{"title":"Solutions of second-order degenerate equations with infinite delay in Banach spaces","authors":"Shangquan Bu, G. Cai","doi":"10.1216/jie.2020.32.259","DOIUrl":"https://doi.org/10.1216/jie.2020.32.259","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"259-274"},"PeriodicalIF":0.8,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48126829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is devoted to the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem (MIND BVP) for the Laplace-Beltrami equation on a compact smooth surface C with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting H1(C) when considering positive constants in the impedance condition. The main purpose is to consider the MIND BVP in a nonclassical setting of the Bessel potential space Hp(C), for s > 1/p, 1 < p < ∞. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet-Neumann, Dirichletimpedance and Neumann-impedance BVPs for the Laplacian in the half-plane. The model mixed Dirichlet-Neumann BVP was investigated by R. Duduchava and M. Tsaava (2018). The other two are investigated in the present paper. This allows to write a necessary and sufficient condition for the Fredholmness of the MIND BVP and to indicate a large set of the space parameters s > 1/p and 1 < p < ∞ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting H1(C) for arbitrary complex values of the nonzero constant in the impedance condition. 1. Formulation of the problem. Let S ⊂ R be some smooth, closed, orientable surface, bordering a compact inner Ω and outer Ω− := R Ω+ domains. By C we denote a subsurface of S, which has two faces C− and C and inherits the orientation from S: C 2010 AMS Mathematics subject classification. Primary 35J57; Secondary 45E10, 47B35.
{"title":"Mixed impedance boundary value problems for the Laplace–Beltrami equation","authors":"L. Castro, R. Duduchava, F. Speck","doi":"10.1216/jie.2020.32.275","DOIUrl":"https://doi.org/10.1216/jie.2020.32.275","url":null,"abstract":"This work is devoted to the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem (MIND BVP) for the Laplace-Beltrami equation on a compact smooth surface C with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting H1(C) when considering positive constants in the impedance condition. The main purpose is to consider the MIND BVP in a nonclassical setting of the Bessel potential space Hp(C), for s > 1/p, 1 < p < ∞. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet-Neumann, Dirichletimpedance and Neumann-impedance BVPs for the Laplacian in the half-plane. The model mixed Dirichlet-Neumann BVP was investigated by R. Duduchava and M. Tsaava (2018). The other two are investigated in the present paper. This allows to write a necessary and sufficient condition for the Fredholmness of the MIND BVP and to indicate a large set of the space parameters s > 1/p and 1 < p < ∞ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting H1(C) for arbitrary complex values of the nonzero constant in the impedance condition. 1. Formulation of the problem. Let S ⊂ R be some smooth, closed, orientable surface, bordering a compact inner Ω and outer Ω− := R Ω+ domains. By C we denote a subsurface of S, which has two faces C− and C and inherits the orientation from S: C 2010 AMS Mathematics subject classification. Primary 35J57; Secondary 45E10, 47B35.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44918579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$ddot{text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. This article develops the mapping properties of generalized Abel operators $alpha {_aD_x^{-s}}+beta {_xD_b^{-s}}$ in fractional Sobolev spaces, where $0
{"title":"Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions","authors":"Yulong Li","doi":"10.1216/jie.2021.33.327","DOIUrl":"https://doi.org/10.1216/jie.2021.33.327","url":null,"abstract":"The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$ddot{text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. This article develops the mapping properties of generalized Abel operators $alpha {_aD_x^{-s}}+beta {_xD_b^{-s}}$ in fractional Sobolev spaces, where $0<alpha,beta$, $alpha+beta=1$, $ 0<s<1$ and $ {_aD_x^{-s}}$, $ {_xD_b^{-s}}$ are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of $(alpha {_aD_x^{-s}}+beta {_xD_b^{-s}})u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u(x)$ while letting $f(x)$ become smoother and imposing homogeneous boundary restrictions $u(a)=u(b)=0$.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45547035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polyconvolution of Hartley integral transforms $H_2$ and integral equations","authors":"N. M. Khoa, T. V. Thắng","doi":"10.1216/jie.2020.32.171","DOIUrl":"https://doi.org/10.1216/jie.2020.32.171","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"171-180"},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46170612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a class of noncompact weakly singular Volterra integral equations: theory and application to fractional differential equations with variable coefficient","authors":"M. Toranj-Simin, M. Hadizadeh","doi":"10.1216/jie.2020.32.193","DOIUrl":"https://doi.org/10.1216/jie.2020.32.193","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43160675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Traveling wave solutions for a SEIR epidemic model in combination with random dispersal and nonlocal dispersal","authors":"Xin Wu, R. Yuan, Baochuan Tian","doi":"10.1216/jie.2020.32.213","DOIUrl":"https://doi.org/10.1216/jie.2020.32.213","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"213-237"},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47158842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Square-mean almost automorphic solution of a stochastic cellular neural network on time scales","authors":"Soniya Dhama, Syed Abbas","doi":"10.1216/jie.2020.32.151","DOIUrl":"https://doi.org/10.1216/jie.2020.32.151","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"151-170"},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46499071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence results for neutral integro-differential equations with nonlocal conditions","authors":"Jianbo Zhu, Xianlong Fu","doi":"10.1216/jie.2020.32.239","DOIUrl":"https://doi.org/10.1216/jie.2020.32.239","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47440501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Valuation of non-recourse stock loan using an integral equation approach","authors":"Nhat Le, M. Ngo","doi":"10.1216/jie.2020.32.181","DOIUrl":"https://doi.org/10.1216/jie.2020.32.181","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47442971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An optimized FD extrapolated scheme based on POD for the 2D integro-differential equation of parabolic type","authors":"Zhendong Luo","doi":"10.1216/jie.2020.32.35","DOIUrl":"https://doi.org/10.1216/jie.2020.32.35","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"35-50"},"PeriodicalIF":0.8,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45387340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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