Strong convergence of Noor iteration in $L^p$ spaces via $p$-uniform convexity
Main Article Content
Abstract
This paper investigates the strong convergence of Noor iteration for nonexpansive operators in $L^p$ spaces, where $1 < p < \infty$. By exploiting the $p$-uniform convexity of $L^p$, we derive explicit $p$-dependent constants in the asymptotic regularity inequalities and obtain quantitative convergence rates. We establish sufficient conditions for strong convergence, including the case of affine operators, demi-compact operators, and norm convergence. As an application, we analyze the $p$-Laplace equation for $p>1$ and validate our theoretical findings through comprehensive numerical experiments. The numerical results demonstrate the efficiency of the method and the importance of parameter selection across different nonlinearity regimes. Our work provides a quantitative refinement of known results in the literature and offers practical insights for solving nonlinear partial differential equations.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
[1] K. Ball, E. Carlen, and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), pp. 463--482.
[2] H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (3) (1996), pp. 367--426.
[3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[4] F. E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Rational Mech. Anal., 24 (1967), pp. 82--90.
[5] R. E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math., 38 (1981), 304-314
[6] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), pp. 396--414.
[7] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.
[8] O. Hanner, On the uniform convexity of $L^p$ and $ℓ^p$, Ark. Mat., 3 (1956), pp. 239--244.
[9] R. E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998.
[10] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), pp. 217--229.
[11] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), pp. 274--276.
[12] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, 1970.
[13] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 14(2) (1966),276--284.
[14] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), pp. 1127--1138.