On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients
Asymptotic formulas as x →∞ are obtained for a fundamental system of solutions to equations of the form l ( y ) : = ( − 1 ) n ( p ( x ) y ( n ) ) ( n ) + q ( x ) y = λ y , x ∈ [ 1 , ∞ ) , where p is a locally integrable function representable as p ( x ) = ( 1 + r ( x ) ) − 1 , r ∈ L 1 ( 1 , ∞ ) , an...
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| Veröffentlicht in: | Mathematical Notes 2018-07, Vol.104 (1-2), p.244-252 |
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| creator | Konechnaya, N. N. Mirzoev, K. A. Shkalikov, A. A. |
| description | Asymptotic formulas as
x
→∞ are obtained for a fundamental system of solutions to equations of the form
l
(
y
)
:
=
(
−
1
)
n
(
p
(
x
)
y
(
n
)
)
(
n
)
+
q
(
x
)
y
=
λ
y
,
x
∈
[
1
,
∞
)
, where p is a locally integrable function representable as
p
(
x
)
=
(
1
+
r
(
x
)
)
−
1
,
r
∈
L
1
(
1
,
∞
)
, and
q
is a distribution such that
q
=
σ
(
k
)
for a fixed integer
k
, 0 ≤
k
≤
n
, and a function σ satisfying the conditions
σ
∈
L
1
(
1
,
∞
)
i
f
k
<
n
,
|
σ
|
(
1
+
|
r
|
)
(
1
+
|
σ
|
)
∈
L
1
(
1
,
∞
)
i
f
k
=
n
. Similar results are obtained for functions representable as
p
(
x
)
=
x
2
n
+
v
(
1
+
r
(
x
)
)
−
1
,
q
=
σ
(
k
)
,
σ
(
x
)
=
x
k
+
v
(
β
+
s
(
x
)
)
, for fixed
k
, 0 ≤
k
≤
n
, where the functions
r
and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression
l
(
y
) (for real functions
p
and
q
) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case
n
= 1. |
| doi_str_mv | 10.1134/S0001434618070258 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2116621797</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2116621797</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-861276362c0dc701c3e8455469b9e6196323292ff09ef64a89b3df3a1f378e653</originalsourceid><addsrcrecordid>eNp1kE9LAzEQxYMoWKsfwFvA82om2U2yx1rrHyj00Hpe0m3Spmw3bZK19NubsoIH8TQM7_feMA-heyCPACx_mhNCIGc5B0kEoYW8QAMoBMukFPwSDc5ydtav0U0I27QBBzJAetbiuNF4FE67fXTR1vhZb9SXdR47g-eu6aJ1bcDR4cXRZQvtd_jFGqO9bqNVDZ4cOtUjRxs3eG7bddcoj8dOG2Nrm7Bwi66MaoK--5lD9Pk6WYzfs-ns7WM8mmY1Ax4zyYEKzjityaoWBGqmZV4UOS-XpeZQckYZLakxpNSG50qWS7YyTIFhQmpesCF66HP33h06HWK1dZ1v08mKpoc5BVGKREFP1d6F4LWp9t7ulD9VQKpzm9WfNpOH9p6Q2Hat_W_y_6ZvBHR1ug</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2116621797</pqid></control><display><type>article</type><title>On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients</title><source>SpringerLink Journals - AutoHoldings</source><creator>Konechnaya, N. N. ; Mirzoev, K. A. ; Shkalikov, A. A.</creator><creatorcontrib>Konechnaya, N. N. ; Mirzoev, K. A. ; Shkalikov, A. A.</creatorcontrib><description>Asymptotic formulas as
x
→∞ are obtained for a fundamental system of solutions to equations of the form
l
(
y
)
:
=
(
−
1
)
n
(
p
(
x
)
y
(
n
)
)
(
n
)
+
q
(
x
)
y
=
λ
y
,
x
∈
[
1
,
∞
)
, where p is a locally integrable function representable as
p
(
x
)
=
(
1
+
r
(
x
)
)
−
1
,
r
∈
L
1
(
1
,
∞
)
, and
q
is a distribution such that
q
=
σ
(
k
)
for a fixed integer
k
, 0 ≤
k
≤
n
, and a function σ satisfying the conditions
σ
∈
L
1
(
1
,
∞
)
i
f
k
<
n
,
|
σ
|
(
1
+
|
r
|
)
(
1
+
|
σ
|
)
∈
L
1
(
1
,
∞
)
i
