Abstract
We prove an existence and uniqueness theorem for solving the operator equation F(x) + G(x) = 0, where F is a continuous and Gâteaux differentiable operator and the operator G satisfies Lipschitz condition on an open convex subset of a Banach space. As corollaries, a recent theorem of Argyros (2003) and the classical convergence theorem for modified Newton iterates are deduced. We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator.
Highlights
We prove an existence and uniqueness theorem for solving the operator equation F(x) + G(x) = 0, where F is a continuous and Gateaux differentiable operator and the operator G satisfies Lipschitz condition on an open convex subset of a Banach space
We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator
This paper considers the problem of approximating a locally unique solution x∗ of the equation F(x) + G(x) = 0, where F and G are continuous operators defined on an open convex subset D of a Banach space X with values in a Banach space Y
Summary
We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator
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