# Ground State Homoclinic Solutions for a Class of Superquadratic Fourth-Order Differential Equations

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2021-07-09

AUTHORS ABSTRACT

In the present paper, we consider the fourth-order differential equation u(4)(x)+ωu′′(x)+a(x)u(x)=f(x,u(x)),∀x∈R(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in {\mathbb {R}}\quad \quad \quad \quad (1) \end{aligned}\end{document}in which ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} represents a constant, a∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in C({\mathbb {R}},{\mathbb {R}})$$\end{document} and f∈C(R2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C({\mathbb {R}}^{2},{\mathbb {R}})$$\end{document}. We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and F(x,u)=∫0uf(x,t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x,u)=\int ^{u}_{0}f(x,t)dt$$\end{document} satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved. More... »

PAGES

1-20

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s12591-021-00576-6

DOI

http://dx.doi.org/10.1007/s12591-021-00576-6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1139571674

Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service:

[
{
"@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json",
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Mathematical Sciences",
"type": "DefinedTerm"
},
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Pure Mathematics",
"type": "DefinedTerm"
}
],
"author": [
{
"affiliation": {
"alternateName": "Department of Mathematics, Faculty of Sciences, Monastir University of Monastir, 5000, Monastir, Tunisia",
"id": "http://www.grid.ac/institutes/grid.411838.7",
"name": [
"Department of Mathematics, Faculty of Sciences, Monastir University of Monastir, 5000, Monastir, Tunisia"
],
"type": "Organization"
},
"familyName": "Timoumi",
"givenName": "Mohsen",
"id": "sg:person.011137336207.07",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011137336207.07"
],
"type": "Person"
}
],
"citation": [
{
"id": "sg:pub.10.1007/s12190-008-0045-4",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1043343461",
"https://doi.org/10.1007/s12190-008-0045-4"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-1-4612-4146-1",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1022709270",
"https://doi.org/10.1007/978-1-4612-4146-1"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s11766-009-1948-z",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1001033495",
"https://doi.org/10.1007/s11766-009-1948-z"
],
"type": "CreativeWork"
}
],
"datePublished": "2021-07-09",
"datePublishedReg": "2021-07-09",
"description": "In the present paper, we consider the fourth-order differential equation u(4)(x)+\u03c9u\u2032\u2032(x)+a(x)u(x)=f(x,u(x)),\u2200x\u2208R(1)\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\\begin{aligned} u^{(4)}(x)+\\omega u''(x)+a(x)u(x)=f(x,u(x)),\\ \\forall x\\in {\\mathbb {R}}\\quad \\quad \\quad \\quad (1) \\end{aligned}\\end{document}in which \u03c9\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega$$\\end{document} represents a constant, a\u2208C(R,R)\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in C({\\mathbb {R}},{\\mathbb {R}})$$\\end{document} and f\u2208C(R2,R)\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f\\in C({\\mathbb {R}}^{2},{\\mathbb {R}})$$\\end{document}. We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and F(x,u)=\u222b0uf(x,t)dt\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F(x,u)=\\int ^{u}_{0}f(x,t)dt$$\\end{document} satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved.",
"genre": "article",
"id": "sg:pub.10.1007/s12591-021-00576-6",
"inLanguage": "en",
"isAccessibleForFree": false,
"isPartOf": [
{
"id": "sg:journal.1136107",
"issn": [
"0971-3514",
"0974-6870"
],
"name": "Differential Equations and Dynamical Systems",
"publisher": "Springer Nature",
"type": "Periodical"
}
],
"keywords": [
"present paper",
"paper",
"kind",
"Luan",
"literature",
"fourth-order differential equation",
"differential equations",
"existence",
"Zou",
"class",
"equations",
"homoclinic solutions",
"solution",
"conditions",
"Ding",
"Schechter",
"results",
"satisfies",
"proof",
"theorem"
],
"name": "Ground State Homoclinic Solutions for a Class of Superquadratic Fourth-Order Differential Equations",
"pagination": "1-20",
"productId": [
{
"name": "dimensions_id",
"type": "PropertyValue",
"value": [
"pub.1139571674"
]
},
{
"name": "doi",
"type": "PropertyValue",
"value": [
"10.1007/s12591-021-00576-6"
]
}
],
"sameAs": [
"https://doi.org/10.1007/s12591-021-00576-6",
"https://app.dimensions.ai/details/publication/pub.1139571674"
],
"sdDataset": "articles",
"sdDatePublished": "2022-05-20T07:39",
"sdPublisher": {
"name": "Springer Nature - SN SciGraph project",
"type": "Organization"
},
"sdSource": "s3://com-springernature-scigraph/baseset/20220519/entities/gbq_results/article/article_898.jsonl",
"type": "ScholarlyArticle",
"url": "https://doi.org/10.1007/s12591-021-00576-6"
}
]

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s12591-021-00576-6'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s12591-021-00576-6'

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s12591-021-00576-6'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s12591-021-00576-6'

This table displays all metadata directly associated to this object as RDF triples.

