## Publications

381 results found

Kassymov A, Ruzhansky M, Torebek BT, 2023, Rayleigh–Faber–Krahn, Lyapunov and Hartmann–Wintner Inequalities for Fractional Elliptic Problems, *Mediterranean Journal of Mathematics*, Vol: 20, ISSN: 1660-5446

In this paper, in the cylindrical domain, we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh–Faber–Krahn inequality. Also, we give Lyapunov and Hartmann–Wintner inequalities for the fractional elliptic boundary value problem.

Huang J, Ruzhansky M, Zhang Q,
et al., 2023, Intrinsic Image Transfer for Illumination Manipulation., *IEEE Trans Pattern Anal Mach Intell*, Vol: 45, Pages: 7444-7456

This article presents a novel intrinsic image transfer (IIT) algorithm for image illumination manipulation, which creates a local image translation between two illumination surfaces. This model is built on an optimization-based framework composed of illumination, reflectance and content photo-realistic losses, respectively. Each loss is first defined on the corresponding sub-layers factorized by an intrinsic image decomposition and then reduced under the well-known spatial-varying illumination illumination-invariant reflectance prior knowledge. We illustrate that all losses, with the aid of an "exemplar" image, can be directly defined on images without the necessity of taking an intrinsic image decomposition, thereby giving a closed-form solution to image illumination manipulation. We also demonstrate its versatility and benefits to several illumination-related tasks: illumination compensation, image enhancement and tone mapping, and high dynamic range (HDR) image compression, and show their high-quality results on natural image datasets.

Rottensteiner D, Ruzhansky M, 2023, An update on the L<sup>p</sup> - L<sup>q</sup> norms of spectral multipliers on unimodular Lie groups, *Archiv der Mathematik*, Vol: 120, Pages: 507-520, ISSN: 0003-889X

This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on Lp-Lq norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians, and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the Lp-Lq norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by the authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the Lp-Lq norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators.

Zhang R, Kumar V, Ruzhansky M, 2023, Liouville type theorems for subelliptic systems on the Heisenberg group with general nonlinearity

In this paper, we establish Liouville type results for semilinear subellipticsystems associated with the sub-Laplacian on the Heisenberg group$\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. Themain technique of the proof is the method of moving planes combined with someintegral inequalities replacing the role of maximum principles. As a specialcase, we obtain the Liouville theorem for the Lane-Emden system on theHeisenberg group $\mathbb{H}^{n}$, which also appears to be a new result in theliterature.

Huang J, Wang H, Wang X,
et al., 2023, Semi-Sparsity for Smoothing Filters., *IEEE Trans Image Process*, Vol: PP

In this paper, we propose a semi-sparsity smoothing method based on a new sparsity-induced minimization scheme. The model is derived from the observations that semi-sparsity prior knowledge is universally applicable in situations where sparsity is not fully admitted such as in the polynomial-smoothing surfaces. We illustrate that such priors can be identified into a generalized L0-norm minimization problem in higher-order gradient domains, giving rise to a new "feature-aware" filter with a powerful simultaneous-fitting ability in both sparse singularities (corners and salient edges) and polynomial-smoothing surfaces. Notice that a direct solver to the proposed model is not available due to the non-convexity and combinatorial nature of L0-norm minimization. Instead, we propose to solve it approximately based on an efficient half-quadratic splitting technique. We demonstrate its versatility and many benefits to a series of signal/image processing and computer vision applications.

Ghosh A, Ruzhansky M, 2023, Sparse bounds for oscillating multipliers on stratified groups

In this article, we address sparse bounds for a class of spectral multipliersthat include oscillating multipliers on stratified Lie groups. Our results canbe applied to obtain weighted bounds for general Riesz means and for solutionsof dispersive equations.

Ruzhansky M, Sebih ME, Tokmagambetov N, 2023, Schrödinger equation with singular position dependent mass

We consider the Schr\"odinger equation with singular position dependenteffective mass and prove that it is very weakly well posed. A uniqueness resultis proved in an appropriate sense, moreover, we prove the consistency with theclassical theory. In particular, this allows one to consider Delta-like or moresingular masses.

