Abstract Let (C(t))t∈ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – coshλt is Saphar (resp. quasi-Fredholm) operator and λt /∉iπ, then A – λ2 is also Saphar (resp. quasi-Fredholm) operator. We show by counter-example that the converse is false in general.
{"title":"On the quasi-Fredholm and Saphar spectrum of strongly continuous Cosine operator function","authors":"H. Boua","doi":"10.2478/mjpaa-2021-0008","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0008","url":null,"abstract":"Abstract Let (C(t))t∈ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – coshλt is Saphar (resp. quasi-Fredholm) operator and λt /∉iπ, then A – λ2 is also Saphar (resp. quasi-Fredholm) operator. We show by counter-example that the converse is false in general.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"7 1","pages":"80 - 87"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69233410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we derive several subordination results and integral means result for certain class of analytic functions with complex order defined by means of q-differential operator. Some interesting corollaries and consequences of our results are also considered
{"title":"Subordination results for a class of analytic functions","authors":"M. Aouf, B. Frasin, G. Murugusundaramoorthy","doi":"10.2478/mjpaa-2021-0004","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0004","url":null,"abstract":"Abstract In this paper, we derive several subordination results and integral means result for certain class of analytic functions with complex order defined by means of q-differential operator. Some interesting corollaries and consequences of our results are also considered","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"7 1","pages":"30 - 42"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42811810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.
{"title":"On a chain of reproducing kernel Cartan subalgebras","authors":"A. Y. Kraidi, K. Kangni","doi":"10.2478/mjpaa-2021-0005","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0005","url":null,"abstract":"Abstract Let 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"7 1","pages":"43 - 49"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49396479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the present paper, we show the existence of solutions of some nonlinear inequalities of the form 〈Au + g(x, u,∇ u), v −u〉 ≥〈 f, v −u〉 with gradient constraint that depend on the solution itself, where A is a Leray-Lions operator defined on Orlicz spaces, g is a nonlinearity and f ∈ L1.
{"title":"Some class of nonlinear inequalities with gradient constraints in Orlicz spaces","authors":"S. Ajagjal, D. Meskine","doi":"10.2478/mjpaa-2020-0022","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0022","url":null,"abstract":"Abstract In the present paper, we show the existence of solutions of some nonlinear inequalities of the form 〈Au + g(x, u,∇ u), v −u〉 ≥〈 f, v −u〉 with gradient constraint that depend on the solution itself, where A is a Leray-Lions operator defined on Orlicz spaces, g is a nonlinearity and f ∈ L1.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"281 - 302"},"PeriodicalIF":0.0,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46235713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present paper is devoted to the numerical approximation for the diffusion equation subject to non-local boundary conditions. For the space discretization, we apply the Legendre-Chebyshev pseudospectral method, so that, the problem under consideration is reduced to a system of ODEs which can be solved by the second order Crank-Nicolson schema. Optimal error estimates for the semi-discrete scheme are derived in L2-norm. Numerical tests are included to demonstrate the effectiveness of the proposed method.
{"title":"Legendre-Chebyshev pseudo-spectral method for the diffusion equation with non-classical boundary conditions","authors":"Abdeldjalil Chattouh, K. Saoudi","doi":"10.2478/mjpaa-2020-0023","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0023","url":null,"abstract":"Abstract The present paper is devoted to the numerical approximation for the diffusion equation subject to non-local boundary conditions. For the space discretization, we apply the Legendre-Chebyshev pseudospectral method, so that, the problem under consideration is reduced to a system of ODEs which can be solved by the second order Crank-Nicolson schema. Optimal error estimates for the semi-discrete scheme are derived in L2-norm. Numerical tests are included to demonstrate the effectiveness of the proposed method.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"303 - 317"},"PeriodicalIF":0.0,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44166274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation -div a(x,u,∇u)=b(x)| u |p-2u+λH(x,u,∇u), - div,,aleft( {x,u,nabla u} right) = bleft( x right){left| u right|^{p - 2}}u + lambda Hleft( {x,u,nabla u} right), where Ω is a bounded smooth domain of N.