85 TRIPLES      22 PREDICATES      47 URIs      36 LITERALS      4 BLANK NODES

Subject Predicate Object
2 anzsrc-for:0101
3 schema:author Nff583602e67243c2abb17c9ce257583e
4 schema:citation sg:pub.10.1007/978-1-4612-4146-1
5 sg:pub.10.1007/s11766-009-1948-z
6 sg:pub.10.1007/s12190-008-0045-4
7 schema:datePublished 2021-07-09
8 schema:datePublishedReg 2021-07-09
9 schema:description In the present paper, we consider the fourth-order differential equation u(4)(x)+ωu′′(x)+a(x)u(x)=f(x,u(x)),∀x∈R(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in {\mathbb {R}}\quad \quad \quad \quad (1) \end{aligned}\end{document}in which ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} represents a constant, a∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in C({\mathbb {R}},{\mathbb {R}})$$\end{document} and f∈C(R2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C({\mathbb {R}}^{2},{\mathbb {R}})$$\end{document}. We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and F(x,u)=∫0uf(x,t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x,u)=\int ^{u}_{0}f(x,t)dt$$\end{document} satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved.
10 schema:genre article
11 schema:inLanguage en
12 schema:isAccessibleForFree false
13 schema:isPartOf sg:journal.1136107
14 schema:keywords Ding
15 Luan
16 Schechter
17 Zou
18 class
19 conditions
20 differential equations
21 equations
22 existence
23 fourth-order differential equation
24 homoclinic solutions
25 kind
26 literature
27 paper
28 present paper
29 proof
30 results
31 satisfies
32 solution
34 theorem
35 schema:name Ground State Homoclinic Solutions for a Class of Superquadratic Fourth-Order Differential Equations
36 schema:pagination 1-20
37 schema:productId N624ccebbf9e941338562df0bc72100dd
38 Ne42d88e3c4064b32b15f301a123832be
39 schema:sameAs https://app.dimensions.ai/details/publication/pub.1139571674
40 https://doi.org/10.1007/s12591-021-00576-6
41 schema:sdDatePublished 2022-05-20T07:39
43 schema:sdPublisher Ne00ee69fe48340669a8e37a5404ac7fd
44 schema:url https://doi.org/10.1007/s12591-021-00576-6
46 sgo:sdDataset articles
47 rdf:type schema:ScholarlyArticle
48 N624ccebbf9e941338562df0bc72100dd schema:name doi
49 schema:value 10.1007/s12591-021-00576-6
50 rdf:type schema:PropertyValue
51 Ne00ee69fe48340669a8e37a5404ac7fd schema:name Springer Nature - SN SciGraph project
52 rdf:type schema:Organization
53 Ne42d88e3c4064b32b15f301a123832be schema:name dimensions_id
54 schema:value pub.1139571674
55 rdf:type schema:PropertyValue
56 Nff583602e67243c2abb17c9ce257583e rdf:first sg:person.011137336207.07
57 rdf:rest rdf:nil
58 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
59 schema:name Mathematical Sciences
60 rdf:type schema:DefinedTerm
61 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
62 schema:name Pure Mathematics
63 rdf:type schema:DefinedTerm
64 sg:journal.1136107 schema:issn 0971-3514
65 0974-6870
66 schema:name Differential Equations and Dynamical Systems
67 schema:publisher Springer Nature
68 rdf:type schema:Periodical
69 sg:person.011137336207.07 schema:affiliation grid-institutes:grid.411838.7
70 schema:familyName Timoumi
71 schema:givenName Mohsen
72 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011137336207.07
73 rdf:type schema:Person
74 sg:pub.10.1007/978-1-4612-4146-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022709270
75 https://doi.org/10.1007/978-1-4612-4146-1
76 rdf:type schema:CreativeWork
77 sg:pub.10.1007/s11766-009-1948-z schema:sameAs https://app.dimensions.ai/details/publication/pub.1001033495
78 https://doi.org/10.1007/s11766-009-1948-z
79 rdf:type schema:CreativeWork
80 sg:pub.10.1007/s12190-008-0045-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1043343461
81 https://doi.org/10.1007/s12190-008-0045-4
82 rdf:type schema:CreativeWork
83 grid-institutes:grid.411838.7 schema:alternateName Department of Mathematics, Faculty of Sciences, Monastir University of Monastir, 5000, Monastir, Tunisia
84 schema:name Department of Mathematics, Faculty of Sciences, Monastir University of Monastir, 5000, Monastir, Tunisia
85 rdf:type schema:Organization