Abstract In this paper, we establish some estimates of the left side of the generalized Gauss-Jacobi quadrature formula for harmonic h-preinvex functions involving Euler’s beta and hypergeometric functions. The obtained results are mainly based on the identity given by M. A. Noor, K. I. Noor and S. Iftikhar in [17].
摘要本文建立了含欧拉函数和超几何函数的调和h-预凸函数的广义Gauss-Jacobi积分公式左侧的一些估计。得到的结果主要基于M. A. Noor、K. I. Noor和S. Iftikhar在b[17]中给出的恒等式。
{"title":"Integral inequalities via harmonically h-convexity","authors":"M. Merad, B. Meftah, A. Souahi","doi":"10.2478/mjpaa-2021-0026","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0026","url":null,"abstract":"Abstract In this paper, we establish some estimates of the left side of the generalized Gauss-Jacobi quadrature formula for harmonic h-preinvex functions involving Euler’s beta and hypergeometric functions. The obtained results are mainly based on the identity given by M. A. Noor, K. I. Noor and S. Iftikhar in [17].","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42529328","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 aim of this paper is to study the onto isometries of the space of strongly Lipschitz p-nuclear operators, introduced by D. Chen and B. Zheng (Nonlinear Anal.,75, 2012). We give some new results about such isometrics and we focus, in particular, on the case F = ℓp*.
{"title":"Some results about Lipschitz p-Nuclear Operators","authors":"Khedidja Bey, A. Belacel","doi":"10.2478/mjpaa-2021-0025","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0025","url":null,"abstract":"Abstract The aim of this paper is to study the onto isometries of the space of strongly Lipschitz p-nuclear operators, introduced by D. Chen and B. Zheng (Nonlinear Anal.,75, 2012). We give some new results about such isometrics and we focus, in particular, on the case F = ℓp*.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47070567","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 some common fixed point theorems satisfying contractive type mapping in the setting of b-metric spaces. The presented theorem is an extension the results of M. Sarwar and M. U. Rahman [23] as well as a generalization of many well-known results in the literature through the context of b-metric spaces. Also, we present a few examples to illustrate the validity of the results obtained in the paper. Finally, results are applied to find the solution for an integral equation.
摘要本文证明了b-度量空间中满足压缩型映射的几个常见不动点定理。本文给出的定理是对M. Sarwar和M. U. Rahman的结果的推广，也是对文献中许多著名结果在b-度量空间中的推广。最后，通过算例说明本文所得结果的有效性。最后，将结果应用于求解一个积分方程。
{"title":"Some fixed point theorems of rational type contraction in b-metric spaces","authors":"Merdaci Seddik, Hamaizia Taieb","doi":"10.2478/mjpaa-2021-0023","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0023","url":null,"abstract":"Abstract In this paper, we prove some common fixed point theorems satisfying contractive type mapping in the setting of b-metric spaces. The presented theorem is an extension the results of M. Sarwar and M. U. Rahman [23] as well as a generalization of many well-known results in the literature through the context of b-metric spaces. Also, we present a few examples to illustrate the validity of the results obtained in the paper. Finally, results are applied to find the solution for an integral equation.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47195342","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 paper aims to propose a new hybrid approach for solving multiobjective optimization problems. This approach is based on a combination of global and local search procedures. The cross-entropy method is used as a stochastic model-based method to solve the multiobjective optimization problem and reach a first elite set of global solutions. In the local search step, an ∈-constraint method converts the multiobjective optimization problem to a series of parameterized single-objective optimization problems. Then, sequential quadratic programming (SQP) is used to solve the derived single-objective optimization problems allowing to reinforce and improve the global results. Numerical examples are used to demonstrate the efficiency and effectiveness of the proposed approach.
{"title":"Combining the cross-entropy algorithm and ∈-constraint method for multiobjective optimization","authors":"Abdelmajid Ezzine, A. Alla, N. Raissi","doi":"10.2478/mjpaa-2021-0019","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0019","url":null,"abstract":"Abstract This paper aims to propose a new hybrid approach for solving multiobjective optimization problems. This approach is based on a combination of global and local search procedures. The cross-entropy method is used as a stochastic model-based method to solve the multiobjective optimization problem and reach a first elite set of global solutions. In the local search step, an ∈-constraint method converts the multiobjective optimization problem to a series of parameterized single-objective optimization problems. Then, sequential quadratic programming (SQP) is used to solve the derived single-objective optimization problems allowing to reinforce and improve the global results. Numerical examples are used to demonstrate the efficiency and effectiveness of the proposed approach.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43196960","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}
Zaineb El kharrazi, Nouh Izem, Mustapha Malek, S. Saoud
Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.
{"title":"A Partition of unity finite element method for valuation American option under Black-Scholes model","authors":"Zaineb El kharrazi, Nouh Izem, Mustapha Malek, S. Saoud","doi":"10.2478/mjpaa-2021-0021","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0021","url":null,"abstract":"Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49101051","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 main goal of this paper is to provide the maximum number of crossing limit cycles of two different families of discontinuous piecewise linear differential systems. More precisely we prove that the systems formed by two regions, where, in one region we define a linear center and in the second region we define a Hamiltonian system without equilibria can exhibit three crossing limit cycles having two or four intersection points with the cubic of separation. After we prove that the systems formed by three regions, where, in two noadjacent regions we define a Hamiltonian system without equilibria, and in the third region we define a center, can exhibit six crossing limit cycles having four and two simultaneously intersection points with the cubic of separation.
