# Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (a)
- convex if for any$\lambda \in \left[0,1\right]$and points $u,v\in E$,$$\psi \left(\lambda u+\left(1-\lambda \right)v\right)\le \lambda \psi \left(u\right)+\left(1-\lambda \right)\psi \left(v\right)$$
- (b)
- strictly convex if the inequality in (a) is strict
- (c)
- totally convex if there exists a function$\phi :\left[0,+\infty \right)\u27f6\left[0,+\infty \right)$varnishing only at zero such that:$\psi \left(u\right)-\psi \left(v\right)\ge \langle {v}^{*},u-v\rangle +\phi (\parallel u-v\parallel ),{v}^{*}\in \partial \psi \left(v\right)$

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

- (a)
- $\partial \psi $is both locally bounded and single-valued on its domain
- (b)
- $\psi $is strictly convex on every subset of$dom\left(\psi \right)$and${\left(\partial \psi \right)}^{-1}$is locally bounded in its domain.

**Definition**

**4.**

- (a)
- the generalized Bregman distance with respect to$\psi $and a subgradient$p$is defined as:$${B}_{\psi}^{p}\left(u,v\right)=\psi \left(u\right)-\psi \left(v\right)-\langle p,u-v\rangle ,p\in \partial \psi \left(v\right)$$$${B}_{\psi}^{p}\left(u,w\right)={B}_{\psi}^{q}\left(u,v\right)+{B}_{\psi}^{p}\left(v,w\right)+\langle u-v,q-p\rangle ,p\in \partial \psi \left(w\right),q\in \partial \psi \left(v\right)$$
- (b)
- the Bregman projection relative to$\psi $of a point$uu\in dom\left(\psi \right)$onto a nonempty, closed, and convex subset$K$, is defined as the unique vector$\prod}_{\psi}^{K}\left(u\right)$satisfying ${B}_{\psi}\left({\displaystyle \prod}_{\psi}^{K}\left(u\right),u\right)=inf\left\{{B}_{\psi}\left(v,u\right):v\in K\right\}$. If$\psi $is totally convex and Gateaux differentiable, then$\prod}_{\psi}^{K}\left(u\right)$is the unique solution contained in$K$of the following variational inequalities (see [21]);

- (i)
- $\langle \nabla \psi \left(u\right)-\nabla \psi \left(z\right),z-w\rangle \ge 0\forall w\in K$
- (ii)
- ${B}_{\psi}\left(w,z\right)+{B}_{\psi}\left(z,u\right)\le {B}_{\psi}\left(w,u\right)\forall w\in K$

**Definition**

**5.**

- (a)
- monotone, if$\langle m-n,u-v\rangle \ge 0\forall u,v\in dom\left(M\right),m\in M\left(u\right),n\in M\left(v\right)$
- (b)
- maximal monotone if$M$is monotone, and the graph of$M$is not contained in the graph of any other monotone map
- (c)
- $\phi $-strongly monotone if there exists a non-negative function$\phi $which varnishes only at zero such that:$$m-n,u-v\ge \phi \left(u-v\right),\forall u,v\in dom\left(M\right),m\in M\left(u\right),n\in M\left(v\right)$$Note that $dom\left(M\right):=\left\{z\in E:M\left(z\right)isnonempty\right\}$.

**Definition**

**6.**

**Definition**

**7.**

**Lemma**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

- (i)
- The function$\psi $is locally bounded from above on$int\left(dom\left(\psi \right)\right)$
- (ii)
- The function$\psi $is locally bounded on$int\left(dom\left(\psi \right)\right)$
- (iii)
- The function$\psi $is locally Lipschitz on$int\left(dom\left(\psi \right)\right)$
- (iv)
- The function$\psi $is continuous on$int\left(dom\left(\psi \right)\right)$.

