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bstract
This study is concerned with the existence of fixed points of Caristi-type mappings motivated by a problem stated by Kirk. First, several existence theorems of maximal and minimal points are established. By using them, some generalized Caristi’s fixed point theorems are proved, which improve Caristi’s fixed point theorem and the results in the studies of Jachymski, Feng and Liu, Khamsi, and Li.
MSC 2010: 06A06; 47H10.
Keywords: maximal and minimal point, Caristi’s fixed point theorem, Caristi-type mapping, partial order
* Correspondence: lzl771218@sina. com
Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China
1 Introduction
In the past decades, Caristi’s fixed point theorem has been generalized and extended in several directions, and the proofs given for Caristi’s result varied and used different techniques, we refer the readers to [1-15].
Recall that T : X ® X is said to be a Caristi-type mapping [14] provided that there exists a function h : [0, +∞) ® [0, +∞) and a function 0 : X ® (-∞, +∞) such that
η(d(x, Tx)) ≤ φ(x) − φ(Tx), ∀ x ∈ X,
where (X, d) is a complete metric space. Let ≼ be a relationship defined on X as fol-
lows
x!y⇔η(d(x,y))≤φ(x)−φ(y), ∀x,y∈X. (1)
Clearly, x ≼ Tx for each x Î X provided that T is a Caristi-type mapping. Therefore, the existence of fixed points of Caristi-type mappings is equivalent to the existence of maximal point of (X, ≼). Assume that h is a continuous, nondecreasing, and subaddi- tive function with h-1({0}) = {0}, then the relationship defined by (1) is a partial order on X. Feng and Liu [12] proved each Caristi-type mapping has a fixed point by investi- gating the existence of maximal point of (X, ≼) provided that 0 is lower semicontinu- ous and bounded below. The additivity of h appearing in [12] guarantees that the relationship ≼ defined by (1) is a partial order on X. However, if h is not subadditive, then the relationship ≼ defined by (1) may not be a partial order on X, and conse- quently the method used there becomes invalid. Recently, Khamsi [13] removed the additivity of h by introducing a partial order on Q as follows
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bstract
This study is concerned with the existence of fixed points of Caristi-type mappings motivated by a problem stated by Kirk. First, several existence theorems of maximal and minimal points are established. By using them, some generalized Caristi’s fixed point theorems are proved, which improve Caristi’s fixed point theorem and the results in the studies of Jachymski, Feng and Liu, Khamsi, and Li.
MSC 2010: 06A06; 47H10.
Keywords: maximal and minimal point, Caristi’s fixed point theorem, Caristi-type mapping, partial order
* Correspondence: lzl771218@sina. com
Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China
1 Introduction
In the past decades, Caristi’s fixed point theorem has been generalized and extended in several directions, and the proofs given for Caristi’s result varied and used different techniques, we refer the readers to [1-15].
Recall that T : X ® X is said to be a Caristi-type mapping [14] provided that there exists a function h : [0, +∞) ® [0, +∞) and a function 0 : X ® (-∞, +∞) such that
η(d(x, Tx)) ≤ φ(x) − φ(Tx), ∀ x ∈ X,
where (X, d) is a complete metric space. Let ≼ be a relationship defined on X as fol-
lows
x!y⇔η(d(x,y))≤φ(x)−φ(y), ∀x,y∈X. (1)
Clearly, x ≼ Tx for each x Î X provided that T is a Caristi-type mapping. Therefore, the existence of fixed points of Caristi-type mappings is equivalent to the existence of maximal point of (X, ≼). Assume that h is a continuous, nondecreasing, and subaddi- tive function with h-1({0}) = {0}, then the relationship defined by (1) is a partial order on X. Feng and Liu [12] proved each Caristi-type mapping has a fixed point by investi- gating the existence of maximal point of (X, ≼) provided that 0 is lower semicontinu- ous and bounded below. The additivity of h appearing in [12] guarantees that the relationship ≼ defined by (1) is a partial order on X. However, if h is not subadditive, then the relationship ≼ defined by (1) may not be a partial order on X, and conse- quently the method used there becomes invalid. Recently, Khamsi [13] removed the additivity of h by introducing a partial order on Q as follows
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