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Ulam-Hyers Stability Results For A Novel Nonlinear Nabla Caputo Fractional Variable-Order Difference System, Danfeng Luo, Thabet Abdeljawad, Zhiguo Luo
Ulam-Hyers Stability Results For A Novel Nonlinear Nabla Caputo Fractional Variable-Order Difference System, Danfeng Luo, Thabet Abdeljawad, Zhiguo Luo
Turkish Journal of Mathematics
This paper is concerned with a kind of nonlinear Nabla Caputo fractional difference system with variable-order and fixed initial valuable. By applying Krasnoselskii's fixed point theorem, we give some sufficient conditions to guarantee the existence results for the considered fractional discrete equations. In addition, we further consider the Ulam-Hyers stability by means of generalized Gronwall inequality. At last, two typical examples are delineated to demonstrate the effectiveness of our theoretical results.
An Improved Oscillation Criteria For First Order Dynamic Equations, Özkan Öcalan
An Improved Oscillation Criteria For First Order Dynamic Equations, Özkan Öcalan
Turkish Journal of Mathematics
In this work, we consider the first-order dynamic equations \begin{equation*} x^{\Delta }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}} \end{equation*} where $p\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) , $ $\tau \in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ and $\tau (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. When the delay term $\tau (t)$ is not necessarily monotone, we present a new sufficient condition for the oscillation of first-order delay dynamic equations on time scales.
On A Coupled Caputo Conformable System Of Pantograph Problems, Sabri T. M. Thabet, Sina Etemad, Shahram Rezapour
On A Coupled Caputo Conformable System Of Pantograph Problems, Sabri T. M. Thabet, Sina Etemad, Shahram Rezapour
Turkish Journal of Mathematics
Our fundamental purpose in the present manuscript is to explore existence and uniqueness criteria for a new coupled Caputo conformable system of pantograph problems in which for the first time, the given boundary conditions are formulated in the Riemann-Liouville conformable framework. To reach the mentioned aims, we utilize different analytical techniques in which some fixed point results play a vital role. In the final part, a simulative example is designed to cover the applicability aspects of theoretical findings available in this research manuscript from a numerical point of view.
Numerical Investigation Of Viscous Effects On The Nonlinear Burgers Equation, Muhammad Imran Khan, Abdul Rauf, Abdullah Shah
Numerical Investigation Of Viscous Effects On The Nonlinear Burgers Equation, Muhammad Imran Khan, Abdul Rauf, Abdullah Shah
Turkish Journal of Mathematics
This research article presents the numerical solution of the viscous Burgers equation. The diagonally implicit fractional step $\theta$ (DIFST) scheme is used for the time discretization and the space derivative is discretized by the conforming finite element method with quadrilateral mesh. The viscosity effect on the shock wave is calculated with an estimation of the $L_2$ error. For comparison of different time discretization schemes, three test problems are computed. The stability and accuracy of the schemes are given by estimating the $L_2$ error norm. Numerical simulation for one- and two-dimensional problems are given and illustrated graphically. The effect of the …
Infinitesimal Bending Of Dna Helices, Miroslav Maksimovic, Ljubica Velimirovic, Marija Najdanovic
Infinitesimal Bending Of Dna Helices, Miroslav Maksimovic, Ljubica Velimirovic, Marija Najdanovic
Turkish Journal of Mathematics
The mathematics of DNA molecules is often studied as a double helix model. This paper focuses on the modelling of infinitesimal bending of DNA helices. The paper checks the flexibility of DNA molecule, i.e. the flexibility of double helix in infinitesimal bending theory. Actually, first, we find infinitesimal bending field of an arbitrary curve on the helicoid, that leaves bent curves on the helicoid, and show that helix bending on the helicoid is not possible. Finally, we deal with the infinitesimal bending of helicoid, using PDEs. Visualization of infinitesimal bending was done in Mathematica.