f
k
=
n
. Similar results are obtained for functions representable as
p
(
x
)
=
x
2
n
+
v
(
1
+
r
(
x
)
)
−
1
,
q
=
σ
(
k
)
,
σ
(
x
)
=
x
k
+
v
(
β
+
s
(
x
)
)
, for fixed
k
, 0 ≤
k
≤
n
, where the functions
r
and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression
l
(
y
) (for real functions
p
and
q
) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case
n
= 1.</description><identifier>ISSN: 0001-4346</identifier><identifier>ISSN: 1067-9073</identifier><identifier>EISSN: 1573-8876</identifier><identifier>DOI: 10.1134/S0001434618070258</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Asymptotic properties ; Differential equations ; Mathematical analysis ; Mathematical functions ; Mathematics ; Mathematics and Statistics ; Operators (mathematics) ; Theorems</subject><ispartof>Mathematical Notes, 2018-07, Vol.104 (1-2), p.244-252</ispartof><rights>Pleiades Publishing, Ltd. 2018</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-861276362c0dc701c3e8455469b9e6196323292ff09ef64a89b3df3a1f378e653</citedby><cites>FETCH-LOGICAL-c316t-861276362c0dc701c3e8455469b9e6196323292ff09ef64a89b3df3a1f378e653</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S0001434618070258$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S0001434618070258$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Konechnaya, N. N.</creatorcontrib><creatorcontrib>Mirzoev, K. A.</creatorcontrib><creatorcontrib>Shkalikov, A. A.</creatorcontrib><title>On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients</title><title>Mathematical Notes</title><addtitle>Math Notes</addtitle><description>Asymptotic formulas as
x
→∞ are obtained for a fundamental system of solutions to equations of the form
l
(
y
)
:
=
(
−
1
)
n
(
p
(
x
)
y
(
n
)
)
(
n
)
+
q
(
x
)
y
=
λ
y
,
x
∈
[
1
,
∞
)
, where p is a locally integrable function representable as
p
(
x
)
=
(
1
+
r
(
x
)
)
−
1
,
r
∈
L
1
(
1
,
∞
)
, and
q
is a distribution such that
q
=
σ
(
k
)
for a fixed integer
k
, 0 ≤
k
≤
n
, and a function σ satisfying the conditions
σ
∈
L
1
(
1
,
∞
)
i
f
k
<
n
,
|
σ
|
(
1
+
|
r
|
)
(
1
+
|
σ
|
)
∈
L
1
(
1
,
∞
)
i
f
k
=
n
. Similar results are obtained for functions representable as
p
(
x
)
=
x
2
n
+
v
(
1
+
r
(
x
)
)
−
1
,
q
=
σ
(
k
)
,
σ
(
x
)
=
x
k
+
v
(
β
+
s
(
x
)
)
, for fixed
k
, 0 ≤
k
≤
n
, where the functions
r
and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression
l
(
y
) (for real functions
p
and
q
) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case
n
= 1.</description><subject>Asymptotic properties</subject><subject>Differential equations</subject><subject>Mathematical analysis</subject><subject>Mathematical functions</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators (mathematics)</subject><subject>Theorems</subject><issn>0001-4346</issn><issn>1067-9073</issn><issn>1573-8876</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp1kE9LAzEQxYMoWKsfwFvA82om2U2yx1rrHyj00Hpe0m3Spmw3bZK19NubsoIH8TQM7_feMA-heyCPACx_mhNCIGc5B0kEoYW8QAMoBMukFPwSDc5ydtav0U0I27QBBzJAetbiuNF4FE67fXTR1vhZb9SXdR47g-eu6aJ1bcDR4cXRZQvtd_jFGqO9bqNVDZ4cOtUjRxs3eG7bddcoj8dOG2Nrm7Bwi66MaoK--5lD9Pk6WYzfs-ns7WM8mmY1Ax4zyYEKzjityaoWBGqmZV4UOS-XpeZQckYZLakxpNSG50qWS7YyTIFhQmpesCF66HP33h06HWK1dZ1v08mKpoc5BVGKREFP1d6F4LWp9t7ulD9VQKpzm9WfNpOH9p6Q2Hat_W_y_6ZvBHR1ug</recordid><startdate>20180701</startdate><enddate>20180701</enddate><creator>Konechnaya, N. N.