Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ2pu=λm(x)u in Ω,u=Δu=…Δ2p−1u=0 on ∂Ω. left{ {matrix{ {{Delta ^{2p}}u = lambda mleft( x right)u,,,in,,Omega ,} cr {u = Delta u = ldots {Delta ^{2p - 1}}u = 0,,,,on,,partial Omega .} cr } } right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.
本文研究了以下非线性特征值问题:在Ω中{Δ2pu=λm(x)u，在∂Ω中u=Δu=…Δ2p−1u=0。左 {{矩阵{{{δ^ p {2}} u = λm 离开u (x ) , ,在 ω,} cr {u =δu = ldots{三角洲^ {2 p - 1}} u = 0 ,, ,在 ,部分 ω。} cr}} 。Ω是有限域具有光滑边界的ℝN∂Ω,N≥1,p∈ℕ*,m∈L∞(Ω)µ{x∈Ω:m (x) > 0}≠0和2Δ菩:=Δ(Δ…(Δu)), 2 p乘以Δ运营商。利用Szulkin定理，证明了至少一个非负特征值的非递减序列的存在性。
{"title":"Existence of solutions for 4p-order PDES","authors":"F. Moradi, N. Moradi, M. Addam, S. E. Habib","doi":"10.2478/mjpaa-2022-0013","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0013","url":null,"abstract":"Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ2pu=λm(x)u in Ω,u=Δu=…Δ2p−1u=0 on ∂Ω. left{ {matrix{ {{Delta ^{2p}}u = lambda mleft( x right)u,,,in,,Omega ,} cr {u = Delta u = ldots {Delta ^{2p - 1}}u = 0,,,,on,,partial Omega .} cr } } right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"179 - 190"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42075184","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}
M. B. Benboubker, E. Nassouri, S. Ouaro, U. Traoré
Abstract In this paper we prove the existence of at least one renormalized solution for the p(x)-Laplacian equation associated with a maximal monotone operator and Radon measure data. The functional setting involves Sobolev spaces with variable exponent W1,p(·)(Ω).
{"title":"Renormalized solutions for a p(·)-Laplacian equation with Neumann nonhomogeneous boundary condition involving diffuse measure data and variable exponent","authors":"M. B. Benboubker, E. Nassouri, S. Ouaro, U. Traoré","doi":"10.2478/mjpaa-2022-0012","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0012","url":null,"abstract":"Abstract In this paper we prove the existence of at least one renormalized solution for the p(x)-Laplacian equation associated with a maximal monotone operator and Radon measure data. The functional setting involves Sobolev spaces with variable exponent W1,p(·)(Ω).","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"163 - 178"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46434747","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}
O. Chakrone, K. Belhadj, M. R. Sidi Ammi, A. Zerouali
Abstract In this paper, we study a class of fractional p(x)-Laplacian Dirichlet problems in a bounded domain with Lipschitz boundary. Using variational methods, we prove in different situations the existence and multiplicity of solutions.
{"title":"Existence and multiplicity results for fractional p(x)-Laplacian Dirichlet problem","authors":"O. Chakrone, K. Belhadj, M. R. Sidi Ammi, A. Zerouali","doi":"10.2478/mjpaa-2022-0011","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0011","url":null,"abstract":"Abstract In this paper, we study a class of fractional p(x)-Laplacian Dirichlet problems in a bounded domain with Lipschitz boundary. Using variational methods, we prove in different situations the existence and multiplicity of solutions.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"148 - 162"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43214316","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 Using generalized Canavati fractional left and right vectorial Taylor formulae we establish mixed fractional Ostrowski, Opial and Grüss type inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms ‖ · ‖ p, 1 ≤ p ≤ ∞. We provide also applications.
{"title":"Generalized Canavati Fractional Ostrowski, Opial and Grüss type inequalities for Banach algebra valued functions","authors":"G. Anastassiou","doi":"10.2478/mjpaa-2022-0010","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0010","url":null,"abstract":"Abstract Using generalized Canavati fractional left and right vectorial Taylor formulae we establish mixed fractional Ostrowski, Opial and Grüss type inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms ‖ · ‖ p, 1 ≤ p ≤ ∞. We provide also applications.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"128 - 147"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46915897","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 We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.
{"title":"Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations","authors":"A. Madrane, F. Benkhaldoun","doi":"10.2478/mjpaa-2022-0009","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0009","url":null,"abstract":"Abstract We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"102 - 127"},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45461283","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}
Samundra Regmi, Christopher I. Argyros, I. Argyros, S. George
Abstract The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.
{"title":"Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions","authors":"Samundra Regmi, Christopher I. Argyros, I. Argyros, S. George","doi":"10.2478/mjpaa-2022-0008","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0008","url":null,"abstract":"Abstract The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"92 - 101"},"PeriodicalIF":0.0,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44546803","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 article, we establish certain sufficient conditions to show the existence of solutions of an initial value problem of fractional-ordinary differential equation in Banach space. Our approach is based on a combination of Mönch fixed point theorem and a suitable measure of non-compactness. Also an example is given to illustrate our approach.
{"title":"Existence result for a fractional differential equation involving a special derivative","authors":"M. Beddani, B. Hedia","doi":"10.2478/mjpaa-2022-0006","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0006","url":null,"abstract":"Abstract In this article, we establish certain sufficient conditions to show the existence of solutions of an initial value problem of fractional-ordinary differential equation in Banach space. Our approach is based on a combination of Mönch fixed point theorem and a suitable measure of non-compactness. Also an example is given to illustrate our approach.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"67 - 77"},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42210786","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 a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.
{"title":"Nonlinear Random Differential Equations with n Sequential Fractional Derivatives","authors":"Yfrah Hafssa, Z. Dahmani","doi":"10.2478/mjpaa-2022-0001","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0001","url":null,"abstract":"Abstract This article deals with a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"1 - 21"},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46634261","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 work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.
{"title":"Pricing American Put Option using RBF-NN: New Simulation of Black-Scholes","authors":"El Kharrazi Zaineb, Saoud Sahar, Mahani Zouhir","doi":"10.2478/mjpaa-2022-0007","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0007","url":null,"abstract":"Abstract The present work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"12 1","pages":"78 - 91"},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41261805","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}
Amnay El Amri, Y. E. Foutayeni, Late El Miloudi Marhrani
Abstract Recently, Karapinar, Aydi and De Hierro introduced the notions of interpolative Kannan-type contractions and interpolative CRR-type contractions and proved that they possess a fixed point when the space is complete. In this paper, we show that the existence of fixed points for interpolative Kannan-type and CRR-type contractions remains true for a metric space which is not necessarily complete and we give a more precise result concerning the behaviour of Picard sequences of arbitrary initial point. We also study some particular classes of interpolative Kannan-type and CRR-type contractions.
{"title":"On some results on interpolative Kannan-type and CRR-type contractions.","authors":"Amnay El Amri, Y. E. Foutayeni, Late El Miloudi Marhrani","doi":"10.2478/mjpaa-2022-0005","DOIUrl":"https://doi.org/10.2478/mjpaa-2022-0005","url":null,"abstract":"Abstract Recently, Karapinar, Aydi and De Hierro introduced the notions of interpolative Kannan-type contractions and interpolative CRR-type contractions and proved that they possess a fixed point when the space is complete. In this paper, we show that the existence of fixed points for interpolative Kannan-type and CRR-type contractions remains true for a metric space which is not necessarily complete and we give a more precise result concerning the behaviour of Picard sequences of arbitrary initial point. We also study some particular classes of interpolative Kannan-type and CRR-type contractions.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"54 - 66"},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42836795","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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