**Lemma**

**2.**

## 3. Main Results

#### 3.1. Split Common Coincidence Point Problem (SCCPP)

**Definition**

**8.**

**Remark**

**2.**

**Definition**

**9.**

#### 3.2. Optimization by Generalized Coincidence Point Problem

**Definition**

**10.**

**Lemma**

**3.**

- (i)
- $T$is monotone.
- (ii)
- $S-T$is$S$-pseudocontractive.

**Proof.**

#### 3.3. Examples of S-Pseudocontraction

- (1)
- Let $E=\mathbb{H}$, a real Hilbert space, $S=\mathbb{I}$, the identity map on $\mathbb{H}$. Then, any pseudocontraction on $\mathbb{H}$ is $S$-pseudocontraction.
- (2)
- Every $J$-pseudocontraction is $S$-psuedocontraction with $S=J+M$, where $M$ is any single-valued monotone map.
- (3)
- Let $E$ be a smooth real Banach space, fix $p>1$. Define ${T}_{p}:E\u27f6{E}^{*}$ and ${S}_{p}:E\u27f6{E}^{*}$ by:$${T}_{p}\left(u\right)=pJ\left(u\right)\phantom{\rule{0ex}{0ex}}{S}_{p}\left(u\right)=\left(p+\mu \right)J\left(u\right)+\parallel u{\parallel}^{p-2}J\left(u\right),\mu 0.$$

**Lemma**

**4.**

**Proof.**

#### 3.4. Approximation of Coincidence Points

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

- ${B}_{{\psi}_{1}}\left(z,{w}_{n+1}\right)\le {B}_{{\psi}_{1}}\left(z,{w}_{n}\right)-{v}_{n}-\parallel {w}_{n}{\parallel}^{2}$
- $\underset{n\to \infty}{\mathrm{lim}}{B}_{{\psi}_{1}}\left(z,{w}_{n}\right)$ exists
- $\left\{{w}_{n}\right\}$ is bounded
- $\underset{n\to \infty}{\mathrm{lim}}\parallel {v}_{n}-{w}_{n}\parallel =0$ and $\left\{{v}_{n}\right\}$ is also bounded.

- $\left\{{g}_{1}({v}_{n})\right\}$ and $\left\{{g}_{2}(A{w}_{n})\right\}$ are bounded
- $\underset{n\to \infty}{\mathrm{lim}}\parallel {g}_{1}\left({v}_{n}\right)-{v}_{n}\parallel =\underset{n\to \infty}{\mathrm{lim}}\parallel {g}_{2}\left(A{w}_{n}\right)-A{w}_{n}\parallel =0$

- (i)
- $W\left({w}_{n}\right)\subseteq sol\left(SCCPP\right)$
- (ii)
- $W\left({w}_{n}\right)$ is a singleton

**Corollary**

**2.**

**Proof.**

- $\underset{n\to \infty}{\mathrm{lim}}{B}_{{\psi}_{1}}\left(z,{w}_{n}\right)$ exists
- $\left\{{w}_{n}\right\}$ is bounded
- $\underset{n\to \infty}{\mathrm{lim}}\parallel {y}_{n}-{w}_{n}\parallel =0$ and $\left\{{y}_{n}\right\}$ is also bounded.
- $\underset{n\to \infty}{\mathrm{lim}}{B}_{{\psi}_{1}}\left({g}_{1}\left({y}_{n}\right),{y}_{n}\right)=\underset{n\to \infty}{\mathrm{lim}}{B}_{{\psi}_{2}}\left({g}_{2}\left(A{w}_{n}\right),A{w}_{n}\right)=0,\mathrm{thus}\left\{{g}_{1}\left({y}_{n}\right)\right\}\mathrm{and}\left\{{g}_{2}\left(A{w}_{n}\right)\right\}$$\mathrm{are}\mathrm{bounded}.\mathrm{Moreover},\underset{n\to \infty}{\mathrm{lim}}\parallel {g}_{1}\left({y}_{n}\right)-{y}_{n}\parallel =\underset{n\to \infty}{\mathrm{lim}}\parallel {g}_{2}\left(A{w}_{n}\right)-A{w}_{n}\parallel =0$

