Abstract In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.
{"title":"A generalized sequential problem of Lane-Emden type via fractional calculus","authors":"Y. Gouari, Z. Dahmani, A. Ndiaye","doi":"10.2478/mjpaa-2020-0013","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0013","url":null,"abstract":"Abstract In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"168 - 183"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42310935","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 have derived some integral transforms of the polynomial weighted incomplete H-functions and incomplete ̄H-functions. The obtained image formulas are of general nature and may, as special cases, give rise to integral transforms involved with the H-functions and ̄H-functions.
{"title":"Certain integral transforms concerning the product of family of polynomials and generalized incomplete functions","authors":"Sapna Meena, S. Bhatter, K. Jangid, S. Purohit","doi":"10.2478/mjpaa-2020-0019","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0019","url":null,"abstract":"Abstract In this article, we have derived some integral transforms of the polynomial weighted incomplete H-functions and incomplete ̄H-functions. The obtained image formulas are of general nature and may, as special cases, give rise to integral transforms involved with the H-functions and ̄H-functions.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"243 - 254"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48184161","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 One of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.
{"title":"Sum of the spaces on ordered setting","authors":"T. Al-shami","doi":"10.2478/mjpaa-2020-0020","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0020","url":null,"abstract":"Abstract One of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"255 - 265"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47983535","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 the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].
{"title":"On some properties of the conformable fractional derivative","authors":"Radouane Azennar, D. Mentagui","doi":"10.2478/mjpaa-2020-0016","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0016","url":null,"abstract":"Abstract In this paper, we prove that the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"210 - 217"},"PeriodicalIF":0.0,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45121680","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 purpose of the present paper is to find the sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk 𝔻. Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.
{"title":"Certain subclasses of Spiral-like univalent functions related with Pascal distribution series","authors":"G. Murugusundaramoorthy","doi":"10.2478/mjpaa-2021-0020","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0020","url":null,"abstract":"Abstract The purpose of the present paper is to find the sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk 𝔻. Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"7 1","pages":"312 - 323"},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44751119","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 K-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled K-fusion frames, and we develop some results on the controlled K-fusion frames for Hilbert spaces, which generalize some well known results of controlled fusion frame case. Also we discuss some characterizations of controlled Bessel K-fusion sequences and of controlled K-fusion frames. Further, we analyze stability conditions of controlled K-fusion frames under perturbation.
{"title":"Controlled K-Fusion Frame for Hilbert Spaces","authors":"Nadia Assila, S. Kabbaj, Brahim Moalige","doi":"10.2478/mjpaa-2021-0011","DOIUrl":"https://doi.org/10.2478/mjpaa-2021-0011","url":null,"abstract":"Abstract K-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled K-fusion frames, and we develop some results on the controlled K-fusion frames for Hilbert spaces, which generalize some well known results of controlled fusion frame case. Also we discuss some characterizations of controlled Bessel K-fusion sequences and of controlled K-fusion frames. Further, we analyze stability conditions of controlled K-fusion frames under perturbation.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"7 1","pages":"116 - 133"},"PeriodicalIF":0.0,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43601597","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 Via Leray-Schauder’s nonlinear alternative, we obtain the existence of a weak solution for a nonlocal problem driven by an operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.
{"title":"Leray-Schauder’s solution for a nonlocal problem in a fractional Orlicz-Sobolev space","authors":"A. Boumazourh, M. Srati","doi":"10.2478/mjpaa-2020-0004","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0004","url":null,"abstract":"Abstract Via Leray-Schauder’s nonlinear alternative, we obtain the existence of a weak solution for a nonlocal problem driven by an operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"42 - 52"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48070118","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 establish several inequalities involving certain generalizations of the hyperbolic functions. The established results serve as generalizations of some known results in the literature. Among other analytical techniques, the procedure makes use of l’Hospital rule for monotonicity.
{"title":"Some Inequalities for Generalized Hyperbolic Functions","authors":"K. Nantomah, E. Prempeh","doi":"10.2478/mjpaa-2020-0007","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0007","url":null,"abstract":"Abstract In this paper, we establish several inequalities involving certain generalizations of the hyperbolic functions. The established results serve as generalizations of some known results in the literature. Among other analytical techniques, the procedure makes use of l’Hospital rule for monotonicity.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"76 - 92"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48307292","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 apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form x˙=y-1(x)y, y˙=-x-f(x)-g(x)y-h(x)y2, dot x = y - 1left( x right)y,,,dot y = - x - fleft( x right) - gleft( x right)y - hleft( x right){y^2}, where l(x) = ∊l1(x) + ∊2l2(x), f (x) = ∊ f1(x) + ∊2 f2(x), g(x) = ∊g1(x) + ∊2g2(x) and h(x) = ∊h1(x) + ∊2h2(x) where lk(x) has degree m and fk(x), gk(x) and hk(x) have degree n for each k = 1, 2, and ∊ is a small parameter.
{"title":"Limit cycles of Liénard polynomial systems type by averaging method","authors":"A. Boulfoul, Nawal Mellahi","doi":"10.2478/mjpaa-2020-0001","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0001","url":null,"abstract":"Abstract We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form x˙=y-1(x)y, y˙=-x-f(x)-g(x)y-h(x)y2, dot x = y - 1left( x right)y,,,dot y = - x - fleft( x right) - gleft( x right)y - hleft( x right){y^2}, where l(x) = ∊l1(x) + ∊2l2(x), f (x) = ∊ f1(x) + ∊2 f2(x), g(x) = ∊g1(x) + ∊2g2(x) and h(x) = ∊h1(x) + ∊2h2(x) where lk(x) has degree m and fk(x), gk(x) and hk(x) have degree n for each k = 1, 2, and ∊ is a small parameter.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"1 - 15"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49414004","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 show the existence of solutions for the nonlinear elliptic equations of the form { -div a(x,u,∇u)=f,u∈W 01Lϕ(Ω)∩L∞(Ω), left{ {matrix{ { - {rm{div}},aleft( {x,u,nabla u} right) = f,} hfill cr {u in W_0^1Lvarphi left( Omega right) cap {L^infty }left( Omega right),} hfill cr } } right. where a(x,s,ξ)⋅ξ≥ϕ¯x-1(ϕ(x,h(| s |)))ϕ(x,| ξ |) aleft( {x,s,xi } right) cdot xi ge bar varphi _x^{ - 1}left( {varphi left( {x,hleft( {left| s right|} right)} right)} right)varphi left( {x,left| xi right|} right) and h : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term f belongs to LN(Ω) or to Lm(Ω), with m=rNr+1 m = {{rN} over {r + 1}} for some r > 0 and φ is a Musielak function satisfying the Δ2-condition.
{"title":"Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces","authors":"M. Bourahma, J. Bennouna, M. El Moumni","doi":"10.2478/mjpaa-2020-0002","DOIUrl":"https://doi.org/10.2478/mjpaa-2020-0002","url":null,"abstract":"Abstract In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form { -div a(x,u,∇u)=f,u∈W 01Lϕ(Ω)∩L∞(Ω), left{ {matrix{ { - {rm{div}},aleft( {x,u,nabla u} right) = f,} hfill cr {u in W_0^1Lvarphi left( Omega right) cap {L^infty }left( Omega right),} hfill cr } } right. where a(x,s,ξ)⋅ξ≥ϕ¯x-1(ϕ(x,h(| s |)))ϕ(x,| ξ |) aleft( {x,s,xi } right) cdot xi ge bar varphi _x^{ - 1}left( {varphi left( {x,hleft( {left| s right|} right)} right)} right)varphi left( {x,left| xi right|} right) and h : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term f belongs to LN(Ω) or to Lm(Ω), with m=rNr+1 m = {{rN} over {r + 1}} for some r > 0 and φ is a Musielak function satisfying the Δ2-condition.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"6 1","pages":"16 - 33"},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43020803","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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