In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent.
Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero X* of a differentiable function F defined on a convex set D of a Hilbert space H, we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation F(x)=f, where F : D(F) ⊆ X → X is a nonlinear monotone operator defined on a real Hilbert space X. It is assumed that F(∧x)=f and that the only available data are fδ with ||f- fδ||≤δ. We prove that the TSNLM converges cubically to a solution of the equation F(x)+α(x-x0)= fδ (such solution is an approximation of ∧x) where x0 is the initial guess. Under a general source condition on x0-∧x, we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.
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