## 4. Application to Inverse Planning Optimization

#### 4.1. An Inverse Planning Optimization Problem (IPOP)

#### 4.2. SCCPP Reformulation of IPOP

- (a)
- $f$ is convex
- (b)
- $f$ is “partly” differentiable, that is, $f$ can be written as a sum of two convex functions ${f}_{1}$ and ${f}_{2}$ such that ${f}_{1}$ or ${f}_{2}$ is differentiable. Without loss of generality, we shall always assume ${f}_{2}$ to be differentiable.
- (c)
- There exists a differentiable convex function $h$ and a positive constant $\theta $, such that $\theta h-{f}_{2}$ is Legendre, totally convex, cofinite, and has Lipschitz continuous gradient

- (i)
- ${T}_{1}$ is strongly monotone, cofinite, sequentially continuous, and ${T}_{1}$ inverse exists
- (ii)
- ${T}_{1}$ is ${S}_{1}$-pseudocontractive, and ${T}_{2}$ is ${S}_{2}$-pseudocontractive.
- (iii)
- ${T}_{2}$ is Lipschitz continuous and $\phi $-strongly monotone
- (iv)
- the mappings ${g}_{1}:={S}_{1}^{-1}\circ {T}_{1}$ and ${g}_{2}:={S}_{2}^{-1}\circ {T}_{2}$ are well defined and single-valued [44]. Also ${\mathbb{I}}_{{E}_{1}}-{g}_{1}$ and ${\mathbb{I}}_{{E}_{2}}-{g}_{2}$ are demi-closed at zero.
- (v)
- a solution of the constrained SCCPP associated with ${T}_{1},{S}_{1}$ and ${T}_{2},{S}_{2}$ solves the constrained IPOP (see Lemma 3 and Remark 1).(i)–(iv) verifies all the assumptions of Corollary 2; hence, by (v), $\left(5\right)$ converges to a solution of the IPOP.

