Note for the Tripled and Quadruple Fixed Points of the Mixed Monotone Mappings

Abstract

Abstract. In this paper, to include more generalized cases, the authors present a modified concept for the tripled and quadruple fixed point of the mixed monotone mappings. Also, they investigate the existence and uniqueness of fixed point of the ordered monotone operator with the Matkowski contractive conditions in the partial ordered metric spaces. As the direct consequences, the existence of coupled fixed point, tripled fixed point and quadruple fixed point are explored at the common frame-

Unified multi-tupled fixed point theorems involving mixed monotone property in ordered metric spaces

“…Corollary 37 (Karapinar and Berinde [63]). Corollary 36 remains true if we replace condition (vii) by the following condition: [30]). Let (X, d, ) be an ordered complete metric space and F : X 4 → X a mapping.…”

“…Selection of * for quadrupled fixed points of Karapinar and Luong [20], Wu and Liu [30] and Berzig and Samet [25] are respectively: Remark 6. To ensure the existence of * -fixed point for a mapping satisfying ι n -mixed monotone property defined on an ordered metric space, the class B n must be restricted to the subclass U ιn (i.e.…”

“…In particular for n = 4, forward cyclic and backward cyclic n-tupled fixed points reduce to quadrupled fixed points of Karapinar and Luong [20] and Wu and Liu [30] respectively. Also, for n = 3, 1-skew cyclic and n-skew cyclic n-tupled fixed points reduce to tripled fixed points of Berinde and Borcut [19] and Wu and Liu [30] respectively.…”

“…Then F has a tripled fixed point (in the sense of Wu and Liu [30]), i.e., there exist x, y, z ∈ X such that F (x, y, z) = x, F (y, z, y) = y and F (z, y, x) = z. Corollary 34 (Berzig and Samet [25]). Let (X, d, ) be an ordered complete metric space and F : X 3 → X a mapping.…”

“…Then F has a quadrupled fixed point (in the sense of Wu and Liu [30] ), i.e., there exist x, y, z, w ∈ X such that [25]). Let (X, d, ) be an ordered complete metric space and F : X 4 → X a mapping.…”