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On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications 各向异性变指数索波列夫空间的集中-紧凑性原理及其应用
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-22 DOI: 10.1007/s13540-024-00246-8
Nabil Chems Eddine, Maria Alessandra Ragusa, Dušan D. Repovš
{"title":"On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications","authors":"Nabil Chems Eddine, Maria Alessandra Ragusa, Dušan D. Repovš","doi":"10.1007/s13540-024-00246-8","DOIUrl":"https://doi.org/10.1007/s13540-024-00246-8","url":null,"abstract":"<p>We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative 带有卡普托导数的分式电报方程的时间相关识别问题
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-22 DOI: 10.1007/s13540-024-00240-0
{"title":"Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative","authors":"","doi":"10.1007/s13540-024-00240-0","DOIUrl":"https://doi.org/10.1007/s13540-024-00240-0","url":null,"abstract":"<h3>Abstract</h3> <p>This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form <span> <span>((D_{t}^{rho })^{2}u(t)+2alpha D_{t}^{rho }u(t)+Au(t)=p( t)q+f(t))</span> </span>, where <span> <span>(0&lt;tle T)</span> </span>, <span> <span>(0&lt;rho &lt;1)</span> </span> and <span> <span>(D_{t}^{rho })</span> </span> is the Caputo derivative. The equation contains a self-adjoint positive operator <em>A</em> and a time-varying multiplier <em>p</em>(<em>t</em>) in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition <span> <span>(B[u(t)] = psi (t))</span> </span> is imposed, where <em>B</em> is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator <em>A</em> and the functional <em>B</em> are discussed.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized Krätzel functions: an analytic study 广义克拉泽尔函数：分析研究
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-22 DOI: 10.1007/s13540-024-00243-x
Ashik A. Kabeer, Dilip Kumar
{"title":"Generalized Krätzel functions: an analytic study","authors":"Ashik A. Kabeer, Dilip Kumar","doi":"10.1007/s13540-024-00243-x","DOIUrl":"https://doi.org/10.1007/s13540-024-00243-x","url":null,"abstract":"<p>The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence, uniqueness and regularity for a semilinear stochastic subdiffusion with integrated multiplicative noise 具有综合乘法噪声的半线性随机子扩散的存在性、唯一性和正则性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-21 DOI: 10.1007/s13540-024-00244-w
Ziqiang Li, Yubin Yan
{"title":"Existence, uniqueness and regularity for a semilinear stochastic subdiffusion with integrated multiplicative noise","authors":"Ziqiang Li, Yubin Yan","doi":"10.1007/s13540-024-00244-w","DOIUrl":"https://doi.org/10.1007/s13540-024-00244-w","url":null,"abstract":"<p>We investigate a semilinear stochastic time-space fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the <span>(psi )</span>-Caputo derivative of order <span>(alpha in (0,1))</span> and the spectral fractional Laplacian of order <span>(beta in (frac{1}{2},1])</span>. The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic behavior for a porous-elastic system with fractional derivative-type internal dissipation 具有分数导数型内耗散的多孔弹性系统的渐近行为
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-21 DOI: 10.1007/s13540-024-00250-y
Wilson Oliveira, Sebastião Cordeiro, Carlos Alberto Raposo da Cunha, Octavio Vera
{"title":"Asymptotic behavior for a porous-elastic system with fractional derivative-type internal dissipation","authors":"Wilson Oliveira, Sebastião Cordeiro, Carlos Alberto Raposo da Cunha, Octavio Vera","doi":"10.1007/s13540-024-00250-y","DOIUrl":"https://doi.org/10.1007/s13540-024-00250-y","url":null,"abstract":"<p>This work deals with the solution and asymptotic analysis for a porous-elastic system with internal damping of the fractional derivative type. We consider an augmented model. The energy function is presented and establishes the dissipativity property of the system. We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem. We present two results for the asymptotic behavior: Strong stability of the <span>(C_0)</span>-semigroup associated with the system using Arendt-Batty and Lyubich-Vũ’s general criterion and polynomial stability applying Borichev and Tomilov’s Theorem.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139915942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of positive solutions for fractional delayed evolution equations of order $$gamma in (1,2)$$ via measure of non-compactness 通过非紧凑性度量看 $$gamma in (1,2)$$ 阶分数延迟演化方程正解的存在性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-20 DOI: 10.1007/s13540-024-00248-6
{"title":"Existence of positive solutions for fractional delayed evolution equations of order $$gamma in (1,2)$$ via measure of non-compactness","authors":"","doi":"10.1007/s13540-024-00248-6","DOIUrl":"https://doi.org/10.1007/s13540-024-00248-6","url":null,"abstract":"<h3>Abstract</h3> <p>The purpose of this paper is to consider the fractional delayed evolution equation of order <span> <span>(gamma in (1,2))</span> </span> in ordered Banach space. In the absence of assumptions about the compactness of cosine families or related sine families, the existence results of positive solutions are studied by using some fixed point theorems and monotone iterative method under the conditions that nonlinear function satisfies the non-compactness measure conditions and some appropriate growth conditions or order conditions.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some boundedness results for Riemann-Liouville tempered fractional integrals 黎曼-刘维尔节制分数积分的一些有界性结果