{"title":"Topological degree methods for a Neumann problem governed by nonlinear elliptic equation","authors":"Adil Abbassi, C. Allalou, Abderrazak Kassidi","doi":"10.2478/mjpaa-2020-0018","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0018","url":null,"abstract":"Abstract In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation -div a(x,u,∇u)=b(x)| u |p-2u+λH(x,u,∇u), - div,,aleft( {x,u,nabla u} right) = bleft( x right){left| u right|^{p - 2}}u + lambda Hleft( {x,u,nabla u} right), where Ω is a bounded smooth domain of N.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"231 - 242"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49503466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Relied on author’s first ever found multivariate Caputo fractional Taylor’s formula (2009, [1], Chapter 13), we develop and prove several multivariate left side Caputo fractional uniform Landau type inequalities.
{"title":"Multivariate Caputo left fractional Landau inequalities","authors":"G. Anastassiou","doi":"10.2478/mjpaa-2020-0021","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0021","url":null,"abstract":"Abstract Relied on author’s first ever found multivariate Caputo fractional Taylor’s formula (2009, [1], Chapter 13), we develop and prove several multivariate left side Caputo fractional uniform Landau type inequalities.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"266 - 280"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47495233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.
{"title":"Boundary Value Problem for Caputo–Fabrizio Random Fractional Differential Equations","authors":"Fouzia Bekada, S. Abbas, M. Benchohra","doi":"10.2478/mjpaa-2020-0017","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0017","url":null,"abstract":"Abstract This article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"218 - 230"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42528926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mohamed Laghzal, A. E. Khalil, M. D. M. Alaoui, A. Touzani
Abstract This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.
{"title":"Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term","authors":"Mohamed Laghzal, A. E. Khalil, M. D. M. Alaoui, A. Touzani","doi":"10.2478/mjpaa-2020-0015","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0015","url":null,"abstract":"Abstract This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"198 - 209"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44808775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . , r0m(am2-bm2)2≤r0m(bm+1-am+1b-a-(m+1)(ab)m2)≤(αa+(1-α)b)m-(aαb1-α)m, matrix{ {r_0^m{{left( {{a^{{m over 2}}} - {b^{{m over 2}}}} right)}^2}} & { le r_0^mleft( {{{{b^{m + 1}} - {a^{m + 1}}} over {b - a}} - left( {m + 1} right){{left( {ab} right)}^{{m over 2}}}} right)} cr {} & { le {{left( {alpha a + left( {1 - alpha } right)b} right)}^m} - {{left( {{a^alpha }{b^{1 - alpha }}} right)}^m},} cr } where r0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.
摘要本文证明了如果a，b>0且0≤α≤1，则对于m=1，2，3，r0m（am2-bm2）2≤r0m（bm+1-am+1b-a-（m+1）（ab）m2）≤（αa+（1-α）b）m-（aαb1-α）m，矩阵｛r_0^m｛left（｛a^｛m over 2｝｝）｝-｛b^｛m over2｝}｝ right）｝^2｝和｛le r_0^m left（｛1-alpha｝right）b｝right）｝^m｝-｛left（｛a^alpha}｛b^{1-alpha}｝right）｝^ m｝，｝cr｝其中r0=min｛α，1–α｝。这是由Kittaneh和Manasrah以及Hirzallah和Kittaneh对Young不等式的两个精化的相当新的推广，这两个精化分别对应于m=1和m=2的情况。作为应用，我们给出了一些关于广义欧氏算子半径和一些众所周知的算子f-连接的数值半径的精细化Young型不等式，以及一些关于正定矩阵的迹、行列式和范数的精细化杨氏型不等式。
{"title":"A new generalization of two refined Young inequalities and applications","authors":"M. Ighachane, M. Akkouchi","doi":"10.2478/mjpaa-2020-0012","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0012","url":null,"abstract":"Abstract In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . , r0m(am2-bm2)2≤r0m(bm+1-am+1b-a-(m+1)(ab)m2)≤(αa+(1-α)b)m-(aαb1-α)m, matrix{ {r_0^m{{left( {{a^{{m over 2}}} - {b^{{m over 2}}}} right)}^2}} & { le r_0^mleft( {{{{b^{m + 1}} - {a^{m + 1}}} over {b - a}} - left( {m + 1} right){{left( {ab} right)}^{{m over 2}}}} right)} cr {} & { le {{left( {alpha a + left( {1 - alpha } right)b} right)}^m} - {{left( {{a^alpha }{b^{1 - alpha }}} right)}^m},} cr } where r0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"155 - 167"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42220962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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