Search

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this paper, we consider a class of generalized operator equilibrium problems and derive a Minty type lemma for this class of problems. Further, we establish some existence theorems for the generalized operator equilibrium problems. The theorems presented in this paper generalize and unify many well-known results in the literature.

Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.

The aim of this paper is to investigate and give a new family of Apell type polynomials, which are related to the Euler, Frobenius-Euler and Apostol-Bernoulli polynomials and numbers and also the generalized Stirling numbers of the second kind etc. The results presented in this paper are based upon the theory of the generating functions. By using functional equations of these generating functions, we drive some identities and relations for these numbers and polynomials. Moreover, we give a computation algorithm these numbers.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices. (i) $M=\left(\begin{array}{cc}AX+YBA (ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\ The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009), Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155 /2013/247028). Some examples are given to illustrate our results.

In this paper, we present some results on the complete convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables by using the Rosenthal-type maximal inequality, Kolmogorov exponential inequality and the truncation method. The results obtained in the paper extend the corresponding ones for weighted sums of negatively associated random variables with identical distribution to the case of arrays of rowwise NSD random variables without identical distribution.

The purpose of this paper is to introduce and study an iterative scheme for solving the split feasibility problems in the setting of $p$-uniformly convex and uniformly smooth Banach spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper extends some recent results done by Jitsupa Deepho and Poom Kumam [Jitsupa Deepho and Poom Kumam, A Modified Halpern’s Iterative Scheme for Solving Split Feasibility Problems, Abstract and Applied Analysis, Volume 2012, Article ID 876069, 8 pages] and some others.

In this paper, we introduce a notion of (?,?,?)-generalized weakly contractive maps in S-metric spaces and prove the existence of fixed points for such maps. We also prove that these maps satisfy property(P). The results presented in this paper extend the results of Khandaqji, Al-Sharif and Al-Khaleel [15] from G-metric spaces to S-metric spaces. We provide examples in support of our results.

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

This paper aims at studying the fundamental properties of fuzzy linear relations between vector spaces. The sum of two fuzzy relations and the scalar multiplication are defined, in a natural way, and some properties of this operations are established. Fuzzy linear relations are investigated and among the results obtained, there should be underlined a characterization of fuzzy linear relations and the fact that the inverse of a fuzzy linear relation is also a fuzzy linear relation. Moreover, the paper shows that the composition of two fuzzy linear relations is a fuzzy linear relation as well. Finally, the article highlights that the family of all fuzzy linear relations is closed under addition and it is closed under scalar multiplication.

The aim of this paper is to continue our investigations in line of our recent paper, Savas [24] and [26]. We introduce the notion of AI-double statistical convergence which includes the new summabilitymethods studied in [24] and [23] as special cases and make certain observations on this new and moregeneral summability method. © 2016, University of Nis. All rights reserved.

In this paper, a monotone Bregan projection algorithm is investigated for solving equilibrium problems and common fixed point problems of a family of closed multi-valued Bregman quasi-strict pseudocontractions. Strong convergence is guaranteed in the framework of reflexive Banach spaces.

Kernel methods are a class of machine learning algorithms which learn and discover patterns in a high (possibly infinite) dimensional feature space obtained by often nonlinear, possibly infinite mapping of an input space. A major problem with kernel methods is their time complexity. For a data set with n input points a time complexity of a kernel method is O(n3), which is intractable for a large data set. A method based on a random Nystr?m features is an approximation method that is able to reduce the time complexity to O(np2+p3) where p is the number of randomly selected input data points. A time complexity of O(p3) comes from the fact that a spectral decomposition needs to be performed on a p x p Gram matrix, and if p is a large number even an approximate algorithm is time consuming. In this paper we will apply the randomized SVD method instead of the spectral decomposition and further reduce the time complexity. An input parameters of a randomized SVD algorithm are p x p Gram matrix and a number m < p. In this case time complexity is O(nm2+p2m+m3), and linear regression is performed on a m-dimensional random features. We will prove that the error of a predictor, learned via this method is almost the same in expectation as the error of a kernel predictor. Aditionally, we will empirically show that this predictor is better than the ONE that uses only Nystr?m method.

The aim of this paper is to introduce a topological model of fractals. Self similar fractals will be approached as inverse limit of finite one dimensional topological spaces with alpha continuous bonding functions. The second approach is to investigate topological graphs in terms nano topological spaces for Lellis Thivagar. From these approximations, the dynamics of Julia sets as a special type of self similar fractals will be studied and some physical properties of fractals through their nano topological graphs will be applied.