#### 4.3. Common Biological and/or Physical Objective Criteria in RTP

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

#### 4.4. Insights on Algorithm Implementation

Forms of${g}_{2}$. |

For Example 1, with $\kappa =1,{q}_{1}=4$ and ${q}_{2}=2$, ${S}_{2}^{-1}\left(x\right)=u\in {\mathbb{R}}^{N}$ such that: $ifj\le \mathbb{T}:$ ${u}_{j}={D}_{p}^{j}+{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}+\sqrt{{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}\right)}^{2}+{\left(\frac{\lambda}{12{a}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}+{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}-\sqrt{{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}\right)}^{2}+{\left(\frac{\lambda}{12{a}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}$ $elseif{x}_{j}\lambda {D}_{max}^{j}:$ ${u}_{j}=\frac{2{b}_{j}{D}_{max}^{j}+{x}_{j}}{2{b}_{j}+\lambda}$ $else$: ${u}_{j}=\frac{{x}_{j}}{\lambda}$ Hence, ${g}_{2}\left(x\right)={S}_{2}^{-1}\left(\lambda x\right)$ For Example 4, with $=1,{q}_{1}=4$, $a=2$, and ${\nabla}^{2}\lambda >1$ ${S}_{2}^{-1}\left(x\right)=u\in {\mathbb{R}}^{N}$ such that: $ifj\le \mathbb{T}:$ ${u}_{j}={D}_{p}^{j}+{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}+\sqrt{{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}\right)}^{2}+{\left(\frac{\lambda}{12{a}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}+{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}-\sqrt{{\left(\frac{{x}_{j}-\lambda {a}_{j}{D}_{p}^{j}}{8{a}_{j}}\right)}^{2}+{\left(\frac{\lambda}{12{a}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}$ else ${u}_{j}={\varphi}_{j}=-\frac{\alpha {n}_{f}}{2\beta}+{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}{x}_{j}+\frac{{\alpha}^{3}{\nabla}^{2}{n}_{f}\lambda}{2\beta}\right)}{8{\beta}^{2}{v}_{j}}+\sqrt{{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}{x}_{j}+\frac{{\alpha}^{3}{\nabla}^{2}{n}_{f}\lambda}{2\beta}\right)}{8{\beta}^{2}{v}_{j}}\right)}^{2}+{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}\lambda -{\alpha}^{2}{v}_{j}\right)}{12{\beta}^{2}{v}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}{x}_{j}+\frac{{\alpha}^{3}{\nabla}^{2}{n}_{f}\lambda}{2\beta}\right)}{8{\beta}^{2}{v}_{j}}-\sqrt{{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}{x}_{j}+\frac{{\alpha}^{3}{\nabla}^{2}{n}_{f}\lambda}{2\beta}\right)}{8{\beta}^{2}{v}_{j}}\right)}^{2}+{\left(\frac{{n}_{f}^{2}\left({\alpha}^{2}{\nabla}^{2}\lambda -{\alpha}^{2}{v}_{j}\right)}{12{\beta}^{2}{v}_{j}}\right)}^{3}}\right)}^{\frac{1}{3}}$ Hence, ${g}_{2}\left(x\right)={S}_{2}^{-1}\left(\lambda x\right)$ For Example 2, with $\kappa =1,a=2$, and ${\alpha}^{2}{n}_{f}>2\beta $ ${S}_{2}^{-1}\left(x\right)\approx u\in {\mathbb{R}}^{N}$ such that: $ifj\le \mathbb{T}:$ $\{\begin{array}{c}{u}_{j}={\varnothing}_{j},\mathrm{if}j\le T\\ {u}_{j}={\varphi}_{j},\mathrm{otherwise}\end{array}$ where ${\varnothing}_{j}$ satisfies ${a}_{j}{\varnothing}_{j}^{3}+{b}_{j}{\varnothing}_{j}^{2}+{c}_{j}{\varnothing}_{j}+{d}_{j}=0$ with ${d}_{j}=-\left({N}_{0}\alpha {v}_{j}+{x}_{j}\right)$ ${a}_{j}=\frac{-{N}_{0}\beta \left({\alpha}^{2}-\frac{2\beta}{{n}_{f}}\right){v}_{j}}{{n}_{f}},{b}_{j}={N}_{0}\alpha \left(\frac{3\beta}{{n}_{f}}-\frac{{\alpha}^{2}}{2}\right){v}_{j},{c}_{j}={N}_{0}{v}_{j}\left({\alpha}^{2}-\frac{2\beta}{{n}_{f}}\right)+\lambda $ Hence, ${g}_{2}\left(x\right)\approx {S}_{2}^{-1}\left(\lambda x\right)$ |