</creator><creator>Mirzoev, K. A.</creator><creator>Shkalikov, A. A.</creator><general>Pleiades Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20180701</creationdate><title>On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients</title><author>Konechnaya, N. N. ; Mirzoev, K. A. ; Shkalikov, A. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-861276362c0dc701c3e8455469b9e6196323292ff09ef64a89b3df3a1f378e653</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Asymptotic properties</topic><topic>Differential equations</topic><topic>Mathematical analysis</topic><topic>Mathematical functions</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators (mathematics)</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Konechnaya, N. N.</creatorcontrib><creatorcontrib>Mirzoev, K. A.</creatorcontrib><creatorcontrib>Shkalikov, A. A.</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematical Notes</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Konechnaya, N. N.</au><au>Mirzoev, K. A.</au><au>Shkalikov, A. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients</atitle><jtitle>Mathematical Notes</jtitle><stitle>Math Notes</stitle><date>2018-07-01</date><risdate>2018</risdate><volume>104</volume><issue>1-2</issue><spage>244</spage><epage>252</epage><pages>244-252</pages><issn>0001-4346</issn><issn>1067-9073</issn><eissn>1573-8876</eissn><abstract>Asymptotic formulas as
x
→∞ are obtained for a fundamental system of solutions to equations of the form
l
(
y
)
:
=
(
−
1
)
n
(
p
(
x
)
y
(
n
)
)
(
n
)
+
q
(
x
)
y
=
λ
y
,
x
∈
[
1
,
∞
)
, where p is a locally integrable function representable as
p
(
x
)
=
(
1
+
r
(
x
)
)
−
1
,
r
∈
L
1
(
1
,
∞
)
, and
q
is a distribution such that
q
=
σ
(
k
)
for a fixed integer
k
, 0 ≤
k
≤
n
, and a function σ satisfying the conditions
σ
∈
L
1
(
1
,
∞
)
i
f
k
<
n
,
|
σ
|
(
1
+
|
r
|
)
(
1
+
|
σ
|
)
∈
L
1
(
1
,
∞
)
i
f
k
=
n
. Similar results are obtained for functions representable as
p
(
x
)
=
x
2
n
+
v
(
1
+
r
(
x
)
)
−
1
,
q
=
σ
(
k
)
,
σ
(
x
)
=
x
k
+
v
(
β
+
s
(
x
)
)
, for fixed
k
, 0 ≤
k
≤
n
, where the functions
r
and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression
l
(
y
) (for real functions
p
and
q
) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case
n
= 1.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S0001434618070258</doi><tpages>9</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0001-4346 |
| ispartof | Mathematical Notes, 2018-07, Vol.104 (1-2), p.244-252 |
| issn | 0001-4346 1067-9073 1573-8876 |
| language | eng |
| recordid | cdi_proquest_journals_2116621797 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Asymptotic properties Differential equations Mathematical analysis Mathematical functions Mathematics Mathematics and Statistics Operators (mathematics) Theorems |
| title | On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients |
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