Article
Mathematics, Interdisciplinary Applications
Vladimir E. Fedorov, Marko Kostic, Tatyana A. Zakharova
Summary: The fractional powers of generators for analytic operator semigroups are used to prove the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. By constructing fractional powers A(?) for an operator A, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation with Gerasimov-Caputo fractional derivatives. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics
Luigi Appolloni, Simone Secchi
Summary: This study investigates the existence of solutions to the fractional nonlinear Schrodinger equation in the Sobolev space, proving the existence of a ground state solution and a multiplicity result in the radially symmetric case.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Michael W. Frazier, Igor E. Verbitsky
Summary: In this study, we investigate the Green's operator of order alpha and the corresponding integral equation solution for a given open set Omega and locally finite measure omega. By analyzing the norm conditions and different values of alpha, we reveal the conditions and properties for the existence of solutions.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics
Shubin Yu, Chunlei Tang, Ziheng Zhang
Summary: In this paper, the existence of solutions for fractional nonlinear Schrodinger equations with prescribed L2-norm constraint is studied. The results show that positive normalized solutions can be obtained for any a > 0, when 2 < p < 2 + 4s/N and R-N \omega is contained in a small ball. Moreover, if omega is the complement of the unit ball in R-N, the existence and multiplicity of radial normalized solutions can also be established for any a > 0.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Engineering, Mechanical
Liangwei Zeng, Milivoj R. Belic, Dumitru Mihalache, Jincheng Shi, Jiawei Li, Siqi Li, Xiaowei Lu, Yi Cai, Jingzhen Li
Summary: We have demonstrated the existence of various types of gap localized modes, including one- and two-dimensional solitons and soliton clusters, as well as vortex modes, in optical media with saturable Kerr nonlinearity and fractional diffraction. We found that soliton clusters with different peak numbers can be stable, and the localized modes at the center of the first and second band gaps are stable. The stability of these modes is confirmed through linear stability analysis and numerical simulations.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
Giovanni Covi, Keijo Monkkonen, Jesse Railo, Gunther Uhlmann
Summary: We study the inverse problem for the fractional Schrodinger equation with a local perturbation by a linear PDO of lower order. We show that the coefficients of the PDO can be uniquely recovered from the exterior Dirichlet-to-Neumann map associated to the perturbed equation. Our study generalizes recent results for zeroth and first order perturbations to higher order perturbations.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Zhongyuan Liu, Ziying Liu, Wenhuan Xu
Summary: This paper deals with the existence of solutions for a slightly subcritical Schrödinger equation with a non-power nonlinearity.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Engineering, Mechanical
Shreya Mitra, Sujoy Poddar, A. Ghose-Choudhury, Sudip Garai
Summary: Using conformable fractional space and time derivatives, a novel class of traveling wave solutions for the Hirota-Schrodinger (HS) equation and the nonlinear Schrodinger equation (NLSE) with quadratic-cubic nonlinearity (QCN) has been obtained. The obtained solutions show interesting dispersive corrections to the propagating waves, with different fractional powers displaying phase shifting, singularity and a flattening of the propagating pulse. Bright one-soliton and singular soliton solutions for the NLSE with QCN have also been discussed. These findings are likely to have significant relevance in the propagation of optical pulses in a highly nonlinear dispersive media.
NONLINEAR DYNAMICS
(2022)
Article
Materials Science, Multidisciplinary
Asif Khan, Amir Ali, Shabir Ahmad, Sayed Saifullah, Kamsing Nonlaopon, Ali Akgul
Summary: In this article, the behaviour of the time fractional nonlinear Schrodinger equation under two different operators are investigated. Numerical and analytical solutions are obtained using the modified double Laplace transform. The error analysis shows that the system depends primarily on time, with small errors observed for small time values. The efficiency of the proposed scheme is verified with examples and further analyzed graphically and numerically.
RESULTS IN PHYSICS
(2022)
Article
Physics, Multidisciplinary
Weijun Chen, Cheng Lian, Yuang Luo
Summary: This study investigates the interaction of Airy beams modeled by a fractional nonlinear cubic-quintic Schrodinger equation. It was found that when the interval between two Airy beams is large enough, they exhibit different interactions based on phase, either attracting or repelling each other. For smaller intervals, single breathing solitons and symmetric breathing soliton pairs are formed. Additionally, the quintic defocusing strength modulates the interaction of Airy beams, affecting factors such as width and repulsion between the beams.