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Palta, J.R.; Mackie, T.R. Intensity-Modulated Radiation Therapy: The State of the Art; Medical Physics Monograph No. 29 American Association of Physicists in Medicine; Medical Physics Publishing: Madison WI, USA, 2003. [Google Scholar]
- Wu, Q.W.; Mohan, R.; Niemierko, A.; Schmidt-Ullrich, R. Optimization of Intensity-Modulated Radiotherapy plans based on the equivalent uniform dose. Int. J. Radiat. Oncol Biol. Phys.
**2002**, 52, 224–235. [Google Scholar] [CrossRef] - Stavrev, P.; Hristov, D.; Warkentin, B.; Sham, E.; Stavreva, N.; Fallone, B.G. Inverse treatment planning by physically constrained minimization of a biological objective function. Med. Phys.
**2003**, 30, 2948–2958. [Google Scholar] [CrossRef] [PubMed] - Xia, P.; Yu, N.; Xing, L.; Sun, X.; Verhey, L.J. Investigation of using power law function as a cost function in inverse planning optimization. Med. Phys.
**2005**, 32, 920–927. [Google Scholar] [CrossRef] [PubMed] - Guo, C.; Zang, P.; Zhang, L.; Gui, Z.; Shu, H. Application of optimization model with piecewise penalty to intensity-modulated radiation therapy. Future Gener. Comput. Syst.
**2018**, 81, 280–290. [Google Scholar] [CrossRef] - Dirscherl, T.; Alvarez-Moret, J.; Bogner, L. Advantage of biological over physical optimization of prostate cancer? Z. Med. Phys.
**2011**, 21, 228–235. [Google Scholar] [CrossRef] - Olafsson, A.; Jeraj, R.; Wright, S.J. Optimization of intensity-modulated radiation therapy with biological objectives. Phys. Med. Biol.
**2005**, 50, 5257–5379. [Google Scholar] [CrossRef] - Hartmann, M.; Bogner, L. Investigation of intensity-modulated radiotherapy optimization with gEUD-based objectives by means of simulated annealing. Med. Phys.
**2008**, 35, 2041–2049. [Google Scholar] [CrossRef] - Romeijn, H.E.; Dempsey, J.F.; Li, J.G. A unifying framework for multi-criteria fluence map optimization models. Phys. Med. Biol.
**2004**, 49, 1991–2013. [Google Scholar] [CrossRef] - Uzan, J.; Nahum, A.E. Radiobiologically guided optimization of the prescription dose and fractionation scheme in radiotherapy using BioSuite. Br. J. Radiol.
**2012**, 85, 1279–1286. [Google Scholar] [CrossRef] [Green Version] - Feng, Z.; Tao, C.; Zhu, J.; Chen, J.; Yu, G.; Qin, S.; Yin, Y.; Li, D. An integrated strategy of biological and physical constraints in biological optimization for cervical cancer. Radiat. Oncol.
**2017**, 12, 64. [Google Scholar] [CrossRef] [Green Version] - Li, X.A.; Alber, M.; Deasy, J.O.; Jackson, A.; Jee, K.K.; Marks, L.B.; Martel, M.K.; Mayo, C.; Moiseenko, V.; Nahum, A.E.; et al. The use and QA of biologically related models for treatment planning: Short report of the TG-166 of the therapy physics committee of the AAPM. Med. Phys.
**2002**, 39, 1386–1409. [Google Scholar] [CrossRef] [Green Version] - Fogliata, A.; Thompson, S.; Stravato, A.; Tomatis, S.; Scorsetti, M.; Cozzi, L. On the gEUD biological optimization objective for organs at risk in photon optimizer of Eclipse treatment planning system. J. Appl. Clin. Med. Phys.
**2018**, 19, 106–114. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kan, M.W.K.; Leung, L.H.T.; Yu, P.K.N. The Use of Biologically Related Model (Eclipse) for the Intensity-Modulated Radiation Therapy Planning of Nasopharyngeal Carcinomas. PLoS ONE
**2014**, 9, e112229. [Google Scholar] [CrossRef] - Senthilkumar, K.; Maria Das, K.J. Comparison of biological-based and dose volume-based intensity-modulated radiotherapy plans generated using the same treatment planning system. J. Cancer Res. Ther.
**2019**, 15, S33–S38. [Google Scholar] [CrossRef] - Sukhikh, E.; Sheino, I.; Vertinsky, A. Biological-based and physical-based optimization for biological evaluation of prostate patients plans. AIP Conf. Proc.
