<p>This study investigates a new algorithm, termed the three-intermixed algorithm, developed from the intermixed algorithm introduced by Yao et al. (Fixed Point Theory Appl 206, 2015). A strong convergence theorem is established for finding a common element in the set of fixed points of nonexpansive mappings. To illustrate our main theorem, we provide an example in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> along with a graphical representation showing the behavior of the sequences <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{x_n\},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{y_n\},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{z_n\}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Beyond theoretical development, the applicability of the main theorem is demonstrated through several important problems, including the convex minimization problem, the split feasibility problem, and both general and new systems of variational inequality problems. In addition, a special case of the main theorem is applied to the network bandwidth allocation problem. A detailed numerical example is presented to show how the proposed iterative scheme can be used to compute a proportionally fair allocation under capacity and minimum rate constraints, thereby confirming the practical efficiency of the method.</p>

A three-intermixed algorithm for common fixed point problems with applications to convex optimization and network allocation

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发表日期：

2026年2月20日

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Kanyanee Saechou,
Atid Kangtunyakarn

摘要

This study investigates a new algorithm, termed the three-intermixed algorithm, developed from the intermixed algorithm introduced by Yao et al. (Fixed Point Theory Appl 206, 2015). A strong convergence theorem is established for finding a common element in the set of fixed points of nonexpansive mappings. To illustrate our main theorem, we provide an example in ${\mathbb {R}}^2$ $R^{2}$ along with a graphical representation showing the behavior of the sequences $\{x_n\},$ ${x_{n}},$ $\{y_n\},$ ${y_{n}},$ and $\{z_n\}.$ ${z_{n}} .$ Beyond theoretical development, the applicability of the main theorem is demonstrated through several important problems, including the convex minimization problem, the split feasibility problem, and both general and new systems of variational inequality problems. In addition, a special case of the main theorem is applied to the network bandwidth allocation problem. A detailed numerical example is presented to show how the proposed iterative scheme can be used to compute a proportionally fair allocation under capacity and minimum rate constraints, thereby confirming the practical efficiency of the method.