stored energy function, or is it limited to bodies subjected to conservative external forces. Moreover, it is applicable to one-, two-, and three-dimensional (indeed, to n-dimensional) problems. As such, it generalizes theories that have been put forth recently. However, in the present study we do not consider complications due to the constraint of local invcrtibility which is commonly assumed to hold in problems of finite elasticity. We hope to address this problem in later work. A number of strong existence theorems for linear elliptic boundary value theorems of elasticity are known, and a comprehensive article on this subject was written by Fichera [13]. By comparison, however, the nonlinear theory is virtually untouched. An excellent summary account of the state of existence theory in nonlinear elasticity as it stood in 1973 can be found in the book of Wang and Truesdell [27]. For one-dimensional problems, the theory is more fully developed primarily because of the fact that, if strong ellipticity is assumed, the operators of one-dimensional elasticity are semi-monotone; i.e., they are monotone in the highest derivatives. As such, they fall into the class of operators studied extensively by B&is [9], Lions [I 81, B rowder [ll], and others. Antman has exploited this fact repeatedly in his studies of elastic rods, plates, and shells (see, for example, [2-61 and the references therein). An exhaustive bibliography on monotone and pseudo-monotone operator theory, with an emphasis on