Ruzhansky M, Sebih ME, Tokmagambetov N, 2023, Heat equation with singular thermal conductivity

In this paper, we study the heat equation with an irregular spatiallydependent thermal conductivity coefficient. We prove that it has a solution inan appropriate very weak sense. Moreover, the uniqueness result and consistencywith the classical solution if the latter exists are shown. Indeed, we allowthe coefficient to be a distribution with a toy example of a Delta-function.

Chatzakou M, Ruzhansky M, 2023, Revised logarithmic Sobolev inequalities of fractional order

In this short note we prove the logarithmic Sobolev inequality withderivatives of fractional order on $\mathbb{R}^n$ with an explicit expressionfor the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ wehave the inequality \[ \int_{\mathbb{R}^n}|f(x)|^2 \log \left(\frac{|f(x)|^2}{\|f\|^{2}_{L^2(\mathbb{R}^n)}}\right)\,dx+\frac{n}{s}(1+\loga)\|f\|_{L^2(\mathbb{R}^n)}^{2}\leqC(n,s,a)\|(-\Delta)^{s/2}f\|^{2}_{L^2(\mathbb{R}^n)} \] with an explicit$C(n,s,a)$ depending on $a$, the order $s$, and the dimension $n$, andinvestigate the behaviour of $C(n,s,a)$ for large $n$. Notably, for large $n$and when $s=1$, the constant $C(n,1,a)$ is asymptotically the same as the sharpconstant of Lieb and Loss. Moreover, we prove a similar type inequality forfunctions $f \in L^q(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n)$ whenever $1<p<n$and $p<q\leq \frac{p(n-1)}{n-p}$.

Cardona D, Grajales B, Ruzhansky M, 2023, On the sharpness of Strichartz estimates and spectrum of compact Lie groups

We prove Strichartz estimates on any compact connected simple Lie group. Inthe diagonal case of Bourgain's exponents $p=q,$ we substantially improve theregularity orders showing the existence of indices $s<s_{0}(d)$ below theSobolev exponent $s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}.$ Motivated by the recentprogress in the field, in the spirit of the analytical number theory methodsdeveloped by Bourgain in the analysis of periodic Schr\"odinger equations, welink the problem of finding Strichartz estimates on compact Lie groups with theproblem of counting the number of representations $r_{s,2}(R)$ of an integernumber $R$ as a sum of $s$ squares, and then, we implicitly use the very wellknown bounds for $r_{s,2}(R)$ from the Hardy-Littlewood-Ramanujan circlemethod. In our analysis, we explicitly compute the parametrisation of thespectrum of the Laplacian (modulo a factor depending on the geometry of thegroup) in terms of sums of squares. As a byproduct, our approach also yieldsexplicit expressions for the spectrum of the Laplacian on all compact connectedsimple Lie groups, providing also a number of sharp results for Strichartzestimates in the borderline case $p=2.$

Kumar V, Ruzhansky M, Zhang H-W, 2023, Smoothing properties of dispersive equations on non-compact symmetric spaces

We establish the Kato-type smoothing property, i.e., global-in-time smoothingestimates with homogeneous weights, for the Schr\"odinger equation onRiemannian symmetric spaces of non-compact type and general rank. These form arich class of manifolds with nonpositive sectional curvature and exponentialvolume growth at infinity, e.g., hyperbolic spaces. We achieve it by provingthe Stein-Weiss inequality and the resolvent estimate of the correspondingFourier multiplier, which are of independent interest. Moreover, we extend thecomparison principles to symmetric spaces and deduce different types ofsmoothing properties for the wave equation, the Klein-Gordon equation, therelativistic and general orders Schr\"odinger equations. In particular, weobserve that some smoothing properties, which are known to fail on theEuclidean plane, hold on the hyperbolic plane.