**2017**, 1882, 20074. [Google Scholar] [CrossRef] - Zhu, J.; Simon, A.; Haigron, P.; Lafond, C.; Acosta, O.; Shu, H.; Castelli, J.; Li, B.; De Crevoisier, R. The benefit of using bladder sub-volume equivalent uniform dose constraints in prostate intensity-modulated radiotherapy planning. Onco Targets Ther.
**2016**, 9, 7537–7544. [Google Scholar] [CrossRef] [Green Version] - Censor, Y.; Elfving, T.; Kopf, N.; Bortfeld, T. The multiple-set split feasibility problem and its application for inverse problems. Inverse Probl.
**2005**, 21, 2071–2084. [Google Scholar] [CrossRef] [Green Version] - Shepard, D.M.; Ferris, M.C.; Olivera, G.H.; Mackie, T.R. Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev.
**1999**, 41, 721–744. [Google Scholar] [CrossRef] [Green Version] - Kiwiel, K.C. Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim.
**1997**, 35, 1142–1168. [Google Scholar] [CrossRef] [Green Version] - Butnariu, D.; Resmerita, E. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal.
**2006**, 2006, 084919. [Google Scholar] [CrossRef] [Green Version] - Roldán-López-de-Hierro, A.F.; Karapinar, E.; Roldán-López-de-Hierro, C.; Martínez-Moreno, J. Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math.
**2015**, 275, 345–355. [Google Scholar] [CrossRef] - Rockafellar, R.T. On the maximal monotonicity of subdifferential mappings. Pac. J. Math.
**1970**, 33, 209–216. [Google Scholar] - Butnariu, D.; Iusem, A.N. Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Computation, 1st ed.; Springer: Dordrecht, The Netherlands, 2000; pp. 1–64. [Google Scholar]
- Reem, D.; Reich, S.; De Pierro, A. Re-examination of Bregman functions and new properties of their divergence. Optimization
**2019**, 68, 279–348. [Google Scholar] [CrossRef] - Chidume, C.E.; Idu, K.O. Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems. Fixed Point Theory Appl.
**2016**, 2016, 97. [Google Scholar] [CrossRef] [Green Version] - Tang, Y.; Bao, Z. New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces. Numer. Algor.
**2019**, 81, 853–878. [Google Scholar] [CrossRef] [Green Version] - Saddeek, A.M.; Hussain, N. Duality fixed points for multivalued generalized K
_{1}J-pseudocontractive Lipschitzian mappings. Acta Math. Univ. Comen.**2019**, 88, 101–112. [Google Scholar] - Chidume, C.E.; Kumam, P.; Adamu, A. A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications. Fixed Point Theory Appl.
**2020**, 2020, 12. [Google Scholar] [CrossRef] - Censor, Y.; Segal, A. The split common fixed point problem for directed operators. J. Convex Anal.
**2009**, 16, 587–600. [Google Scholar] [CrossRef] - Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal.
**2011**, 74, 4083–4087. [Google Scholar] [CrossRef] - Cho, S.Y.; Qin, X.; Kang, S.M. Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim.
**2013**, 57, 1429–1446. [Google Scholar] [CrossRef] - Reich, S.; Tuyen, T.M. Two projection Algorithms for solving the split common fixed point problem. J. Optim. Theory Appl.
**2020**, 186, 148–168. [Google Scholar] [CrossRef] - Kraikaew, R.; Saejung, S. On split common fixed point problems. J. Math. Anal. Appl.
**2014**, 415, 513–524. [Google Scholar] [CrossRef] - Takahashi, W. The split common fixed point problem for generalized demimetric mappings in two Banach spaces. Optimization
**2019**, 68, 411–427. [Google Scholar] [CrossRef] - Moudafi, A. Alternating CQ-algorithm for convex feasibility and split fixed point problems. J. Nonlinear Convex Anal.
**2014**, 15, 809–818. [Google Scholar] - Censor, Y.; Gibali, A.; Reich, S. Algorithms for the Split Variational Inequality Problem. Numer. Algor.