A Study Of Impulsive Discrete Dirac System With Hyperbolic Eigenparameter, Turhan Köprübaşi
A Study Of Impulsive Discrete Dirac System With Hyperbolic Eigenparameter, Turhan Köprübaşi
Turkish Journal of Mathematics
Let $L$ denote the discrete Dirac operator generated in $\ell _{2}\left( %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ^{2}\right) $ by the difference operators of first order% \begin{equation*} \left\{ \begin{array}{cc} {\bigtriangleup y_{n}^{\left( 2\right) }+p_{n}y_{n}^{\left( 1\right) }=\lambda y_{n}^{(1)}} & \\ {\bigtriangleup y_{n-1}^{\left( 1\right) }+q_{n}y_{n}^{\left( 2\right) }=\lambda y_{n}^{(2)}}, \end{array} \text{ }n\in \mathbb{N} \setminus \left\{ k-1,k,k+1\right\} \right. \end{equation*} with boundary and impulsive conditions% \begin{equation*} \begin{array}{c} y_{0}^{(1)}=0\text{ }, \\ \\ \left( \begin{array}{c} y_{k+1}^{(1)} \\ y_{k+2}^{(2)}% \end{array}% \right) =\theta \left( \begin{array}{c} y_{k-1}^{(2)} \\ y_{k-2}^{(1)}% \end{array}% \right) ;\text{ }\theta =\left( \begin{array}{cc} \theta _{1} & \theta _{2} \\ \theta _{3} & \theta _{4}% …
A Generalization Of Parabolic Potentials Associated To Laplace-Bessel Differential Operator And Its Behavior In The Weighted Lebesque Spaces, Çağla Seki̇n
Turkish Journal of Mathematics
In this work we introduce some generalizations of the singular parabolic Riesz and parabolic Bessel potentials. Namely, $\Delta _{\nu }$ being the Laplace-Bessel singular differential operator, we define the families of operators \begin{equation*} H_{\beta ,\nu }^{\alpha }=\left( \frac{\partial }{\partial t}+(-\Delta _{\nu })^{\beta /2}\right) ^{-\alpha /\beta }\text{ and }\mathcal{H}_{\beta ,\nu }^{\alpha }=\left( I+\frac{\partial }{\partial t}+(-\Delta _{\nu })^{\beta /2}\right) ^{-\alpha /\beta }\text{ , (}\alpha ,\beta >0\text{),} \end{equation*} and investigate their properties in the special weighted $L_{p,\nu }$-spaces.
Cyclic And Constacyclic Codes Over The Ring $\Mathbb{Z}_{4}[U]/\Langle U^3-U^2\Rangle$ And Their Gray Image, Mehmet Özen, Fatma Zehra Uzekmek, Eli̇f Segah Öztaş
Cyclic And Constacyclic Codes Over The Ring $\Mathbb{Z}_{4}[U]/\Langle U^3-U^2\Rangle$ And Their Gray Image, Mehmet Özen, Fatma Zehra Uzekmek, Eli̇f Segah Öztaş
Turkish Journal of Mathematics
In this article, the structure of generator polynomial of the cyclic codes with odd length is formed over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ where $u^3=u^2$. With the isomorphism we have defined, the generator polynomial of constacyclic codes with odd length over this ring is created from the generator of the cyclic codes. Additionally, necessary and sufficient conditions for a linear code in this ring to be a self dual code and a LCD code are mentioned. Furthermore, for all units over this ring, $\mathbb{Z}_{4}$-images of $\lambda$-constacyclic codes and also $\mathbb{Z}_{4}$-images of cyclic codes are examined by using related ones from defined three …
Hilbert Series Of Tangent Cones For Gorenstein Monomial Curves In $\Mathbb{A}^{4}(K)$, Anargyros Katsabekis
Hilbert Series Of Tangent Cones For Gorenstein Monomial Curves In $\Mathbb{A}^{4}(K)$, Anargyros Katsabekis
Turkish Journal of Mathematics
In this paper, we study the Hilbert series of the tangent cone of Gorenstein monomial curves in the 4-dimensional affine space. We give an explicit formula for the reduced Hilbert series of the tangent cone of a noncomplete intersection Gorenstein monomial curve whose tangent cone is Cohen-Macaulay.