In this paper, we present a unique polar-like decomposition theorem for rectangular complex matrices. Applying this decomposition, we define on the set of rectangular matrices a new partial ordering called WL(weak L?wner) partial order ? an extension of the GL(generalized L?wner) partial order, and derive some basic properties of the new partial ordering.

In the present paper, the initial value problem for the Schr?dinger equation with time delay in a Hilbert space is investigated. Theorems on stability estimates for the solution of the problem are established. The applications of theorems for three types of Schr?dinger problems are provided.

In this paper, we define the notion of paratopological polygroups and find their topological properties. In particular, we find those properties that make a paratopological polygroup a topological polygroup. We also give an example of topological polygroup and obtain some of their properties.

In this research a new analytical approach is used to solve nonlinear boundary value problems (BVPs) of higher order occurring in nonlinear phenomena. It converts a complex nonlinear problem into zeroth order and first order problem. It consists of initial guess, auxiliary functions (containing unknown convergence controlling parameters) and a homotopy. The unknown parameters are determined by minimizing the residual. Many methods which are explained in this paper are used to determine these parameters. Here Galerkin?s method is used for this purpose. It is applied to solve non-linear BVPs of fourth and fifth order. The results are compared with the already existing methods e.g., Galerkin?s Method with Quintic B-splines, Differential Transform Method (DTM), and Optimal Homotopy AsymptoticMethod (OHAM). It gives efficient and accurate first-order approximate solution. The results achieved by this technique are in excellent concurrence with the exact solution and hence proved that this method is effective and easy to apply.

The purpose of this paper is to propose and investigate a stochastic SIQR epidemic model with saturated incidence rate. Firstly, we give some conditions to guarantee the stochastic SIQR epidemic model has a unique global positive solution. Then we verify that the disease in this model will die out exponentially if Rs 0 < 1, while the disease will be persistent in the mean if Rs 0 > 1. Moreover, by constructing suitable Lyapunov functions, we establish some sufficient conditions for the existence of an ergodic stationary distribution for the model. Finally, we provide some numerical simulations to illustrate the analytical results.

In this paper, we define the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind. Then we give the exponential, combinatorial, permanental and determinantal representations and the Binet formulas of the adjacency-Jacobsthal-Hurwitz numbers of the first and second kind by the aid of the generating functions and the generating matrices of the sequences defined.

In this paper, an iterative method is applied to solve some p-Laplacian boundary value problem involving Riemann-Liouville fractional derivative operator. More precisely, we establish the existence of two positive solutions. Moreover, we prove that these solutions are one maximal and the other is minimal. An example is presented to illustrate our main result. Finally, a numerical method to solve this problem is given.

In this paper we study radicals of a semigroup which is the union of a family of its subsets, indexed by a nonempty set, such that the intersection of two distinct subsets is contained in the set of left zeroes of that semigroup.

Let T be a bounded linear operator on a complex Hilbert space and n,m ? N. Then T is said to be n-normal if T+Tn = TnT+ and (n,m)-normal if T+mTn = TnT+m. In this paper, we study several properties of n-normal, (n,m)-normal operators. In particular, we prove that if T is 2-normal with ?(T) ? (-?(T)) ? {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n,m)-normal, then T is decomposable and Weyl?s theorem holds for f (T), where f is an analytic function on ?(T) which is not constant on each of the components of its domain.

The present paper is concerned with the solution of the coupled generalized Sylvester-transpose matrix equations {A1XB1 + C1XD1 + E1XTF1 = M1, A2XB2 + C2XD2 + E2XTF2 = M2, including the well-known Lyapunov and Sylvester matrix equations. Based on a variant of biconjugate residual (BCR) algorithm, we construct and analyze an efficient algorithm to find the (least Frobenius norm) solution of the general Sylvester-transpose matrix equations within a finite number of iterations in the absence of round-off errors. Two numerical examples are given to examine the performance of the constructed algorithm.

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

In this paper, we are going to obtain numerical solutions of the improved Boussinesq equation with the aid of Galerkin quadratic B-spline finite element method. To test the accuracy and efficiency of the current method, four test problems have been used. These are solitary wave movement, interaction of two solitary waves, wave break-up and blow-up of solutions. Their results have been compared with those available in the literature for different values of space and time steps. Also, the error norms L2 and L1 have been computed and presented in comparison.

In this paper we show many spectral properties that are inherited by m-complex symmetric and m-skew complex symmetric operators and give new results or recapture some known ones for complex symmetric operators.

