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Remarks on Caristi’s fixed point theorem and Kirk’s problem. (English) Zbl 1201.54036

Let (X,d) be a complete metric space. An operator T:XT is said to be a Caristi type mapping if the following condition is satisfied: η(d(x,Tx))ϕ(x)-ϕ(Tx) for all xX, where η:[0,+)(-,+) and ϕ:X(-,+).

In the present paper, the author, motivated by M. A. Khamsi [Nonlinear Anal., Theory Methods Appl. 71, No.  1–2, A, 227–231 (2009; Zbl 1175.54056)], proves the following theorems.

Theorem 1. Suppose that η:[0,+)[0,+) with η(0)=0, ϕ:X(-,+) is lower-semicontinuous on X, and there exist x 0 X and two real numbers a<0, -<β<+, such that ϕ(x)ad(x,x 0 )+β. Suppose that one of the following conditions is satisfied:

(i) a0, η is nonnegative and nondecreasing on W={d(x,y):x,yX}, and there exists c>0 and ε>0 such that η(t)ct for all t{t0:η(t)ε}W;

(ii) a<0, η(t)+at is nonnegative and nondecreasing on W, and there exist c>0 and ε>0 such that η(t)+atct for all t{t0:η(t)+atε}W.

Then each Caristi type mapping T:XX has a fixed point in X.

Theorem 2. Suppose that η:[0,+)[0,+) with η(0)=0, ϕ:X(-,+) is lower-semicontinuous on X and bounded below on each bounded subset of X, and there exist x 0 X and a real number -<a<+ such that

lim inf d(x,x 0 )+ ϕ(x) d(x,x 0 )>a·

Suppose that one of the following conditions is satisfied:

(i) a0, η is nondecreasing on [0,+) and

lim inf t0 + η(t) t>0,

(ii) a<0, η(t)+at is nonnegative and nondecreasing on [0,+) and

lim inf t0 + η(t) t-a·

Then each Caristi type mapping T:XX has a fixed point in X.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
06A06Partial order