Find in Library

Suppose each of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is a maximal monotone, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mspace width="0.166667em"></mspace><mi>B</mi></mrow></semantics></math></inline-formula> is firmly nonexpansive with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In this paper, we have two purposes: the first is finding the zeros of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>A</mi><mi>j</mi></msub><mo>+</mo><mi>B</mi></mrow></semantics></math></inline-formula>, and the second is finding the zeros of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>A</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula>. To address the first problem, we produce fixed-point equations on the original Hilbert space as well as on the product space and find that these equations associate with crucial operators which are called generalized forward–backward splitting operators. To tackle the second problem, we point out that it can be reduced to a special instance of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> by defining new operators on the product space. Iterative schemes are given, which produce convergent sequences and these sequences ultimately lead to solutions for the last two problems.