Maps On $\Mathcal{S}(\Mathcal{H})$ Preserving The Difference Of Noninvertible Algebraic Operators, Zynab Izadi, Rahmat Soltani
Maps On $\Mathcal{S}(\Mathcal{H})$ Preserving The Difference Of Noninvertible Algebraic Operators, Zynab Izadi, Rahmat Soltani
Turkish Journal of Mathematics
The aim of this paper is to present the general structure of nonlinear surjective maps on $\mathcal S(\mathcal H)$ preserving the operator pairs in which their difference is a noninvertible algebraic operator. $\mathcal S(\mathcal H)$ represents the real Jordan algebra of bounded self-adjoint operators acting on an infinite dimensional Hilbert space $\mathcal{ H}$.
Conformal Generic Submersions, Mehmet Aki̇f Akyol
Conformal Generic Submersions, Mehmet Aki̇f Akyol
Turkish Journal of Mathematics
Akyol and Şahin (2017) introduced the notion of conformal semiinvariant submersions from almost Hermitian manifolds. The present paper deals with the study of conformal generic submersions from almost Hermitian manifolds which extends semiinvariant Riemannian submersions, generic Riemannian submersions and conformal semiinvariant submersions in a natural way. We mention some examples of such maps and obtain characterizations and investigate some properties, including the integrability of distributions, the geometry of foliations and totally geodesic foliations. Moreover, we obtain some conditions for such submersions to be totally geodesic and harmonic, respectively.
Order Compact And Unbounded Order Compact Operators, Nazi̇fe Erkurşun Özcan, Ni̇yazi̇ Anil Gezer, Şazi̇ye Ece Özdemi̇r, İrem Mesude Geyi̇kçi̇
Order Compact And Unbounded Order Compact Operators, Nazi̇fe Erkurşun Özcan, Ni̇yazi̇ Anil Gezer, Şazi̇ye Ece Özdemi̇r, İrem Mesude Geyi̇kçi̇
Turkish Journal of Mathematics
We investigate properties of order compact, unbounded order compact and relatively uniformly compact operators acting on vector lattices. An operator is said to be order compact if it maps an arbitrary order bounded net to a net with an order convergent subnet. Analogously, an operator is said to be unbounded order compact if it maps an arbitrary order bounded net to a net with uo-convergent subnet. After exposing the relationships between order compact, unbounded order compact, semicompact and GAM-compact operators; we study those operators mapping an arbitrary order bounded net to a net with a relatively uniformly convergent subnet. By …
Notes On Multivalent Bazilevic Functions Defined By Higher Order Derivatives, Mohamed K. Aouf, Adela O. Mostafa, Teodor Bulboaca
Notes On Multivalent Bazilevic Functions Defined By Higher Order Derivatives, Mohamed K. Aouf, Adela O. Mostafa, Teodor Bulboaca
Turkish Journal of Mathematics
In this paper we consider two subclasses $B(p,q,\alpha,\beta)$ and $B_{1}(p,q,\alpha,\beta)$ of p-valently Bazilevi\'c functions defined by higher order derivatives, and we defined and studied some properties of the images of the functions of these classes by the integral operators $\mathrm{I}_{n,p}$ and $\mathrm{J}_{n,p}$ for multivalent functions, defined by using higher order derivatives.
