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A Coupled Fixed Point Problem Under a Finite Number of Equality Constraints

  • Praveen Agarwal
  • Mohamed Jleli
  • Bessem Samet
Chapter

Abstract

Let \((E,\Vert \cdot \Vert )\) be a Banach space with a cone P. Let \(F,\varphi _i: E\times E\rightarrow E\) (\(i=1,2,\ldots ,r\)) be a finite number of mappings. In this chapter, we provide sufficient conditions for the existence and uniqueness of solutions to the problem: Find \((x,y)\in E\times E\) such that
$$\begin{aligned} \left\{ \begin{array}{lll} F(x,y)&{}=&{}x,\\ F(y,x)&{}=&{}y,\\ \varphi _i(x,y)&{}=&{}0_E,\,\, i=1,2,\ldots ,r, \end{array} \right. \end{aligned}$$
where \(0_E\) is the zero vector of E. The main reference for this chapter is the paper [4].

References

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    Ait Mansour, A., Malivert, C., Thera, M.: Semicontinuity of vector-valued mappings. Optimization 56(1–2), 241–252 (2007)MathSciNetCrossRefGoogle Scholar
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    Guo, D., Je Cho, Y., Zhu, J.: Partial Ordering Methods in Nonlinear Problems. Nova Publishers, New York (2004)Google Scholar
  3. 3.
    Jleli, M., Samet, B.: A fixed point problem under two constraint inequalities. Fixed Point Theory Appl. 2016, 18 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jleli, M., Samet, B.: A Coupled fixed point problem under a finite number of equality constraints in a Banach space partially ordered by a cone. Fixed Point Theory (in Press)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAnand International College of EngineeringJaipurIndia
  2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia

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