On the existence of fixed points that belong to the zero set of a certain function
Karapinar et al. Fixed Point Theory and Applications
On the existence of fixed points that belong to the zero set of a certain function
Erdal Karapinar 0 1 2 5 6
Donal O'Regan 0 1 2 4 5
Bessem Samet 0 1 2 3 7
0 11451 , Saudi Arabia
1 University , P.O. Box 2455, Riyadh
2 College of Science , King Saud
3 Department of Mathematics
4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland , Galway , Ireland
5 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University , Jeddah , Saudi Arabia
6 Department of Mathematics, Atilim University , Incek, Ankara, 06836 , Turkey
7 Department of Mathematics, College of Science, King Saud University , P.O. Box 2455, Riyadh, 11451 , Saudi Arabia
Let T : X → X be a given operator and FT be the set of its fixed points. For a certain function ϕ : X → [0, ∞), we say that FT is ϕ-admissible if FT is nonempty and FT ⊆ Zϕ , where Zϕ is the zero set of ϕ. In this paper, we study the ϕ-admissibility of a new class of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong fixed point theorem.
ϕ-admissible; fixed point; homotopy result; partial metric
1 Introduction
Let T : X → X be a given operator. The set of fixed points of T is denoted by FT , that
Zϕ = x ∈ X : ϕ(x) = .
FT = {x ∈ X : Tx = x}.
(F) max{a, b} ≤ F(a, b, c), for all a, b, c ≥ ;
(F) F(a, , ) = a, for all a ≥ ;
(F) F is continuous.
As examples, the following functions belong to F :
. F(a, b, c) = a + b + c,
. F(a, b, c) = max{a, b} + ln(c + ),
. F(a, b, c) = a + b + c(c + ),
. F(a, b, c) = (a + b)ec,
N
. F(a, b, c) = (a + b)(c + )n, n ∈ .
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( ) ψ is upper semi-continuous from the right;
( ) ψ (t) < t, for all t > .
For given functions ϕ : X → [, ∞), F ∈ F , and ψ ∈ , we denote by T (ϕ, F, ψ ) the class
of operators T : X → X satisfying
The aim of this paper is to study the ϕ-admissibility of the set FT , where T belongs to
the class of operators T (ϕ, F, ψ ), (F, ψ ) ∈ F × . As applications, we obtain an homotopy
result and a partial metric version of the Boyd-Wong fixed point theorem.
2 Main result
Theorem . Let (X, d) be a complete metric space and T : X → X be a given operator.
Suppose that the following conditions hold:
(i) there exist ϕ : X → [, ∞), F ∈ F , and ψ ∈ such that T ∈ T (ϕ, F, ψ );
(ii) ϕ is lower semi-continuous.
Then the set FT is ϕ-admissible. Moreover, the operator T has a unique fixed point.
F , ϕ(ξ ), ϕ(ξ ) ≤ ψ F , ϕ(ξ ), ϕ(ξ ) .
which is impossible from (.). Consequently, we have
Using the above equality and (F), we obtain
FT ⊆ Zϕ.
Consequently, we have
Now, we have to prove that FT is a nonempty set. Let x be an arbitrary element of X.
Consider the Picard sequence {xn} ⊂ X defined by
where T n is the nth iterate of T . If for some N ∈ N we have xN = xN+, then xN will be an
element of FT . As a result we can suppose that
xn = T nx, n ∈ N = {, , , . . .},
d(xn, xn+) > ,
n ∈ N.
Using (.), we have
F d(Txn, Txn–), ϕ(Txn), ϕ(Txn–)
≤ ψ F d(xn, xn–), ϕ(xn), ϕ(xn–) , n ∈ N∗;
here N∗ = {, , . . .}. If for some N ∈ N∗, we have
F d(xN , xN–), ϕ(xN ), ϕ(xN–) = ,
then property (F) yields
d(xN , xN–) ≤ F d(xN , xN–), ϕ(xN ), ϕ(xN–) = ,
which is a contradiction with (.). Thus
F d(xn, xn–), ϕ(xn), ϕ(xn–) > , n ∈ N∗.
Using (.), (.), the definition of the sequence {xn}, and ( ), we have
⎨⎧ F(d(xn+, xn), ϕ(xn+), ϕ(xn)) ≤ ψ (F(d(xn, xn–), ϕ(xn), ϕ(xn–))),
n ∈ N∗.
Suppose now that c > . Using the properties ( )-( ), we deduce from (.) that
Using (F) and (.), we get
Next, we show that {xn} is a Cauchy sequence in the metric space (X, d). Suppose that {xn}
is not a Cauchy sequence. Then there exists ε > for which we can find two sequences of
positive integers {m(k)} and {n(k)} such that, for all k ∈ N,
n(k) > m(k) > k,
Using (.), for all k ∈ N we have
≤ d(xm(k), xn(k)–) + d(xn(k)–, xn(k))
ε ≤ d(xm(k), xn(k)) < ε + d(xn(k)–, xn(k)), k ∈ N.
d(xm(k), xn(k)) ≥
for k ∈ N.
Using the properties (F)-(F), (.), and (.), we get
Using the above limit and ( ), we obtain
On the other hand, using (.) and (F), for all k ∈ N we have
which is a contradiction. As a consequence, {xn} is a Cauchy sequence. Since (X, d) is a
complete metric space, there is a z ∈ X such that
≤ ϕ(z) ≤ linm→i∞nf ϕ(xn) = ,
z ∈ Zϕ.
Since ϕ is lower semi-continuous, it follows from (.) and (.) that
d(xn+, Tz) ≤ ψ F d(xn, z), ϕ(xn), , n ∈ N.
Also using the continuity of F , (F), (.), and (. (...truncated)