Cardona D, Chatzakou M, Delgado J, et al., 2023, Degenerate Schrödinger equations with irregular potentials

In this work we investigate a class of degenerate Schr\"odinger equationsassociated to degenerate elliptic operators with irregular potentials on $\Ran$by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. Weestablish the well-posedness for the corresponding degenerate Schr\"odinger anddegenerate parabolic equations. When the subelliticity is available on thedegenerate elliptic operator we deduce spectral properties for a class ofdegenerate Hamiltonians. We also study the $L^p$ mapping properties foroperators with symbols in the $S(m^{-\beta},g)$ classes in the spirit ofclassical Fefferman's $L^p$-bounds for the $(\rho, \delta)$ calculus. Finally,within our $S(m,g)$-classes, sharp $L^p$-estimates and Schatten properties forSchr\"odinger operators for H\"ormander sums of squares are also investigated.

Wang X, Huang J, Chatzakou M, et al., 2023, A Light-weight CNN Model for Efficient Parkinson's Disease Diagnostics

In recent years, deep learning methods have achieved great success in variousfields due to their strong performance in practical applications. In thispaper, we present a light-weight neural network for Parkinson's diseasediagnostics, in which a series of hand-drawn data are collected to distinguishParkinson's disease patients from healthy control subjects. The proposed modelconsists of a convolution neural network (CNN) cascading to long-short-termmemory (LSTM) to adapt the characteristics of collected time-series signals. Tomake full use of their advantages, a multilayered LSTM model is firstly used toenrich features which are then concatenated with raw data and fed into ashallow one-dimensional (1D) CNN model for efficient classification.Experimental results show that the proposed model achieves a high-qualitydiagnostic result over multiple evaluation metrics with much fewer parametersand operations, outperforming conventional methods such as support vectormachine (SVM), random forest (RF), lightgbm (LGB) and CNN-based methods.

Cardona D, Ruzhansky M, 2023, Oscillating singular integral operators on compact Lie groups revisited, *Mathematische Zeitschrift*, Vol: 303, ISSN: 0025-5874

Fefferman (Acta Math 24:9–36, 1970, Theorem 2′) has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian Δ , namely, operators of the form Tθ(-Δ):=(1-Δ)-nθ4ei(1-Δ)θ2,0≤θ<1.The aim of this work is to extend Fefferman’s result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace–Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.

Borikhanov MB, Ruzhansky M, Torebek BT, 2023, Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation, *Fractional Calculus and Applied Analysis*, Vol: 26, Pages: 111-146, ISSN: 1311-0454

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities D0|tαu+(-Δ)psu=γ|u|m-1u+μ|u|q-2u,γ,μ∈R,m>0,q>1, involving time-fractional Caputo derivative D0|tα and space-fractional p-Laplacian operator (-Δ)ps. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of γ, μ, m and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

Cobos SG, Restrepo JE, Ruzhansky M, 2023, $L^p-L^{q}$ estimates for non-local heat and wave type equations on locally compact groups

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimatesfor the solutions of heat and wave type equations (new in this setting) on alocally compact separable unimodular group $G$ by using a non-localintegro-differential operator in time and any positive left invariant operator(maybe unbounded and either with discrete or continuous spectrum) on $G$. Wealso provide asymptotic estimates (large-time behavior) for the solutions whichin some cases can be claimed to be sharp. Illustrative examples are given forseveral groups.

Ruzhansky M, Shaimardan S, Tokmagambetov N, 2023, Some estimates for Mittag-Leffler function in quantum calculus and applications

The study of the Mittag-Leffler function and its various generalizations hasbecome a very popular topic in mathematics and its applications. In the presentpaper we prove the following estimate for the $q$-Mittag-Leffler function:\begin{eqnarray*} \frac{1}{1+\Gamma_q\left(1-\alpha\right)z}\leqe_{\alpha,1}\left(-z;q\right)\leq\frac{1}{1+\Gamma_q\left(\alpha+1\right)^{-1}z}.\end{eqnarray*} for all $0 < \alpha < 1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct andinverse problems for time-fractional pseudo-parabolic equations in quantumcalculus for a large class of positive operators with discrete spectrum.

