On inclusion properties of discrete Morrey spacesGunawan, Hendra; Hakim, Denny Ivanal; Idris, Mochammad
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2122
AbstractWe discuss a necessary condition for inclusion relations of weak type discrete Morrey spaces which can be seen as an extensionof the results in [H. Gunawan, E. Kikianty and C. Schwanke,Discrete Morrey spaces and their inclusion properties,Math. Nachr. 291 2018, 8–9, 1283–1296]and[D. D. Haroske and L. Skrzypczak,Morrey sequence spaces: Pitt’s theorem and compact embeddings,Constr. Approx. 51 2020, 3, 505–535].We also prove a proper inclusion from weak type discrete Morrey spaces into discrete Morrey spaces.In addition, we give a necessary condition for this inclusion.Some connections between the inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.
On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier seriesNagy, Károly; Salim, Mohamed
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2109
AbstractGoginava proved that the maximal operator σα,*{\sigma^{\alpha,*}}(0<α<1{0<\alpha<1}) of two-dimensional Marcinkiewicz type (C,α){(C,\alpha)}means is bounded from the two-dimensional dyadic martingale Hardy space Hp(G2){H_{p}(G^{2})}to the space Lp(G2){L^{p}(G^{2})}for p>22+α{p>\frac{2}{2+\alpha}}. Moreover, he showed that assumption p>22+α{p>\frac{2}{2+\alpha}}is essential for the boundedness of the maximal operator σα,*{\sigma^{\alpha,*}}. It was shown that at the point p0=22+α{p_{0}=\frac{2}{2+\alpha}}the maximal operator σα,*{\sigma^{\alpha,*}}isbounded from the dyadic Hardy space H2/(2+α)(G2){H_{2/(2+\alpha)}(G^{2})}to the space weak-L2/(2+α)(G2){L^{2/(2+\alpha)}(G^{2})}. The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σα,*{\sigma^{\alpha,*}}means (0<α<1{0<\alpha<1}) in the endpoint case p0=22+α{p_{0}=\frac{2}{2+\alpha}}. In particular, the optimal condition on the weights is given which provides the boundedness from H2/(2+α)(G2){H_{2/(2+\alpha)}(G^{2})}to L2/(2+α)(G2){L^{2/(2+\alpha)}(G^{2})}.Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H2/(2+α)(G2){H_{2/(2+\alpha)}(G^{2})}.
Several characterizations of Bessel functions and their applicationsNahid, Tabinda; Ali, Mahvish
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2108
AbstractThe present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.
The incomplete exponential pRp (α,β;z) function with applicationsPal, Ankit; Kumar Jana, Ranjan; Khammash, Ghazi S.; Shukla, Ajay K.
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2112
AbstractIn this paper, we are motivated to establish the generalization of a Rqp(α,β;z){{}_{p}R_{q}(\alpha,\beta;z)}function in terms of incomplete exponential functions and obtain some properties of an incomplete exponential Rqp(α,β;z){{}_{p}R_{q}(\alpha,\beta;z)}function. Further we give generating relations for incomplete exponential Rqp(α,β;z){{}_{p}R_{q}(\alpha,\beta;z)}function. Some applications related to ground water pumping (hydrology) modelling and probability theory are also discussed.
When are multiplicative semi-derivations additive?Siddeeque, Mohammad Aslam; Khan, Nazim
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2121
AbstractLet R be an associative ring. A multiplicative semi-derivation d is a map on R satisfyingd(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y) and d(g(x))=g(d(x)){d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))}for all x,y∈R{x,y\in R}, where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on Mn(ℂ){M_{n}(\mathbb{C})}, the algebra of all n×n{n\times n}matrices over the field ℂ{\mathbb{C}}of complex numbers, is an additive derivation.
On a structure of the set of positive solutions to second-order equations with a super-linear non-linearityŠremr, Jiří
2022 Georgian Mathematical Journal
doi: 10.1515/gmj-2021-2117
AbstractWe study the existence and multiplicity of positive solutions to the periodic problemu′′=p(t)u-q(t,u)u+f(t);u(0)=u(ω),u′(0)=u′(ω),u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega),where p,f∈L([0,ω])p,f\in L([0,\omega])and q:[0,ω]×R→Rq\colon[0,\omega]\times\mathbb{R}\to\mathbb{R}is a Carathéodory function.By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.