Results in Mathematics | 2021
On $$\\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces
Abstract
<jats:p>We introduce the class of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive and quasi <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive operators on <jats:italic>r</jats:italic>-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive operators coincide with so-called <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-averaged operators. For our more general setting, we show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-averaged operators form a subset of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive. Moreover, we will see that quasi <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for <jats:italic>r</jats:italic>-uniformly convex Banach spaces that the weak cluster points of the iterates <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_{n+1}:=Tx_{n}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo>:</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mi>T</mml:mi>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> belong to the fixed point set <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{Fix}\\,}}T$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mrow>\n <mml:mspace />\n <mml:mi>Fix</mml:mi>\n <mml:mspace />\n </mml:mrow>\n <mml:mi>T</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> whenever the operator <jats:italic>T</jats:italic> is nonexpansive and quasi <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>α</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{Fix}\\,}}T$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mrow>\n <mml:mspace />\n <mml:mi>Fix</mml:mi>\n <mml:mspace />\n </mml:mrow>\n <mml:mi>T</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. Further, the projections <jats:inline-formula><jats:alternatives><jats:tex-math>$$P_{{{\\,\\mathrm{Fix}\\,}}T}x_n$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>P</mml:mi>\n <mml:mrow>\n <mml:mrow>\n <mml:mspace />\n <mml:mi>Fix</mml:mi>\n <mml:mspace />\n </mml:mrow>\n <mml:mi>T</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:msub>\n <mml:mi>x</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_p$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$p \\in (1,\\infty ) \\backslash \\{2\\}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>p</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>∞</mml:mi>\n <mml:mo>)</mml:mo>\n <mml:mo>\\</mml:mo>\n <mml:mo>{</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>}</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> spaces on probability measure spaces.</jats:p>