**2012**, 59, 301–323. [Google Scholar] [CrossRef] - Jirakitpuwapat, W.; Kumam, P.; Cho, Y.J.; Sitthithakerngkiet, K. A General Algorithm for the Split Common Fixed Point Problem with Its Applications to Signal Processing. Mathematics
**2019**, 7, 226. [Google Scholar] [CrossRef] [Green Version] - Moudafi, A. A three-operator splitting algorithm for null-point problems. Fixed Point Theory
**2020**, 21, 685–692. [Google Scholar] [CrossRef] - Wega, G.B.; Zegeye, H. A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces. J. Fixed Point Theory Appl.
**2020**, 22, 57. [Google Scholar] [CrossRef] - Rouhani, B.D.; Mohebbi, V. Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space. J. Optim. Theory Appl.
**2020**, 186, 34–147. [Google Scholar] [CrossRef] - Chidume, C.E.; Adamu, A.; Nnakwe, M.O. Strong convergence of an inertial algorithm for maximal monotone inclusions with applications. Fixed Point Theory Appl.
**2020**, 13. [Google Scholar] [CrossRef] - Hoffmann, A.L.; Siem, A.Y.D.; den Hertog, D.; Kaanders, J.H.A.M.; Huizenga, H. Convex reformulation of biologically-based multi-criteria intensity-modulated radiation therapy optimization including fractionation effects. Phys. Med. Biol.
**2008**, 53, 6345–6362. [Google Scholar] [CrossRef] [Green Version] - Bauschke, H.H.; Wang, X.; Yao, L. General resolvents for monotone operators; characterization and extension. In Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems; Censor, Y., Jiang, M., Wang, G., Eds.; Medical Physics Publishing: Madison, WI, USA, 2010; pp. 57–74. [Google Scholar]
- Wang, J.Z.; Guerrero, M.; Li, X.A. How low is the α/β ratio for prostate cancer? Int. J. Radiat. Oncol. Biol. Phys.
**2003**, 55, 194–203. [Google Scholar] [CrossRef] - Qi, X.S.; Schultz, C.J.; Li, X.A. An estimation of radiobiologic parameters from clinical outcomes for radiation treatment planning of brain tumor. Int. J. Radiat. Oncol. Biol. Phys.
**2006**, 64, 1570–1580. [Google Scholar] [CrossRef] - Qi, X.S.; White, J.; Li, X.A. Is α/β for breast cancer really low? Radiother. Oncol.
**2011**, 100, 282–288. [Google Scholar] [CrossRef] [PubMed] - Tai, A.; Erickson, B.; Khater, K.A.; Li, X.A. Estimate of radiobiologic parameters from clinical data for biologically based treatment planning for liver irradiation. Int. J. Radiat. Oncol. Biol. Phys.
**2008**, 70, 900–907. [Google Scholar] [CrossRef] - Van Leeuwen, C.M.; Oei, A.L.; Crezee, J.; Bel, A.; Franken, N.A.P.; Stalpers, L.J.A.; Kok, H.P. The alfa and beta of tumours: A review of parameters of linear quadratic model derived from clinical radiotherapy studies. Radiat. Oncol.
**2018**, 13, 96. [Google Scholar] [CrossRef] - Alber, M.; Nusslin, F. A representation of an NTCP function for local complication mechanisms. Phys. Med. Biol.
**2001**, 46, 439–447. [Google Scholar] [CrossRef] - Bauschke, H.H.; Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed.; Springer: Cham, Switzerland, 2017; pp. 393–446. [Google Scholar]
- Aragon Artacho, F.J.; Campoy, R. Computing the resolvents of the sum of maximally monotone operators with the averaged alternating modified reflections algorithm. J. Optim. Theory Appl.
**2019**, 181, 709–726. [Google Scholar] [CrossRef] [Green Version] - Combettes, P.L. Iterative construction of the resolvents of a sum of maximal monotone operators. J. Convex Anal.
**2009**, 16, 727–748. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chidume, C.E.; Okereke, L.C.
Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization. *Symmetry* **2020**, *12*, 2086.
https://doi.org/10.3390/sym12122086

**AMA Style**

Chidume CE, Okereke LC.
Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization. *Symmetry*. 2020; 12(12):2086.
https://doi.org/10.3390/sym12122086

**Chicago/Turabian Style**

Chidume, Charles E., and Lois C. Okereke.
2020. "Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization" *Symmetry* 12, no. 12: 2086.
https://doi.org/10.3390/sym12122086