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-16 DOI: 10.1007/s13540-024-00247-7
César E. Torres Ledesma, Hernán A. Cuti Gutierrez, Jesús P. Avalos Rodríguez, Willy Zubiaga Vera
{"title":"Some boundedness results for Riemann-Liouville tempered fractional integrals","authors":"César E. Torres Ledesma, Hernán A. Cuti Gutierrez, Jesús P. Avalos Rodríguez, Willy Zubiaga Vera","doi":"10.1007/s13540-024-00247-7","DOIUrl":"https://doi.org/10.1007/s13540-024-00247-7","url":null,"abstract":"<p>In this work we generalize some results of the Riemann-Liouville fractional calculus for the tempered case, namely, we deal with some boundedness results of Riemann-Liouville tempered fractional integrals on continuous function space and Lebesgue spaces in bounded intervals and on the real line. Moreover, the limit behavior of the Riemann-Liouville tempered fractional integrals approaching to the Riemann-Liouville fractional integrals is considered.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Representations of solutions of systems of time-fractional pseudo-differential equations 时分数伪微分方程系统解的表示法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-16 DOI: 10.1007/s13540-024-00241-z
Sabir Umarov
{"title":"Representations of solutions of systems of time-fractional pseudo-differential equations","authors":"Sabir Umarov","doi":"10.1007/s13540-024-00241-z","DOIUrl":"https://doi.org/10.1007/s13540-024-00241-z","url":null,"abstract":"<p>Systems of fractional order differential and pseudo-differential equations are used in modeling of various dynamical processes. In the analysis of such models, including stability analysis, asymptotic behaviors, etc., it is useful to have a representation formulas for the solution. In this paper we prove the existence and uniqueness theorems and derive representation formulas for the solution of general systems of fractional multi-order linear pseudo-differential equations through the matrix-valued Mittag-Leffler function. Examples illustrating the obtained results are also provided.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity 具有奇异非线性的临界分数拉普拉斯方程正解的存在性和多重性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-16 DOI: 10.1007/s13540-024-00242-y
{"title":"Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity","authors":"","doi":"10.1007/s13540-024-00242-y","DOIUrl":"https://doi.org/10.1007/s13540-024-00242-y","url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we consider the following problem <span> <span>$$begin{aligned} {left{ begin{array}{ll} (-varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+lambda u^{-gamma }, &amp;{} text { in } varOmega , u&gt;0, text { in } varOmega , quad u=0, &amp;{} text { on } partial varOmega , end{array}right. } end{aligned}$$</span> </span>where <span> <span>(varOmega subset {mathbb {R}}^{N}(N &gt; 2s))</span> </span> is a smooth bounded domain, <span> <span>(sin (0,1))</span> </span>, <span> <span>(lambda )</span> </span> is a positive constant, <span> <span>(0&lt;gamma &lt;1)</span> </span>, <span> <span>(2_{s}^{*}=frac{2 N}{N-2s})</span> </span> and <span> <span>((-varDelta )^{s} )</span> </span> is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity <em>g</em>.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative 具有黎曼-刘维尔分数导数的时变分数动力系统的可达性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-02-14 DOI: 10.1007/s13540-024-00245-9
K. S. Vishnukumar, M. Vellappandi, V. Govindaraj
{"title":"Reachability of time-varying fractional dynamical systems with Riemann-Liouville fractional derivative","authors":"K. S. Vishnukumar, M. Vellappandi, V. Govindaraj","doi":"10.1007/s13540-024-00245-9","DOIUrl":"https://doi.org/10.1007/s13540-024-00245-9","url":null,"abstract":"<p>This study examines the reachability of a time-varying fractional dynamical system with Riemann-Liouville fractional derivative. The state transition matrix is used to solve the time-varying systems. Using the reachability Grammian matrix, the reachability linear time-varying fractional dynamical system is discussed. The existence and uniqueness of a solution of a nonlinear time-varying fractional dynamical system is established, and sufficient conditions for the reachability of nonlinear time-varying fractional dynamical systems are obtained with the help of Banach fixed point theorem. The reachability results are proved for a time-varying integro-fractional dynamical system for a particular case. A successive approximation method is proposed to give numerical solutions to the reachability problems.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139739417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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