Abstract
AbstractIn this paper, we investigate the conditions on the control mappings $\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$
ψ
,
φ
:
(
0
,
∞
)
→
R
that guarantee the existence of the fixed points of the mapping $T:X\rightarrow P(X)$
T
:
X
→
P
(
X
)
satisfying the following inequalities: $$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(d(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$
ψ
(
H
(
T
x
,
T
y
)
)
≤
φ
(
d
(
x
,
y
)
)
∀
x
,
y
∈
X
,
provided that
H
(
T
x
,
T
y
)
>
0
,
and $$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(A(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$
ψ
(
H
(
T
x
,
T
y
)
)
≤
φ
(
A
(
x
,
y
)
)
∀
x
,
y
∈
X
,
provided that
H
(
T
x
,
T
y
)
>
0
,
where $A(x,y)=\max \{ d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 \} $
A
(
x
,
y
)
=
max
{
d
(
x
,
y
)
,
d
(
x
,
T
x
)
,
d
(
y
,
T
y
)
,
(
d
(
x
,
T
y
)
+
d
(
T
x
,
y
)
)
/
2
}
, and $(X, d)$
(
X
,
d
)
is a metric space. The obtained fixed point results improve many earlier results on the set-valued contractions. As an application, we consider the existence of the solutions of an FDE.
Funder
Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Algebra and Number Theory,Analysis
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