Research Article
A Family of Measures of Noncompactness in
the Locally Sobolev Spaces and Its Applications to Some
Nonlinear Volterra Integrodifferential Equations
H. Mehravaran, M. Khanehgir , and R. Allahyari
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Correspondence should be addressed to M. Khanehgir; mkhanehgir@gmail.com
Received 5 October 2017; Accepted 15 January 2018; Published 15 February 2018
Academic Editor: Ji Gao
Copyright © 2018 H. Mehravaran et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study a new family of measures of noncompactness in the locally Sobolev space
,1
loc
(R
+
), equipped with a suitable topology.
As an application of the technique associated with this family of measures of noncompactness, we study the existence of solutions
for a class of nonlinear Volterra integrodiferential equations. Further, we give an illustrative example to verify the efectiveness and
applicability of our results.
1. Introduction
A locally integrable function is a function which is integrable
on every compact subset of its domain of defnition. Te
importance of such functions lies in the fact that their
function space is similar to
spaces, but its members are not
required to satisfy any growth restriction on their behavior
at infnity: in other words, locally integrable functions can
grow arbitrarily fast at infnity but are still manageable in a
way similar to ordinary integrable functions.
Sobolev spaces [1], that is, the class of functions with
derivatives in
, play an important role in the analysis.
In the recent times, there have been considerable eforts to
study these spaces. Te signifcant impact of them arises from
the fact that solutions of partial diferential equations are
naturally found in Sobolev spaces. Tey also can be applied
to approximation theory, calculus of variations, diferential
geometry, spectral theory, and so on.
Te degree of noncompactness of a set is measured
by means of functions called measures of noncompactness.
Measures of noncompactness are very useful tools in Banach
spaces. Tey are widely used in fxed point theory, diferential
equations, functional equations, integral and integrodiferen-
tial equations, optimization, and so on [2–8].
In this paper, we defne two topological structures on the
locally Sobolev space
,1
loc
(R
+
): the Fr´ echet metric topology
F
loc
given by a sequence of seminorms and the topology
loc
generated by the family of projections,
:
,1
loc
(R
+
) →
,1
([0,]
), ≥0, (1)
where the spaces
,1
([0,]
) are furnished with weak
topologies. Next, we introduce a weak measure of noncom-
pactness in
,1
([0,]
). Tereafer, we defne a family
of measures of noncompactness {
}
≥0
in
,1
loc
(R
+
) with
the topology
loc
and we investigate the basic properties of
{
}
≥0
.
As an application, we study the existence of solutions for
a class of nonlinear Volterra integrodiferential equations
(
1
,...,
)=((
1
,...,
),
∫
1
0
⋅⋅⋅∫
0
((
1
,...,
),(
1
,...,
),((
1
,...,
)),
1
((
1
,...,
)),...,
((
1
,...,
)),
0
(
1
,...,
))
1
⋅⋅⋅
).
(2)
Hindawi
Journal of Mathematics
Volume 2018, Article ID 3579079, 13 pages
https://doi.org/10.1155/2018/3579079