# Semi-implicit iterative schemes with perturbed operators for infinite accretive mappings and infinite nonexpansive mappings and their applications to parabolic systems

Volume 10, Issue 3, pp 902--921 Publication Date: March 20, 2017       Article History
• 630 Downloads
• 710 Views

### Authors

Li Wei - School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, Hebei, China. Ravi P. Agarwal - Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX78363, USA. - Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia. Yaqin Zheng - College of Science, Agricultural University of Hebei, Baoding 071001, Hebei, China.

### Abstract

In a real uniformly convex and uniformly smooth Banach space, we first prove a new path convergence theorem and then present some new semi-implicit iterative schemes with errors which are proved to be convergent strongly to the common element of the set of zero points of infinite m-accretive mappings and the set of fixed points of infinite nonexpansive mappings. The superposition of perturbed operators are considered in the construction of the iterative schemes and new proof techniques are employed compared to some of the recent work. Some examples are listed and computational experiments are conducted, which guarantee the effectiveness of the proposed iterative schemes. Moreover, a kind of parabolic systems is exemplified, which sets up the relationship among iterative schemes, nonlinear systems and variational inequalities.

### Keywords

• M-accretive mapping
• $\tau_i$-strongly accretive mapping
• contractive mapping
• $\lambda_i$-strictly pseudocontractive mapping
• semi-implicit iterative scheme
• parabolic systems.

•  47H05
•  47H09
•  47H10

### References

• [1] R. P. Agarwal, D. O’Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, Springer, New York (2009)

• [2] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), 9 pages.

• [3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste Romania, Bucharest; Noordhoff International Publishing, Leiden (1976)

• [4] R. E. Bruck Jr., Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc., 179 (1973), 251–262.

• [5] L.-C. Ceng, Q. H. Ansari, S. Schaible, J.-C. Yao, Hybrid viscosity approximation method for zeros of m-accretive operators in Banach spaces, Numer. Funct. Anal. Optim., 33 (2012), 142–165.

• [6] L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numer. Funct. Anal. Optim., 29 (2008), 987–1033.

• [7] L.-C. Ceng, A. R. Khan, Q. H. Ansari, J.-C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal., 70 (2009), 1830–1840.

• [8] L.-C. Ceng, H.-K. Xu, J.-C. Yao, Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators, Numer. Funct. Anal. Optim., 29 (2008), 324–345.

• [9] H.-H. Cui, M.-L. Su, On sufficient conditions ensuring the norm convergence of an iterative sequence to zeros of accretive operators, Appl. Math. Comput., 258 (2015), 67–71.

• [10] P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912.

• [11] Y.-L. Song, L.-C. Ceng, A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces, J. Global Optim., 57 (2013), 1327–1348.

• [12] W. Takahashi, Proximal point algorithms and four resolvents of nonlinear operators of monotone type in Banach spaces, Taiwanese J. Math., 12 (2008), 1883–1910.

• [13] S.-H. Wang, P. Zhang, Some results on an infinite family of accretive operators in a reflexive Banach space, Fixed Point Theory Appl., 2015 (2015 ), 11 pages.

• [14] L. Wei, R. P. Agarwal, Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems, Fixed Point Theory Appl., 2016 (2016 ), 23 pages.

• [15] L. Wei, R. P. Agarwal, P. Y. J. Wong, New method for the existence and uniqueness of solution of nonlinear parabolic equation, Bound. Value Probl., 2015 (2015 ), 18 pages.

• [16] L.Wei, Y.-C. Ba, R. P. Agarwal, New ergodic convergence theorems for non-expansive mappings andm-accretive mappings, J. Inequal. Appl., 2016 (2016 ), 20 pages.

• [17] L. Wei, R.-L. Tan, Iterative scheme with errors for common zeros of finite accretive mappings and nonlinear elliptic system, Abstr. Appl. Anal., 2014 (2014 ), 9 pages.

• [18] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.

• [19] H.-K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl., 314 (2006), 631–643.