Exact And Nonstandard Finite Difference Schemes For The Burgers Equation B(2,2), Canan Köroğlu, Ayhan Aydin
Exact And Nonstandard Finite Difference Schemes For The Burgers Equation B(2,2), Canan Köroğlu, Ayhan Aydin
Turkish Journal of Mathematics
In this paper, we consider the Burgers equation B(2,2). Exact and nonstandard finite difference schemes (NSFD) for the Burgers equation B(2,2) are designed. First, two exact finite difference schemes for the Burgers equation B(2,2) are proposed using traveling wave solution. Then, two NSFD schemes are represented for this equation. These two NSFD schemes are compared with a standard finite difference (SFD) scheme. Numerical results show that the NSFD schemes are accurate and efficient in the numerical simulation of the kink-wave solution of the B(2,2) equation. We see that although the SFD scheme yields numerical instability for large step sizes, NSFD …
Close-To-Convexity Of A Class Of Harmonic Mappings Defined By A Third-Order Differential Inequality, Eli̇f Yaşar, Si̇bel Yalçin Tokgöz
Close-To-Convexity Of A Class Of Harmonic Mappings Defined By A Third-Order Differential Inequality, Eli̇f Yaşar, Si̇bel Yalçin Tokgöz
Turkish Journal of Mathematics
In this paper, we consider a class of normalized harmonic functions in the unit disk satisfying a third-order differential inequality and we investigate several properties of this class such as close-to-convexity, coefficient bounds, growth estimates, sufficient coefficient condition, and convolution. Moreover, as an application, we construct harmonic polynomials involving Gaussian hypergeometric function which belong to the considered class. We also provide examples illustrating graphically with the help of Maple.
Dynnikov Coordinates On Punctured Torus, Alev Meral
Dynnikov Coordinates On Punctured Torus, Alev Meral
Turkish Journal of Mathematics
We generalize the Dynnikov coordinate system previously defined on the standard punctured disk to an orientable surface of genus-1 with n punctures and one boundary component.
Portfolio Optimization With Two Quasiconvex Risk Measures, Çağin Ararat
Portfolio Optimization With Two Quasiconvex Risk Measures, Çağin Ararat
Turkish Journal of Mathematics
We study a static portfolio optimization problem with two risk measures: a principle risk measure in the objective function and a secondary risk measure whose value is controlled in the constraints. This problem is of interest when it is necessary to consider the risk preferences of two parties, such as a portfolio manager and a regulator, at the same time. A special case of this problem where the risk measures are assumed to be coherent (positively homogeneous) is studied recently in a joint work of the author. The present paper extends the analysis to a more general setting by assuming …
Gauss-Bonnet Theorems And The Lorentzian Heisenberg Group, Tong Wu, Sining Wei, Yong Wang
Gauss-Bonnet Theorems And The Lorentzian Heisenberg Group, Tong Wu, Sining Wei, Yong Wang
Turkish Journal of Mathematics
In this paper, we compute sub-Riemannian limits of Gaussian curvature for a C$^{2}$-smooth surface in the Lorentzian Heisenberg group for the second Lorentzian metric and the third Lorentzian metric and signed geodesic curvature for C$^{2}$-smooth curves on surfaces. We get Gauss-Bonnet theorems in the Lorentzian Heisenberg group for the second Lorentzian metric and the third Lorentzian metric.
On Rings Whose Jacobson Radical Coincides With Upper Nilradical, Guanglin Ma, Yao Wang, Yanli Ren
On Rings Whose Jacobson Radical Coincides With Upper Nilradical, Guanglin Ma, Yao Wang, Yanli Ren
Turkish Journal of Mathematics
We call a ring~R is JN if whose Jacobson radical coincides with upper nilradical, and R is right SR if each element r∈ R can be written as r=s+r where s is an element from the right socle and r is a regular element of~R. SR rings is a class of special subrings of JN rings, which is the extension of soclean rings. We give their some characterizations and examples, and investigate the relationship between JN rings, SR rings and related rings, respectively.
A New Implicit-Explicit Local Differential Method For Boundary Value Problems, Hüseyi̇n Tunç, Murat Sari
A New Implicit-Explicit Local Differential Method For Boundary Value Problems, Hüseyi̇n Tunç, Murat Sari
Turkish Journal of Mathematics
In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM …
On A Fifth-Order Nonselfadjoint Boundary Value Problem, Eki̇n Uğurlu, Kenan Taş
On A Fifth-Order Nonselfadjoint Boundary Value Problem, Eki̇n Uğurlu, Kenan Taş
Turkish Journal of Mathematics
In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative operator.
