On the Maximality of the Sum of Two Maximal Monotone Operators

  • Yuqing Chen
  • Yeol Je ChoEmail author
  • Themistocles M. Rassias
Part of the Springer Optimization and Its Applications book series (SOIA, volume 135)


Let E be a real reflexive Banach space, E be the dual space of E and \(T: D(T)\subseteq E\to 2^{E^{*}}\), \(S:D(S)\subseteq E\to 2^{E^*}\) be two maximal monotone operators such that D(T) ∩ D(S) ≠ ∅. Assume that there exist x0 ∈ E, r > 0, λ0 > 0 such that inffTx(f, x − x0) is lower bounded on each bounded subset of D(T) and, if, for each y ∈ B(x0, r), g ∈ E, x n  ∈ D(T) and λ n  ∈ (0, λ0) with \(g\in Tx_n+ S_{\lambda _n}x_n+Jx_n\) for each n = 1, 2, ⋯, \(\{R_{\lambda _n}^Sx_n\}_{=1}^{\infty }\) is bounded, then we have
$$\displaystyle \inf _{n\geq 1}(S_{\lambda _n}x_n,R_{\lambda _n}^Sx_n-y)>-\infty , $$
where \(R_{\lambda }^S\) is the Yosida resolvent of S, then T + S is maximal monotone. Also, we construct a degree theory for the sum of two maximal monotone operators, where the sum may not be maximal monotone, and the degree theory is also applied to study the operator equation 0 ∈ (T + S)x. Finally, we give some applications of the main results to nonlinear partial differential equations.



The authors thank the referee for helpful comments and suggestions on this manuscript. Also, we wish to express our thanks to Professor Mihai Turinici for reading the paper and providing very helpful comments.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Yuqing Chen
    • 1
  • Yeol Je Cho
    • 2
    • 3
    Email author
  • Themistocles M. Rassias
    • 4
  1. 1.Guangdong University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of Mathematics Education and RINSGyeongsang National UniversityJinjuKorea
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan
  4. 4.Department of MathematicsNational Technical University of AthensAthensGreece

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