Chatzakou M, Restrepo JE, Ruzhansky M, 2023, Heat and wave type equations with non-local operators, II. Hilbert spaces and graded Lie groups

We study heat and wave type equations on a separable Hilbert space$\mathcal{H}$ by considering non-local operators in time with any positivedensely defined linear operator with discrete spectrum. We show the explicitrepresentation of the solution and analyse the time-decay rate in a scale ofsuitable Sobolev space. We perform similar analysis on multi-term heat andmulti-wave type equations. The main tool here is the Fourier analysis which canbe developed in a separable Hilbert space based on the linear operatorinvolved. As an application, the same Cauchy problems are considered andanalysed in the setting of a graded Lie group. In this case our analysis relieson the group Fourier analysis. An extra ingredient in this framework allows, inthe case of heat type equations, to establish $L^p$-$L^q$ estimates for$1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Liegroup groups. Examples and applications of the developed theory are given,either in terms of self-adjoint operators on compact or non-compact manifolds,or in the case of particular settings of graded Lie groups. The results of thispaper significantly extend in different directions the results of Part I, whereoperators on compact Lie groups were considered. We note that the resultsobtained in this paper are also new already in the Euclidean setting of$\mathbb{R}^n$.

Kassymov A, Ruzhansky M, Suragan D, 2023, Hardy inequalities on metric measure spaces, III: The case q ≤ p ≤ 0 and applications, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 479, ISSN: 1364-5021

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q≤p<0) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475, 20180310 (doi:10.1098/rspa.2018.0310); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477, 20210136 (doi:10.1098/rspa.2021.0136)), which treated the cases 1<p≤q<∞ and p>q, respectively.

Cardona D, Delgado J, Grajales B, et al., 2023, Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria

Let $A$ and $B$ be invariant linear operators with respect to a decomposition$\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces offinite dimension. We give necessary and sufficient conditions for thecontrollability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms ofthe (global) matrix-valued symbols $\sigma_A$ and $\sigma_B$ of $A$ and $B,$respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$.Then, we present some applications including the controllability of the Cauchyproblem on compact manifolds for elliptic operators and the controllability offractional diffusion models for H\"ormander sub-Laplacians on compact Liegroups. We also give conditions for the controllibility of wave andSchr\"odinger equations in these settings.

Cardona D, Delgado J, Ruzhansky M, 2023, Boundedness of the dyadic maximal function on graded Lie groups

Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C.Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function\begin{equation*} \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right|\end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $d\sigma(y)$is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove ananalogue of this result on arbitrary graded Lie groups. More precisely, to anyfinite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$we associate the corresponding dyadic maximal function$\mathcal{M}_D^{d\sigma}$ using the homogeneous structure of the group. Then,we prove a criterion in terms of the order (at zero and at infinity) of thegroup Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to afixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of$\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1<p\leq \infty.$

Karimov E, Ruzhansky M, Toshtemirov B, 2023, Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative, *Mathematical Methods in the Applied Sciences*, Vol: 46, Pages: 54-70, ISSN: 0170-4214

In a rectangular domain, a boundary-value problem is considered for a mixed equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial-boundary problems for the heat equation with the regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.

Cardona D, Chatzakou M, Ruzhansky M, et al., 2023, Schatten-von Neumann properties for Hörmander classes on compact Lie groups

Let $G$ be a compact Lie group of dimension $n.$ In this work we characterisethe membership of classical pseudo-differential operators on $G$ in the traceclass ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals$S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schattencharacterisations of elliptic pseudo-differential operators of$(\rho,\delta)$-type for the large range $0\leq \delta<\rho\leq 1.$ Additionalnecessary and sufficient conditions are given in terms of the matrix-valuedsymbols of the operators, which are global functions on the phase space$G\times \widehat{G},$ with the momentum variables belonging to the unitarydual $\widehat{G}$ of $G$. In terms of the parameters $(\rho,\delta),$ on thetorus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing theexistence of atypical operators in the exotic class$\Psi^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ belonging to all theSchatten ideals. Additional order criteria are given in the setting ofclassical pseudo-differential operators. We present also some open problems inthis setting.