Estimations On Some Hybrid Exponential Sums Related To Kloosterman Sums, Yingjie Cheng, Xiwang Cao, Liqin Qian, Jinlong Wan
Estimations On Some Hybrid Exponential Sums Related To Kloosterman Sums, Yingjie Cheng, Xiwang Cao, Liqin Qian, Jinlong Wan
Turkish Journal of Mathematics
In this paper, we revisit the bounds of the mixed exponential sums introduced by Lv and Zhang (2020). Moreover, we give some estimations for some new hybrid exponential sums related to Kloosterman sums over finite fields of odd characteristic by using the properties of Jacobi sums and Gaussian sums.
On Some Generalizations Of The Beta Function In Several Variables, Mustapha Raissouli, Salma El-Soubhy
On Some Generalizations Of The Beta Function In Several Variables, Mustapha Raissouli, Salma El-Soubhy
Turkish Journal of Mathematics
Intensive studies aiming to extend the gamma and beta functions and to establish some properties for these extensions have been recently carried out. In this paper, we first introduce a generalized gamma function in n variables. Afterwards, two generalized beta functions in several variables are introduced and their properties are discussed. Among others, we investigate recurrence relationships, Mellin transform properties, and partial differential equations involving these generalized functions. At the end, some results about partial derivatives of these extended functions are presented as well.
On Elements Whose Moore-Penrose Inverse Is Idempotent In A ${\Ast}$-Ring, Haiyang Zhu, Jianlong Chen, Yukun Zhou
On Elements Whose Moore-Penrose Inverse Is Idempotent In A ${\Ast}$-Ring, Haiyang Zhu, Jianlong Chen, Yukun Zhou
Turkish Journal of Mathematics
In this paper, we investigate the elements whose Moore-Penrose inverse is idempotent in a ${\ast}$-ring. Let $R$ be a ${\ast}$-ring and $a\in R^\dagger$. Firstly, we give a concise characterization for the idempotency of $a^\dagger$ as follows: $a\in R^\dagger$ and $a^\dagger$ is idempotent if and only if $a\in R^{\#}$ and $a^2=aa^*a$, which connects Moore-Penrose invertibility and group invertibility. Secondly, we generalize the results of Baksalary and Trenkler from complex matrices to ${\ast}$-rings. More equivalent conditions which ensure the idempotency of $a^\dagger$ are given. Particularly, we provide the characterizations for both $a$ and $a^\dagger$ being idempotent. Finally, the equivalent conditions under which …
The Lebesgue Constants On Projective Spaces, Alexander Kushpel
The Lebesgue Constants On Projective Spaces, Alexander Kushpel
Turkish Journal of Mathematics
We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier-Laplace projections on the real projective spaces $\mathrm{P}^{d}(\mathbb{R})$. In particular, these results extend sharp asymptotic found by Fejer [2] in the case of $\mathbb{S}^{1}$ in 1910 and by Gronwall [4] in 1914 in the case of $\mathbb{S}^{2}$. The case of spheres, $\mathbb{S}^{d}$, complex and quaternionic projective spaces, $\mathrm{P}^{d}(\mathbb{C})$, $% \mathrm{P}^{d}(\mathbb{H})$ and the Cayley elliptic plane $\mathrm{P}^{16}(% \mathrm{Cay})$ was considered by Kushpel [8].
Completeness Conditions Of Systems Of Bessel Functions In Weighted $L^2$-Spaces In Terms Of Entire Functions, Ruslan Khats'
Completeness Conditions Of Systems Of Bessel Functions In Weighted $L^2$-Spaces In Terms Of Entire Functions, Ruslan Khats'
Turkish Journal of Mathematics
Let $J_{\nu}$ be the Bessel function of the first kind of index $\nu\ge 1/2$, $p\in\mathbb R$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system $\big\{x^{-p-1}\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$ in the weighted space $L^2((0;1);x^{2p} dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$.
Existence Of Nonnegative Solutions For Discrete Robin Boundary Value Problems With Sign-Changing Weight, Yan Zhu
Turkish Journal of Mathematics
In this paper,~we are concerned with the following discrete problem first $$\left\{ \begin{array}{ll} -\Delta^{2}u(t-1)=\lambda p(t)f(u(t)), &t\in[1,N-1]_{\mathbb{Z}},\\ \Delta u(0)=u(N)=0,\\ \end{array} \right. $$ where $N>2$~is an integer,~$\lambda>0$~is a parameter,~$p:[1,N-1]_{\mathbb{Z}}\rightarrow\mathbb{R}$~is a sign-changing function,~$f:[0,+\infty)\rightarrow[0,+\infty)$~is a continuous and nondecreasing function.~$\Delta u(t)=u(t+1)-u(t)$,~$\Delta^{2}u(t)=\Delta(\Delta u(t))$.~By using the iterative method and Schauder's fixed point theorem,~we will show the existence of nonnegative solutions to the above problem. Furthermore, we obtain the existence of nonnegative solutions for discrete Robin systems with indefinite weights.
New Oscillation Criteria For Differential Equations With Sublinear And Superlinear Neutral Terms, Ali Muhib, Elmetwally M. Elabbasy, Osama Moaaz
New Oscillation Criteria For Differential Equations With Sublinear And Superlinear Neutral Terms, Ali Muhib, Elmetwally M. Elabbasy, Osama Moaaz
Turkish Journal of Mathematics
The aim of this article is to establish some new oscillation criteria for the differential equation of even-order of the form \begin{equation*} (r\left( l\right) (y^{\left( n-1\right) }\left( l\right) )^{\alpha })^{\prime }+f(l,x(\tau (l)))=0, \end{equation*} where $y\left( l\right) =x\left( l\right) +p\left( l\right) x^{\beta }\left( \sigma _{1}\left( l\right) \right) +h\left( l\right) x^{\delta }\left( \sigma _{2}\left( l\right) \right) $. By using Riccati transformations, we present new conditions for oscillation of the studied equation. Furthermore, two illustrative examples showing applicability of the new results are included.
Volume Properties And Some Characterizations Of Ellipsoids In ${\Bbb E}^{N+1}$, Dong-Soo Kim, Incheon Kim, Young Ho Kim
Volume Properties And Some Characterizations Of Ellipsoids In ${\Bbb E}^{N+1}$, Dong-Soo Kim, Incheon Kim, Young Ho Kim
Turkish Journal of Mathematics
Suppose that $M$ is a strictly convex and closed hypersurface in ${\Bbb E}^{n+1}$ with the origin $o$ in its interior. We consider the homogeneous function $g$ of positive degree $d$ satisfying $M=g^{-1}(1)$. Then, for a positive number $h$ the level hypersurface $g^{-1}(h)$ of $g$ is a homothetic hypersurface of $M$ with respect to the origin $o$. In this paper, for tangent hyperplanes $\Phi_h$ to $g^{-1}(h)$ ($0
On Ordered $\Gamma$-Hypersemigroups, Minimal Bi-Ideals, And Minimal Left Ideals, Niovi Kehayopulu
On Ordered $\Gamma$-Hypersemigroups, Minimal Bi-Ideals, And Minimal Left Ideals, Niovi Kehayopulu
Turkish Journal of Mathematics
The definition of ordered $\Gamma$-hypersemigroups and the definitions of regular and intra-regular ordered $\Gamma$-hypersemigroups in the existing bibliography should be corrected. Care should be given to the definitions of bi-$\Gamma$-hyperideals and quasi-$\Gamma$-hyperideals as well. The main results are a characterization of minimal bi-ideals of an ordered $\Gamma$-hypersemigroup $S$ in terms of $\cal B$-simple bi-ideals of $S$ and a characterization of minimal left (resp. right) ideals of an ordered $\Gamma$-hypersemigroup $S$ in terms of left (resp. right) simple subsemigroups of $S$.