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Articles 211 - 240 of 2494
Full-Text Articles in Physical Sciences and Mathematics
K−Uniformly Multivalent Functions Involving Liu-Owa Q−Integral Operator, Asena Çeti̇nkaya
K−Uniformly Multivalent Functions Involving Liu-Owa Q−Integral Operator, Asena Çeti̇nkaya
Turkish Journal of Mathematics
In this paper, we introduce $q-$analogue of Liu-Owa integral operator and define a subclass of $k-$uniformly multivalent starlike functions of order $\gamma, (0\leq\gamma< p; p\in\mathbb{N})$ by using the Liu-Owa $q-$integral operator. We examine coefficient estimates, growth and distortion bounds for the functions belonging to the subclass of $k-$uniformly multivalent starlike functions of order $\gamma$. Moreover, we determine radii of $k-$uniformly starlikeness, convexity and close-to-convexity for the functions belonging to this subclass.
Variational Geometry For Surfaces In Conformally Flat Space, Najma Mosadegh, Esmaiel Abedi
Variational Geometry For Surfaces In Conformally Flat Space, Najma Mosadegh, Esmaiel Abedi
Turkish Journal of Mathematics
In this paper, it is shown that a closed surface in 3-dimensional harmonic conformally flat space is minimal if the sign of the mean curvature does not change. Also, it is determined that the critical point of mean curvature functional of the surface is homeomorphic to the sphere.
Efficient Nyberg-Rueppel Type Of Ntru Digital Signature Algorithm, Ferdi̇ Elverdi̇, Sedat Akleylek, Bariş Bülent Kirlar
Efficient Nyberg-Rueppel Type Of Ntru Digital Signature Algorithm, Ferdi̇ Elverdi̇, Sedat Akleylek, Bariş Bülent Kirlar
Turkish Journal of Mathematics
Message recovery is an important property in Nyberg-Rueppel type digital signature algorithms. However, the security of Nyberg-Rueppel type digital signature algorithms depends on the hard problems which might be vulnerable to quantum attacks. Therefore, quantum resistant Nyberg-Rueppel type digital signature algorithms with message recovery property are needed. Since NTRU-based cryptosystems are one of the best studied quantum-resistant schemes, using traditional NTRU encryption scheme has several advantages on the message recovery property. In this paper, we define Nyberg-Rueppel type of NTRU digital signature algorithm. It is carried out by combining NTRU-based encryption and signature algorithms. In the proposed scheme, efficient message …
A New Subclass Of Certain Analytic Univalent Functions Associated With Hypergeometric Functions, Alaatti̇n Akyar
A New Subclass Of Certain Analytic Univalent Functions Associated With Hypergeometric Functions, Alaatti̇n Akyar
Turkish Journal of Mathematics
The main objective of the present paper is to give with using the linear operator theory and also hypergeometric representations of related functions a new special subclass $\mathcal{TS}_{p}(2^{-r},2^{-1}), r\in \mathbb{ Z }^{+}$ of uniformly convex functions and in addition a suitable subclass of starlike functions with negative Taylor coefficients. Furthermore, the provided trailblazer outcomes in presented study are generalized to certain functions classes with fixed finitely many Taylor coefficients.
Examination Of Eigenvalues And Spectral Singularities Of A Discrete Dirac Operator With An Interaction Point, Şeri̇fenur Cebesoy
Examination Of Eigenvalues And Spectral Singularities Of A Discrete Dirac Operator With An Interaction Point, Şeri̇fenur Cebesoy
Turkish Journal of Mathematics
In this paper, the main content is the consideration of the concepts of eigenvalues and spectral singularities of an operator generated by a discrete Dirac system in $\ell_{2}(\mathbb{Z},\mathbb{C}^{2})$ with an interior interaction point. Defining a transfer matrix $ M $ enables us to present a relationship between the $ M_{22} $ component of this matrix and Jost functions of mentioned Dirac operator so that its eigenvalues and spectral properties can be studied. Finally, some special cases are examined where the impulsive condition possesses certain symmetries.
Various Operators In Relation To Fractional Order Calculus And Some Of Their Applications To Normalized Analytic Functions In The Open Unit Disk, Hüseyi̇n Irmak
Various Operators In Relation To Fractional Order Calculus And Some Of Their Applications To Normalized Analytic Functions In The Open Unit Disk, Hüseyi̇n Irmak
Turkish Journal of Mathematics
The main object of this scientific work is firstly to introduce various operators of fractional calculus (that is that fractional integral and fractional derivative(s)) in certain domains of the complex plane, then to determine certain results correlating with normalized analytic functions, which are analytic in certain domains in the complex plane, as a few applications of those operators, and also to present a number of extensive implications of them as special results.
Numerical Solution Of A Singularly Perturbed Fredholm Integro Differential Equation With Robin Boundary Condition, Muhammet Enes Durmaz, Gabi̇l Ami̇rali̇, Mustafa Kudu
Numerical Solution Of A Singularly Perturbed Fredholm Integro Differential Equation With Robin Boundary Condition, Muhammet Enes Durmaz, Gabi̇l Ami̇rali̇, Mustafa Kudu
Turkish Journal of Mathematics
In this paper, we deal with singularly perturbed Fredholm integro differential equation (SPFIDE) with mixed boundary conditions. By using interpolating quadrature rules and exponential basis function, fitted second order difference scheme has been constructed on a Shishkin mesh. The stability and convergence of the difference scheme have been analyzed in the discrete maximum norm. Some numerical examples have been solved and numerical outcomes are obtained.
Classification Of Some Geometric Structures On 4-Dimensional Riemannian Lie Group, Esmaeil Peyghan, Davood Seifipour
Classification Of Some Geometric Structures On 4-Dimensional Riemannian Lie Group, Esmaeil Peyghan, Davood Seifipour
Turkish Journal of Mathematics
In this paper we study the spectral geometry of a $4$-dimensional Lie group. The main focus of this paper is to study the $2$-Stein and $2$-Osserman structures on a $4$-dimensional Riemannian Lie group. In this paper, we study the spectrum and trace of Jacobi operator and also we study the characteristic polynomial of generalized Jacobi operator on the non-abelian $4$-dimensional Lie group $G$, whenever $G$ is equipped with an orthonormal left invariant Riemannian metric $g$. The Lie algebra structures in dimension four have key role in this paper. It is known that in the classification of $4$-dimensional non-abelian Lie algebras …
Sober Spaces, Mehmet Baran, Hassan Abughalwa
Sober Spaces, Mehmet Baran, Hassan Abughalwa
Turkish Journal of Mathematics
The goal of this paper is to introduce various forms of sober objects in a topological category and investigate the relationships among these various forms. Moreover, we characterize quasi-sober objects, $T_{0}$ objects and each of various forms of sober objects in some topological categories and give some invariance properties of them. Finally, we compare our results with some well-known results in the category of topological spaces.
Improved Inequalities Related To The A-Numerical Radius For Commutators Of Operator, Kais Feki
Improved Inequalities Related To The A-Numerical Radius For Commutators Of Operator, Kais Feki
Turkish Journal of Mathematics
Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathbb{B}_{A}(\mathcal{H})$ be the subspace of all operators which admit $A$-adjoints operators. In this paper, we establish some inequalities involving the commutator and the anticommutator of operators in semi-Hilbert spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Mainly, among other inequalities, we prove that for $T, S\in\mathbb{B}_{A}(\mathcal{H})$ we have \begin{align*} \omega_A(TS \pm ST) \leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where $$f_A(X,Y)=\ Y\ _A\sqrt{\omega_A^2(X)-\frac{\left \,\left\ \frac{X+X^{\sharp_A}}{2}\right\ _A^2-\left\ \frac{X-X^{\sharp_A}}{2i}\right\ _A^2\right }{2}}.$$ This covers and improves the well-known inequalities of Fong and Holbrook. Here $\omega_A(\cdot)$ and $\ \cdot\ _A$ …
Finite Ordered $\Gamma$-Hypersemigroups Constructed By Ordered $\Gamma$-Semigroups, Niovi Kehayopulu
Finite Ordered $\Gamma$-Hypersemigroups Constructed By Ordered $\Gamma$-Semigroups, Niovi Kehayopulu
Turkish Journal of Mathematics
In the investigation of ordered $\Gamma$-hypersemigroups we often need counterexamples (of finite order) given by a table of multiplication and a figure that are impossible to make by hand and very difficult to write programs as well. So it is useful to have examples of ordered $\Gamma$-semigroups for which is much more easier to write programs and then from these examples to obtain corresponding examples of ordered $\Gamma$-hypersemigroups. In this respect we show that from every example of a regular, intra-regular, right (left) regular, right (left) quasi-regular, semisimple, right (left) simple, simple, strongly regular ordered $\Gamma$-semigroup given by a table …
Uniformly Convergent Finite Difference Method For Reaction-Diffusion Type Third Order Singularly Perturbed Delay Differential Equation, Rajendran Mahendran, Veerasamy Subburayan
Uniformly Convergent Finite Difference Method For Reaction-Diffusion Type Third Order Singularly Perturbed Delay Differential Equation, Rajendran Mahendran, Veerasamy Subburayan
Turkish Journal of Mathematics
A class of third order reaction-diffusion type singularly perturbed ordinary delay differential equations is considered in this article. A fitted finite difference method on Shishkin mesh is suggested to solve the problem. Moreover, we present a class of nonlinear problems. An error estimation is obtained based on the maximum norm and it is of almost first order convergence. Numerical results are given to support theoretical claims.
Determination Of A Differential Pencil Frominterior Spectral Data On A Union Of Two Closed Intervals, İbrahi̇m Adalar
Determination Of A Differential Pencil Frominterior Spectral Data On A Union Of Two Closed Intervals, İbrahi̇m Adalar
Turkish Journal of Mathematics
In this paper, we consider a quadratic pencil of Sturm-Liouville operator on closed sets. We study an interior-inverse problem for this kind operator and give a uniqueness theorem with an appropriate example.
Discrete Impulsive Sturm-Liouville Equation With Hyperboliceigenparameter, Turhan Köprübaşi, Yelda Aygar Küçükevci̇li̇oğlu
Discrete Impulsive Sturm-Liouville Equation With Hyperboliceigenparameter, Turhan Köprübaşi, Yelda Aygar Küçükevci̇li̇oğlu
Turkish Journal of Mathematics
Let $L$ denote the selfadjoint difference operator of second order with boundary and impulsive conditions generated in $\ell _{2}\left( %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion \right) $ by \begin{equation*} a_{n-1}y_{n-1}+b_{n}y_{n}+a_{n}y_{n+1}=\left( 2\cosh z\right) y_{n}\text{ },% \text{ }n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion \setminus \left\{ k-1,k,k+1\right\} , \end{equation*}% \begin{equation*} \begin{array}{c} y_{0}=0\text{ }, \\ \left\{ \begin{array}{c} y_{k+1}=\theta _{1}y_{k-1} \\ \bigtriangleup y_{k+1}=\theta _{2}\bigtriangledown y_{k-1} \end{array}% \right. ,\text{ }\theta _{1},\theta _{2}\in %TCIMACRO{\U{211d}}% %BeginExpansion \mathbb{R}, %EndExpansion \end{array}% \end{equation*} where $\left\{ a_{n}\right\} _{n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion },$ $\left\{ b_{n}\right\} _{n\in %TCIMACRO{\U{2115} } %BeginExpansion \mathbb{N} %EndExpansion }$ are real sequences and $\bigtriangleup ,\bigtriangledown $ are respectively forward …
Scattering Properties Of Impulsive Difference Dirac Equations, Şeyda Solmaz, Elgi̇z Bayram
Scattering Properties Of Impulsive Difference Dirac Equations, Şeyda Solmaz, Elgi̇z Bayram
Turkish Journal of Mathematics
In this paper, we explore the Jost solutions and the scattering matrix of the impulsive difference Dirac systems (IDDS) on the whole axis and study their analytic and asymptotic properties. Furthermore, characteristic properties of the scattering matrix of the IDDS have been examined.
Scattering Theory Of The Quadratic Eigenparameter Depending Impulsive Sturm-Liouville Equations, Güler Başak Öznur, Elgi̇z Bairamov
Scattering Theory Of The Quadratic Eigenparameter Depending Impulsive Sturm-Liouville Equations, Güler Başak Öznur, Elgi̇z Bairamov
Turkish Journal of Mathematics
We handle an impulsive Sturm-Liouville boundary value problem. We find the Jost solution, Jost function, and scattering function of this problem and examine the properties of scattering function. We also study eigenvalues and resolvent operator of this problem. Finally, we exemplify our work by taking a different problem.
Existence Of Solutions For An Infinite System Of Tempered Fractional Order Boundary Value Problems In The Spaces Of Tempered Sequences, Khuddush Mahammad, Rajendra Prasad Kapula, Leela Doddi
Existence Of Solutions For An Infinite System Of Tempered Fractional Order Boundary Value Problems In The Spaces Of Tempered Sequences, Khuddush Mahammad, Rajendra Prasad Kapula, Leela Doddi
Turkish Journal of Mathematics
This paper deals with infinite system of nonlinear two-point tempered fractional order boundary value problems $$ \begin{aligned} {}^\mathtt{RL}_{~0}\mathbb{D}^{δ_2,\ell}_\mathtt{z}\Big[\mathtt{p}_\mathtt{j}(\mathtt{z})&{}^\mathtt{RL}_{~0}\mathbb{D}^{δ_1,\ell}_\mathtt{z} \vartheta_\mathtt{j}(\mathtt{z})\Big]=λ_\mathtt{j}\varphi\big(\mathtt{z},\vartheta(\mathtt{z})\big),\, \mathtt{z}\in[0,\mathtt{T}], δ_1,δ_2\in(1,2),\\ &\hskip0.25cm\vartheta_\mathtt{j}(0)=\lim_{\mathtt{z}\to0}\left[{}^\mathtt{RL}_{~0}\mathbb{D}^{δ_1,\ell}_\mathtt{z}(e^{\ell\mathtt{z}}\vartheta_\mathtt{j}(\mathtt{z}))\right]=0,\\ &e^{\ell\mathtt{T}}\vartheta_\mathtt{j}(\mathtt{T})=\lim_{\mathtt{z}\to\mathtt{T}}\left[{}^\mathtt{RL}_{~0}\mathbb{D}^{δ_1,\ell}_\mathtt{z}(e^{\ell\mathtt{z}}\vartheta_\mathtt{j}(\mathtt{z}))\right]=0, \end{aligned} $$ where $\mathtt{j}\in\{1,2,3,\cdot\cdot\cdot\},\,\ell\ge 0,$ ${}^\mathtt{RL}_{~0}\mathbb{D}^{\star,\ell}_\mathtt{z}$ denotes the Riemann--Liouville tempered fractional derivative of order $\star\in\{δ_1,δ_2\}$, $\vartheta(\mathtt{z})=\left(\vartheta_\mathtt{j}(\mathtt{z})\right)_{\mathtt{j}=1}^{\infty},$ $\varphi_\mathtt{j}:[0,\mathtt{T}]\to[0,\mathtt{T}]$ are continuous and we derive sufficient conditions for the existence of solutions to the system via the Hausdorff measure of noncompactness and Meir-Keeler fixed point theorem in tempered sequence spaces.
Global Existence And Energy Decay For A Coupled System Of Kirchhoff Beam Equations With Weakly Damping And Logarithmic Source, Ducival Carvalho Pereira, Carlos Alberto Raposo Da Cunha, Adriano Pedreira Cattai
Global Existence And Energy Decay For A Coupled System Of Kirchhoff Beam Equations With Weakly Damping And Logarithmic Source, Ducival Carvalho Pereira, Carlos Alberto Raposo Da Cunha, Adriano Pedreira Cattai
Turkish Journal of Mathematics
This paper deals with the global solutions and exponential stability for a coupled system of Kirchhoff beam weakly damping and with a logarithmic source. We apply the potential well and establish the global well-posedness by using the Faedo--Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created from the Nehari manifold. Moreover, by using Nakao's lemma, we prove the exponential stability of the solution.
Boundary Value Problems For A Second-Order $(P,Q) $-Difference Equation With Integral Conditions, İlker Gençtürk
Boundary Value Problems For A Second-Order $(P,Q) $-Difference Equation With Integral Conditions, İlker Gençtürk
Turkish Journal of Mathematics
Our purpose in this paper is to obtain some new existence results of solutions for a boundary value problem for a $ (p,q) $-difference equations with integral conditions, by using fixed point theorems. Examples illustrating the main results are also presented.
A Discussion On The Existence And Uniqueness Analysis For The Coupled Two-Term Fractional Differential Equations, Sachin Kumar Verma, Ramesh Kumar Vats, Avadhesh Kumar, Velusamy Vijayakumar, Anurag Shukla
A Discussion On The Existence And Uniqueness Analysis For The Coupled Two-Term Fractional Differential Equations, Sachin Kumar Verma, Ramesh Kumar Vats, Avadhesh Kumar, Velusamy Vijayakumar, Anurag Shukla
Turkish Journal of Mathematics
This paper mainly concentrates on the study of a new boundary value problem of coupled nonlinear two-term fractional differential system. We make use of the theories on fractional calculus and fixed point approach to derive the existence and uniqueness results of the considered two-term fractional systems. To confirm the application of the stated outcomes, two examples are provided.
On A Solvable System Of Rational Difference Equations Of Higher Order, Merve Kara, Yasi̇n Yazlik
On A Solvable System Of Rational Difference Equations Of Higher Order, Merve Kara, Yasi̇n Yazlik
Turkish Journal of Mathematics
In this paper, we present that the following system of difference equations $$ x_{n}=\frac{x_{n-k}z_{n-l}}{b_{n}x_{n-k}+a_{n}z_{n-k-l}}, \ y_{n}=\frac{y_{n-k}x_{n-l}}{d_{n}y_{n-k}+c_{n}x_{n-k-l}}, \ z_{n}=\frac{z_{n-k}y_{n-l}}{f_{n}z_{n-k}+e_{n}y_{n-k-l}}, $$ where $n\in \mathbb{N}_{0}$, $k,l\in\mathbb{N}$, the initial values $x_{-i},y_{-i},z_{-i}$ are real numbers, for $i \in \overline{1,k+l}$, and sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are non-zero real numbers, for all $n\in \mathbb{N}_{0}$, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case …
On The Existence For Parametric Boundary Value Problems Of A Coupled System Of Nonlinear Fractional Hybrid Differential Equations, Yige Zhao, Yibing Sun
On The Existence For Parametric Boundary Value Problems Of A Coupled System Of Nonlinear Fractional Hybrid Differential Equations, Yige Zhao, Yibing Sun
Turkish Journal of Mathematics
In this paper, we consider the existence and uniqueness for parametric boundary value problems of a coupled system of nonlinear fractional hybrid differential equations. By the fixed point theorem in Banach algebra, an existence theorem for parametric boundary value problems of a coupled system of nonlinear fractional hybrid differential equations is given. Further, a uniqueness result for parametric boundary value problems of a coupled system of nonlinear fractional hybrid differential equations is proved due to Banach's contraction principle. Further, we give three examples to verify the main results.
On Extended Interpolative Single And Multivalued $F$-Contractions, İsa Yildirim
On Extended Interpolative Single And Multivalued $F$-Contractions, İsa Yildirim
Turkish Journal of Mathematics
The main objective of this paper is to study an extended interpolative single and multivalued Hardy-Rogers type $F$-contractions in complete metric spaces. We prove some fixed point theorems for such mappings. Further, we give an application to integral equations to verify our main results. The results presented in this paper improve the recent works of Karapinar et al. [12] and Mohammadi et al. [16].
A Refinement Of Newton And Maclaurin Inequalities Through Abstract Convexity, Gülteki̇n Tinaztepe, Ramazan Tinaztepe
A Refinement Of Newton And Maclaurin Inequalities Through Abstract Convexity, Gülteki̇n Tinaztepe, Ramazan Tinaztepe
Turkish Journal of Mathematics
In this study, the refinements of Maclaurin's and Newton's inequalities are given. These refinements are obtained by applying the results on optimality conditions of abstract convex functions. When doing this, we obtain lower bounds for the solutions of some special rational equations.
Elliptical Kinematics Of The Accretive Surface Growth, Zehra Özdemi̇r, Gül Güner
Elliptical Kinematics Of The Accretive Surface Growth, Zehra Özdemi̇r, Gül Güner
Turkish Journal of Mathematics
The stresses within the soft tissue are not constant for some shell surfaces. They vary with position along the mantle edge. In this paper, we show that elliptical geometry is more convenient to describe this type of surface. Thus, we introduce the elliptical kinematics along an initial curve and construct some accretive surfaces with an elliptical cross-section. In fact, these surfaces are not only curves with an elliptical cross-sectional curve, but also the material points of the surface follow an elliptical trajectory during their formation. This situation can be easily explained through elliptical motion and elliptical quaternion algebra. Then, we …
Non-Solvable Groups All Of Whose Indices Are Odd-Square-Free, Sajjad Mahmood Robati, Roghayeh Hafezieh Balaman
Non-Solvable Groups All Of Whose Indices Are Odd-Square-Free, Sajjad Mahmood Robati, Roghayeh Hafezieh Balaman
Turkish Journal of Mathematics
Given a finite group $G$ and $x\in G$, the class size of $x$ in $G$ is called odd-square-free if it is not divisible by the square of any odd prime number. In this paper, we show that if $G$ is a nonsolvable finite group, all of whose class sizes are odd-square-free, then we have some control on the structure of $G$, which is an answer to the dual of the question mentioned by Huppert in [5].
Relation Between Matrices And The Suborbital Graphs By The Special Number Sequences, Ümmügülsün Akbaba, Ali̇ Hi̇kmet Değer
Relation Between Matrices And The Suborbital Graphs By The Special Number Sequences, Ümmügülsün Akbaba, Ali̇ Hi̇kmet Değer
Turkish Journal of Mathematics
under circuit and forest conditions. Special number sequences and special vertex values of minimal length paths in suborbital graphs have been associated in our previous studies. In these associations, matrix connections of the special continued fractions $\mathcal K (-1/-k)$, where $k\in \mathbb{Z}^{+}, \ k\geq 2$ with the values of the special number sequences are used and new identities are obtained. In this study, by producing new matrices, new identities related to Fibonacci, Lucas, Pell, and Pell-Lucas number sequences are found by using both recurrence relations and matrix connections of the continued fractions. In addition, the farthest vertex values of the …
Composition Laws On The Fricke Surface And Markov Triples, Abdurrahman Muhammed Uludağ, Esra Ünal Yilmaz
Composition Laws On The Fricke Surface And Markov Triples, Abdurrahman Muhammed Uludağ, Esra Ünal Yilmaz
Turkish Journal of Mathematics
We determine some composition laws related to the Fricke surface and the "double" Fricke surface. This latter surface admits the squares of Markov triples as its solutions.
Minimal Generators Of Annihilators Of Even Neat Elements In The Exterior Algebra Footnotesize, Songül Esi̇n
Minimal Generators Of Annihilators Of Even Neat Elements In The Exterior Algebra Footnotesize, Songül Esi̇n
Turkish Journal of Mathematics
This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated.
Necessary Conditions For Extended Spectral Decomposable Multivalued Linear Operators, Yosra Barkaoui, Maher Mnif
Necessary Conditions For Extended Spectral Decomposable Multivalued Linear Operators, Yosra Barkaoui, Maher Mnif
Turkish Journal of Mathematics
In this paper, we use subsets of the Riemann sphere and specific types of invariant linear subspaces to introduce the extended spectral decomposable multivalued linear operators (linear relations) in Banach spaces. We also introduce the extended Bishop's property, the extended relatively single-valued extension property and the extended Dunford's property. More importantly, we show that these properties are three necessary conditions for a linear relation to be extended spectral decomposable.