Cardona D, Ruzhansky M, 2023, Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups, *Complex Variables and Elliptic Equations*, ISSN: 1747-6933

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the (Formula presented.) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case (Formula presented.) Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated.

Ghosh S, Kumar V, Ruzhansky M, 2023, Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups, *Mathematische Annalen*, ISSN: 0025-5831

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional p-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional p-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case p= 2 with G being the Heisenberg group.

Kumar V, Ruzhansky M, 2023, L<sup>p</sup>-L<sup>q</sup> Boundedness of (k, a)-Fourier Multipliers with Applications to Nonlinear Equations, *International Mathematics Research Notices*, Vol: 2023, Pages: 1073-1093, ISSN: 1073-7928

The (k, a)-generalised Fourier transform is the unitary operator defined using the a-deformed Dunkl harmonic oscillator. The main aim of this paper is to prove Lp-Lq boundedness of (k, a)-generalised Fourier multipliers. To show the boundedness, we first establish Paley inequality and Hausdorff-Young-Paley inequality for (k, a)-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.

Shaimardan S, Karimov E, Ruzhansky M, et al., 2022, The Prabhakar fractional $q$-integral and $q$-differential operators, and their properties

In this paper, we have introduced the Prabhakar fractional $q$-integral and$q$-differential operators. We first study the semi-group property of thePrabhakar fractional $q$-integral operator, which allowed us to introduce thecorresponding $q$-differential operator. Formulas for compositions of$q$-integral and $q$-differential operators are also presented. We show theboundedness of the Prabhakar fractional $q$-integral operator in the class of$q$-integrable functions.

Ruzhansky M, Shriwastawa A, Tiwari B, 2022, Hardy inequalities on metric measure spaces, IV: The case $p=1$

In this paper, we investigate the two-weight Hardy inequalities on metricmeasure space possessing polar decompositions for the case $p=1$ and $1 \leq q<\infty.$ This result complements the Hardy inequalities obtained in \cite{RV}in the case $1< p\le q<\infty.$ The case $p=1$ requires a different argumentand does not follow as the limit of known inequalities for $p>1.$ As abyproduct, we also obtain the best constant in the established inequality. Wegive examples obtaining new weighted Hardy inequalities on homogeneous Liegroups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case$p=1$ and $1\le q<\infty.$

Ruzhansky M, Shaimardan S, 2022, Direct and inverse source problems for heat equation in quantum calculus

In this paper we explore the weak solutions of the Cauchy problem and aninverse source problem for the heat equation in the quantum calculus,formulated in abstract Hilbert spaces. For this we use the Fourier seriesexpansions. Moreover, we prove the existence, uniqueness and stability of theweak solution of the inverse problem with a final determination condition. Wegive some examples such as the q-Sturm-Liouville problem, the q-Besseloperator, the q-deformed Hamiltonian, the fractional Sturm-Liouville operator,and the restricted fractional Laplacian, covered by our analysis.

Karimov E, Ruzhansky M, Shaimardan S, 2022, Fractional Cauchy problems associated with the bi-ordinal Hilfer fractional $q$-derivative

To study the existence and uniqueness of solutions to Cauchy-type problemsfor fractional q-difference equations with the bi-ordinal Hilfer fractionalq-derivative which is an extension of the Hilfer fractional q-derivative. Anapproach is based on the equivalence of the nonlinear Cauchy-type problem witha nonlinear Volterra q-integral equation of the second kind. Applying an analogof Banach fixed point theorem we prove the uniqueness and the existence of thesolution. Moreover, we present an explicit solution to the q-analog of theCauchy problem for the linear case.

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