{"title":"Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics","authors":"Loubna Damene, R. Benterki","doi":"10.2478/mjpaa-2021-0017","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0017","url":null,"abstract":"Abstract The main goal of this paper is to provide the maximum number of crossing limit cycles of two different families of discontinuous piecewise linear differential systems. More precisely we prove that the systems formed by two regions, where, in one region we define a linear center and in the second region we define a Hamiltonian system without equilibria can exhibit three crossing limit cycles having two or four intersection points with the cubic of separation. After we prove that the systems formed by three regions, where, in two noadjacent regions we define a Hamiltonian system without equilibria, and in the third region we define a center, can exhibit six crossing limit cycles having four and two simultaneously intersection points with the cubic of separation.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44710320","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 prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.
{"title":"Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L1-data","authors":"M. Moumni, Deval Sidi Mohamed","doi":"10.2478/mjpaa-2021-0018","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0018","url":null,"abstract":"Abstract We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46072881","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 ≤ v ≤ 1. Then for all positive integer m (1) - For v ∈ v∈[ 0,12n ] v in left[ {0,{1 over {{2^n}}}} right] , we have (avb1-v)m+∑k=1n2k-1vm(bm-(ab2k-1-1)m2k)2≤(va+(1-v)b)m. {left( {{a^v}{b^{1 - v}}} right)^m} + sumlimits_{k = 1}^n {{2^{k - 1}}{v^m}{{left( {sqrt {{b^m}} - root {{2^k}} of {left( {a{b^{2k - 1}} - 1} right)m} } right)}^2} le {{left( {va + left( {1 - v} right)b} right)}^m}.} (2) - For v ∈ v∈[ 2n-12n,1 ] v in left[ {{{{2^n} - 1} over {{2^n}}},1} right] , we have (avb1-v)m+∑k=1n2k-1(1-v)m(am-(ba2k-1-1)m2k)2≤(va+(1-v)b)m, {left( {{a^v}{b^{1 - v}}} right)^m} + sumlimits_{k = 1}^n {{2^{k - 1}}{{left( {1 - v} right)}^m}{{left( {sqrt {{a^m}} - root {{2^k}} of {left( {b{a^{2k - 1}} - 1} right)m} } right)}^2} le {{left( {va + left( {1 - v} right)b} right)}^m},} we also prove two similar inequalities for the cases v ∈ v∈[ 2n-12n,12 ] v in left[ {{{{2^n} - 1} over {{2^n}}},{1 over 2}} right] and v ∈ v∈[ 12,2n+12n ] v in left[ {{1 over 2},{{{2^n} + 1} over {{2^n}}}} right] . These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.
{"title":"Further generalized refinement of Young’s inequalities for τ -mesurable operators","authors":"M. Ighachane, M. Akkouchi","doi":"10.2478/mjpaa-2021-0015","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0015","url":null,"abstract":"Abstract In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m (1) - For v ∈ v∈[ 0,12n ] v in left[ {0,{1 over {{2^n}}}} right] , we have (avb1-v)m+∑k=1n2k-1vm(bm-(ab2k-1-1)m2k)2≤(va+(1-v)b)m. {left( {{a^v}{b^{1 - v}}} right)^m} + sumlimits_{k = 1}^n {{2^{k - 1}}{v^m}{{left( {sqrt {{b^m}} - root {{2^k}} of {left( {a{b^{2k - 1}} - 1} right)m} } right)}^2} le {{left( {va + left( {1 - v} right)b} right)}^m}.} (2) - For v ∈ v∈[ 2n-12n,1 ] v in left[ {{{{2^n} - 1} over {{2^n}}},1} right] , we have (avb1-v)m+∑k=1n2k-1(1-v)m(am-(ba2k-1-1)m2k)2≤(va+(1-v)b)m, {left( {{a^v}{b^{1 - v}}} right)^m} + sumlimits_{k = 1}^n {{2^{k - 1}}{{left( {1 - v} right)}^m}{{left( {sqrt {{a^m}} - root {{2^k}} of {left( {b{a^{2k - 1}} - 1} right)m} } right)}^2} le {{left( {va + left( {1 - v} right)b} right)}^m},} we also prove two similar inequalities for the cases v ∈ v∈[ 2n-12n,12 ] v in left[ {{{{2^n} - 1} over {{2^n}}},{1 over 2}} right] and v ∈ v∈[ 12,2n+12n ] v in left[ {{1 over 2},{{{2^n} + 1} over {{2^n}}}} right] . These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44487637","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 paper deals with a new class of mixed fractional differential equations with integral boundary conditions. We show an important equivalence result between our problem and nonlinear integral Fredholm equation of the second kind. The existence and uniqueness of a positive solution are proved using Guo-Krasnoselskii’s fixed point theorem and Banach’s contraction mapping principle. Different types of Ulam-Hyers stability are discussed. Three examples are also given to show the applicability of our results.
{"title":"A new class of mixed fractional differential equations with integral boundary conditions","authors":"Djiab Somia, N. Brahim","doi":"10.2478/mjpaa-2021-0016","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0016","url":null,"abstract":"Abstract This paper deals with a new class of mixed fractional differential equations with integral boundary conditions. We show an important equivalence result between our problem and nonlinear integral Fredholm equation of the second kind. The existence and uniqueness of a positive solution are proved using Guo-Krasnoselskii’s fixed point theorem and Banach’s contraction mapping principle. Different types of Ulam-Hyers stability are discussed. Three examples are also given to show the applicability of our results.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44637123","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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