Article
Mathematics, Applied
Qing Yang, Chuanzhi Bai
Summary: In this paper, we investigate a fractional Kirchhoff type problem and prove the existence of sign-changing ground state solutions using constraint variational method and analysis techniques.
Article
Mathematics, Applied
Lidia Aceto, Paolo Novati
Summary: This paper investigates the approximation of fractional powers of self-adjoint positive operators. It improves the existing error estimates for the scalar case and extends the analysis to operators. Numerical experiments are conducted to validate the obtained estimates.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Lidia Aceto, Paolo Novati
Summary: This paper investigates rational approximations to the fractional powers of self-adjoint positive operators generated from the Gauss-Laguerre rules. Practical error estimates are derived for a priori selection of Laguerre points, and numerical experiments are conducted to demonstrate the effectiveness of the approaches and the reliability of the estimates.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Pedro J. J. Miana, Natalia Romero
Summary: In this paper, the Catalan sequence is defined and a generating function for it is introduced. The powers of the generating function are then expressed in terms of the Catalan triangle numbers. New formulas involving the spectrum of c*(j) and the expression of c-*(j) in terms of Catalan polynomials are derived. Finally, some examples are provided to illustrate the results and future research directions are suggested.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Petr N. Vabishchevich
Summary: We studied the Cauchy problem for a first-order evolutionary equation with fractional powers of an operator and developed computational algorithms based on approximations of operator functions. The results show that using exponent splitting schemes can provide approximate solutions in time. Numerical experiments were conducted to verify the effectiveness of this method.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2023)
Article
Mathematics, Applied
Tomas Caraballo, Alexandre N. Carvalho, Jose A. Langa, Alexandre N. Oliveira-Sousa
Summary: In this paper, we study the stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove the robustness of nonuniform hyperbolicity for linear evolution processes and provide conditions for the uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. An example of an evolution process in a Banach space that exhibits nonuniform exponential dichotomy is presented, and the permanence of nonuniform hyperbolicity under perturbations is studied. Finally, we prove the persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.
ASYMPTOTIC ANALYSIS
(2022)
Article
Mathematics, Applied
M. C. Bortolan, A. N. Carvalho, J. A. Langa, G. Raugel
Summary: This work investigates Morse-Smale semigroups under nonautonomous perturbations and introduces the concept of Morse-Smale evolution processes of hyperbolic type. The stability of the phase diagram of the attractors is proven, with intersecting stable and unstable manifolds. The complete proofs of local and global lambda-lemmas in the infinite-dimensional case, originally due to D. Henry, are included here for completeness.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Maykel Belluzi, Marcelo J. D. Nascimento, Karina Schiabel
Summary: The study focuses on fractional powers of a cascade system of partial differential equations and the linear operators associated with them. It discusses the local solvability of the fractional equation with subcritical nonlinearity, using a cascade system of Schrodinger equation as an example. A connection between the fractional system and the original system is established, along with the proof of convergence of linear semigroups obtained by the fractional power operator to the original linear semigroup as the power alpha approaches 1.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Everaldo M. Bonotto, Marcelo J. D. Nascimento, Eric B. Santiago
Summary: This paper investigates the long-time dynamics of solutions of an evolution system and proves the local and global well-posedness using the uniform sectorial operators theory. Additionally, the existence, regularity, and upper semicontinuity of pullback attractors are demonstrated.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Maykel Belluzi, Tomas Caraballo, Marcelo J. D. Nascimento, Karina Schiabel
Summary: In this paper, we investigate nonautonomous semilinear parabolic problems with time-dependent linear operators and obtain regularity results and estimates for the solutions in stronger function spaces. We then apply these results to a nonautonomous reaction-diffusion equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Maykel Belluzi, Tomas Caraballo, Marcelo J. D. Nascimento, Karina Schiabel
Summary: In this paper, we study the local mild solution of a semilinear parabolic problem and prove that it is actually a strong solution by exploring properties of the semigroup generated by the operator -A. We also apply these results to a reaction-diffusion equation with a nonlinearity satisfying a polynomial growth condition, and determine the parameter range for which the problem still has a strong solution.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Mathematics, Applied
Alexandre N. Carvalho, Arthur C. Cunha, Jose A. Langa, James C. Robinson
