Volume 29, Issue 1

For the perturbed controlled nonlinear delay functional differential equation with the mixed initial condition a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of main interval. In the formula the effects of perturbations of the delay parameters containing in the phase coordinates and controls, the initial vector, the initial and control functions are detected.

This paper deals with the existence and multiplicity of solutions for the p ⁢ ( x ) p(x) -Laplacian Robin problem without the well-known Ambrosetti–Rabinowitz type growth conditions. By means of the variational approach (with the Cerami condition), existence and multiplicity results of solutions are established under weaker conditions.

We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

We discuss a necessary condition for inclusion relations of weak type discrete Morrey spaces which can be seen as an extension of 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.

Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory.

Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by p ⁢ ( x ) p(x) -biharmonic operator. Our technical approach is based on variational methods. Some applications illustrate the obtained results. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

Goginava proved that the maximal operator σ α , * {\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 H p ⁢ ( G 2 ) {H_{p}(G^{2})} to the space L p ⁢ ( G 2 ) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}} . Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}} . It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) {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 σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0<\alpha<1} ) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} .

The 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.

In this paper, we are motivated to establish the generalization of a R q p ⁢ ( α , β ; z ) {{}_{p}R_{q}(\alpha,\beta;z)} function in terms of incomplete exponential functions and obtain some properties of an incomplete exponential R q p ⁢ ( α , β ; z ) {{}_{p}R_{q}(\alpha,\beta;z)} function. Further we give generating relations for incomplete exponential R q p ⁢ ( 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.

We study the higher integrability of weights satisfying a reverse Hölder inequality ( ⨏ I u β ) 1 β ≤ B ⁢ ( ⨏ I u α ) 1 α {\biggl{(}\fint_{I}u^{\beta}\biggr{)}^{\frac{1}{\beta}}}\leq B{\biggl{(}\fint_{I}u^{\alpha}\biggr{)}^{\frac{1}{\alpha}}} for some B > 1 B>1 and given α < β \alpha<\beta , in the limit cases when α ∈ { - ∞ , 0 } \alpha\in\{-\infty,0\} and/or β ∈ { 0 , + ∞ } \beta\in\{0,+\infty\} . The results apply to the Gehring and Muckenhoupt weights and their corresponding limit classes.

Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ ( x ⁢ y ) = d ⁢ ( x ) ⁢ g ⁢ ( y ) + x ⁢ d ⁢ ( 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 M n ⁢ ( 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.

We study the existence and multiplicity of positive solutions to the periodic problem u ′′ = 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 → R q\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.

In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .