Abstract
This paper details an existence and uniqueness theorem for solving an operator equation of the form F ( x ) = 0 , where F is a Gateaux differentiable operator defined on an open convex subset of a Banach space proved. From the main theorem, an earlier theorem of Argyros follows as a consequence. Other corollaries constitute the semilocal versions of the theorems due to Ozban and Weerakoon and Fernando in a general Banach space. Our main theorem leads to the existence of solutions for a class of nonlinear Urysohn-type integral equations in the n-dimensional Euclidean space.
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