Summary: We provide a simple proof of a result by Mane (1981) that states a compact subset A of a Banach space, which is negatively invariant under a map S, is finite-dimensional if DS(x) = C(x) + L(x) where C is compact and L is a contraction. Additionally, we demonstrate that if S is both compact and differentiable, A is finite-dimensional. Furthermore, we present some results concerning the (box-counting) dimension of such sets assuming a 'smoothing property' and the Kolmogorov epsilon-entropy of the embedding of Z into X. Finally, we apply these results to an abstract semilinear parabolic equation and the two-dimensional Navier-Stokes equations on a periodic domain.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Flank D. M. Bezerra, Alexandre N. Carvalho, Lucas A. Santos
Summary: In this paper, the well-posedness of the Cauchy problem associated with a third-order evolution equation is discussed. The equation involves mathematical concepts and conditions such as separable Hilbert space, unbounded sectorial operator, etc.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Statistics & Probability
Tomas Caraballo, Jose A. Langa, Alexandre N. Carvalho, Alexandre N. Oliveira-Sousa
Summary: In this work, the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems are studied. The existence and permanence of unstable sets of hyperbolic solutions are studied first, and then used to establish the lower semicontinuity of nonautonomous random attractors and the persistence of the gradient structure under nonautonomous random perturbations. The abstract results are applied to a stochastic differential equation and in a damped wave equation with a perturbation on the damping.
STOCHASTICS AND DYNAMICS
(2022)
Article
Mathematics
Flank D. M. Bezerra, Marcelo J. D. Nascimento
Summary: This article focuses on the long-time dynamics of a class of semilinear thermoelastic systems with variable thermal coefficient. The main results establish the existence, regularity, and upper semicontinuity of the pullback attractors with respect to the coefficients of thermal expansion of the material under nonlinear forces and suitable conditions of growth and dissipativity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Alexandre N. Carvalho, Luciano R. N. Rocha, Jose A. Langa, Rafael Obaya
Summary: In this work, we investigate the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation. We provide a non-autonomous structure that completely describes the dynamics of this model and give a Morse decomposition for the skew-product attractor. Our findings indicate that the complexity of the isolated invariant sets is related to the complexity of the attractor, and when beta is asymptotically almost periodic, the isolated invariant sets will be almost periodic hyperbolic global solutions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Mathematics, Applied
Jakub Banaskiewicz, Alexandre N. Carvalho, Juan Garcia-Fuentes, Piotr Kalita
Summary: This paper studies the dynamics of slowly non-dissipative systems using the approach of unbounded attractors. It provides abstract results on the existence and properties of unbounded attractors, as well as the properties of unbounded omega-limit sets in slowly non-dissipative settings. The paper also develops the pullback non-autonomous counterpart of the unbounded attractor theory.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Physics, Mathematical
Tomasz Dlotko
Summary: A growing interest in considering hybrid systems of equations describing more complicated physical phenomena has been observed in the past 10 years. This article discusses the existence-uniqueness of solutions to the Navier-Stokes-Cahn-Hilliard system using the semigroup approach, and explains the limitation of maximal regularity of local solutions imposed by the chosen boundary conditions.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Heraclio Lopez-Lazaro, Marcelo J. D. Nascimento, Obidio Rubio
Summary: In this article, a semilinear mathematical model with delay term for thermal conduction problems defined on a one-dimensional moving boundary domain is studied. The goals of this work are to prove the existence, regularity, and finite fractal dimension of the pullback attractors on tempered universes that depend on a non-increasing function η: R -> (0, +infinity). (C) 2022 Elsevier Ltd. All rights reserved.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Physics, Mathematical
Tomas Caraballo, Alexandre N. Carvalho, Heraclio Lopez-Lazaro
Summary: We present a global modification of the Ladyzhenskaya equations for incompressible non-Newtonian fluids. The modification involves a cut-off function and an additional artificial smoothing dissipation term, and aims at comparative analysis between the modified and non-modified systems. We demonstrate the existence and regularity of weak solutions, the existence of global attractors, the estimation of fractal dimension of global attractors, and the relationship of the autonomous dynamics between the modified and non-modified systems.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Geunsu Choi, Mingu Jung, Sun Kwang Kim, Miguel Martin
Summary: This paper studies weak-star quasi norm attaining operators and proves that the set of such operators is dense in the space of bounded linear operators regardless of the choice of Banach spaces. It is also shown that weak-star quasi norm attaining operators have distinct properties from other types of norm attaining operators, although they may share some equivalent properties under certain conditions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maria Lorente, Francisco J. Martin-Reyes, Israel P. Rivera-Rios
