Around averaged mappings

<jats:p>This paper is intended for a general mathematical audience. The examples show how the study of existence of fixed points of averaged mappings <jats:inline-formula><jats:alternatives><jats:tex-math>$$T_{\lambda }= (1-\lambda )I+ \lambda T$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$0<\lambda <1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and <jats:italic>I</jats:italic> is the identity operator, can help in the study of existence of fixed points of mappings <jats:italic>T</jats:italic>.</jats:p>

[1]  Simeon Reich,et al.  Averaged mappings in the Hilbert ball , 1985 .

[2]  V. Berinde,et al.  Approximating fixed points of enriched contractions in Banach spaces , 2019, Journal of Fixed Point Theory and Applications.

[3]  W. A. Kirk.  Krasnoselskii's iteration process in hyperbolic space , 1982 .

[4]  J. Górnicki.  On some mappings with a unique fixed point , 2019, Journal of Fixed Point Theory and Applications.

[5]  F. Browder,et al.  The solution by iteration of nonlinear functional equations in Banach spaces , 1966 .

[6]  J. Górnicki.  Remarks on asymptotic regularity and fixed points , 2019, Journal of Fixed Point Theory and Applications.

[7]  S. Ishikawa.  Fixed points and iteration of a nonexpansive mapping in a Banach space , 1976 .

[8]  J. S. W. Wong,et al.  On nonlinear contractions , 1969 .

[9]  V. Berinde.  Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces , 2019, Carpathian Journal of Mathematics.

[10]  A note on the fixed point theorem of Górnicki , 2019, Journal of Fixed Point Theory and Applications.

[11]  Vasile Berinde,et al.  Kannan's fixed point approximation for solving split feasibility and variational inequality problems , 2021, J. Comput. Appl. Math..

[12]  V. Berinde.  Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition , 2020 .

[13]  L. V. Nguyen.  On fixed points of asymptotically regular mappings , 2020, Rendiconti del Circolo Matematico di Palermo Series 2.

[14]  Helmut Schaefer,et al.  Über die Methode sukzessiver Approximationen. , 1957 .

[15]  V. Berinde,et al.  Approximating fixed points of enriched Chatterjea contractions by Krasnoselskij iterative method in Banach spaces , 2019, 1909.03494.

[16]  M. Edelstein,et al.  A Remark on a Theorem of M. A. Krasnoselski , 1966 .

[17]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[18]  Itai Shafrir,et al.  The asymptotic behavior of firmly nonexpansive mappings , 1987 .