Summary: In this paper, we provide quantitative one-sided estimates that recover the dependences in the classical setting. We estimate the one-sided maximal function in Lorentz spaces and demonstrate the applicability of the conjugation method for commutators in this context.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Fernando Cobos, Luz M. Fernandez-Cabrera
Summary: We provide a necessary and sufficient condition for the weak compactness of bilinear operators interpolated using the real method. However, this characterization does not hold for interpolated operators using the complex method.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ovgue Gurel Yilmaz, Sofiya Ostrovska, Mehmet Turan
Summary: The Lupas q-analogue Rn,q, the first q-version of the Bernstein polynomials, was originally proposed by A. Lupas in 1987 but gained popularity 20 years later when q-analogues of classical operators in approximation theory became a focus of intensive research. This work investigates the continuity of operators Rn,q with respect to the parameter q in both the strong operator topology and the uniform operator topology, considering both fixed and infinite n.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
M. Agranovsky, A. Koldobsky, D. Ryabogin, V. Yaskin
Summary: This article modifies the concept of polynomial integrability for even dimensions and proves that ellipsoids are the only convex infinitely smooth bodies satisfying this property.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Abel Komalovics, Lajos Molnar
Summary: In this paper, a parametric family of two-variable maps on positive cones of C*-algebras is defined and studied from various perspectives. The square roots of the values of these maps under a faithful tracial positive linear functional are considered as a family of potential distance measures. The study explores the problem of well-definedness and whether these distance measures are true metrics, and also provides some related trace characterizations. Several difficult open questions are formulated.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Frederic Bayart
Summary: The passage describes the construction of an operator on a separable Hilbert space that is 5-hypercyclic for all δ in the range (ε, 1) and is not 5-hypercyclic for all δ in the range (0, ε).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Helene Frankowska, Nikolai P. Osmolovskii
Summary: This paper investigates second-order optimality conditions for the minimization problem of a C2 function f on a general set K in a Banach space X. Both necessary and sufficient conditions are discussed, with the sufficiency condition requiring additional assumptions. The paper demonstrates the validity of these assumptions for the case when the set K is an intersection of sets described by smooth inequalities and equalities, such as in mathematical programming problems. The novelty of the approach lies in the arbitrary nature of the set K and the straightforward proofs.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ole Fredrik Brevig, Kristian Seip
Summary: This paper studies the Hankel operator on the Hardy space and discusses its minimal and maximal norms, as well as the relationship between the maximal norm and the properties of the function.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Alexander Meskhi
Summary: Rubio de Francia's extrapolation theorem is established for new weighted grand Morrey spaces Mp),lambda,theta w (X) with weights w beyond the Muckenhoupt Ap classes. This result implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. The necessary conditions for the boundedness of the Hardy-Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maud Szusterman
Summary: In this work, the necessary conditions on the structure of the boundary of a convex body K to satisfy all inequalities are investigated. A new solution for the 3-dimensional case is obtained in particular.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Rami Ayoush, Michal Wojciechowski
Summary: In this article, lower bounds for the lower Hausdorff dimension of finite measures are provided under certain restrictions on their quaternionic spherical harmonics expansions. This estimate is analogous to a result previously obtained by the authors for complex spheres.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
F. G. Abdullayev, V. V. Savchuk
Summary: This paper investigates the convergence and theorem proof of the Takenaka-Malmquist system and Fejer-type operator on the unit circle, and provides relevant results on the class of holomorphic functions representable by Cauchy-type integrals with bounded densities.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Sofiya Ostrovska, Mikhail I. Ostrovskii
Summary: This work aims to establish new results on the structure of transportation cost spaces. The main outcome of this paper states that if a metric space X contains an isometric copy of L1 in its transportation cost space, then it also contains a 1-complemented isometric copy of $1.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Pilar Rueda, Enrique A. Sanchez Perez
Summary: We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. Also, we demonstrate the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined, and provide concrete examples involving the relevant space L0(mu).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