In this paper, we obtain the spectrum of signed complete and complete bipartite graphs whose negative edges form a matching. Moreover, we construct a family of signed complete graphs having symmetric spectrum.

This paper introduces the definition of a Lie rough group as a natural development of the concepts of a smooth manifold and a rough group on an approximation space. Furthermore, the properties of Lie rough groups are discussed. It is shown that every Lie rough group is a topological rough group, and that the product of two Lie rough groups is again a Lie rough group. We define the concepts of Lie rough subgroups and Lie rough normal subgroups. Finally, our aim is to give an example by using definition of Lie rough homomorphism sets G and H.

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The first order and the second order of accuracy difference schemes approximately solving this problem are presented. The convergence estimates for the solutions of these difference schemes are obtained. Theoretical results are supported by numerical example.

The aim of this paper is to describe the elements in rings with involution which are EP. Especially, contact is established between EP elements and the solutions of certain equations. In addition, we reduce the preliminary requirements met in some existing results.

In this paper, soft bigeneralized topological spaces are introduced. Besides, we defined soft open set, soft closed set, soft closure set, soft interior set, soft basis and soft neighborhood on the soft bigeneralized topological spaces. Furthermore, some important theorems are proved and interesting examples are given.

Let $G^{\sigma}$ be an oriented graph and $S(G^{\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\sigma}$. The skew-rank of $G^{\sigma}$, denoted by $sr(G^{\sigma})$, is the rank of $S(G^{\sigma})$. Denote by $d(G)=|E(G)|-|V(G)|+\theta(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\theta(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^{\sigma})\leq r(G)+2d(G)$ for an oriented graph $G^{\sigma}$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^{\sigma})$ of an oriented graph $G^{\sigma}$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^{\sigma})\geq r(G)-2d(G)$ for an oriented graph $G^{\sigma}$ and characterize the graphs whose skew-rank attain the lower bound., Comment: 12 pages

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

A signed graph consists of a (simple) graph G=(V,E) together with a function ? : E ? {+,-} called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed ?-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed ?-graphs and we study the spectral characterization of the signed ?-graphs containing a triangle.

In this paper, we introduce the incomplete Horadam numbers Wn(k), and hyper-Horadam numbers W(k)n, which generalize the Horadam?s numbers defined by the recurrence Wn = pWn-1 + qWn-2, with W0 = a and W1 = b. We give some combinatorial properties. As an application, we evaluate a lower and upper bounds for the spectral norms of r-circulant matrices associated with these two generalizations. Moreover, we establish a new bounds for the spectral norms of r-circulant matrices associated with Horadam?s numbers in terms of incomplete Horadam and hyper-Horadam numbers.

In this paper, we introduce the concept of fuzzy soft filters and study some of their properties. Also, we study the notion of convergence of fuzzy soft filters in fuzzy soft topological spaces. We prove the existence of product fuzzy soft filters.

In [3], it was shown that there are no warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold of a warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function and show that there exists a class of non-trivial warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold is pointwise slant by giving some examples. We present a characterization theorem and establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant function for such warped product submanifolds of a locally product Riemannian manifold. The equality case is also considered.

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ? Ai ? M (i = 1,...,n) for some scalars m < M and ? = (w1,...,wn) is a weight vector with wi ? 0 and ?n,i=1 wi=1, then ?p (?n,i=1 wiAi)? ?p?p(Pt(?,A)) and ?p (?n,i=1 wiAi) ? ?p?p(?(?,A)), where p > 0,? = max {(M+m)2/4Mm,(M+m)2/42p Mm}, ? is a positive unital linear map and t ? [-1,1]\{0}.

In this paper, Bernstein polynomials method (BPM) and their operational matrices are adopted to obtain approximate analytical solutions of variational problems. The operational matrix of differentiation is introduced and utilized to reduce the calculus of variations problems to the solution of system of algebraic equations. The solutions are obtained in the form of rapidly convergent finite series with easily computable terms. Comparison between the present method and the homotopy perturbation method (HPM), the non-polynomial spline method and the B-spline collocation method are made to show the effectiveness and efficiency for obtaining approximate solutions of the calculus of variations problems. Moreover, convergence analysis based on residual function is investigated to verified the numerical results.

In this paper we will investigate the problem of finding the open-locating-dominating number for some classes of planar graphs - convex polytopes. We considered Dn, Tn, Bn, Cn, En and Rn classes of convex polytopes known from the literature. The exact values of open-locating-dominating number for Dn and Rn polytopes are presented, along with the upper bounds for Tn, Bn, Cn, and En polytopes.

A strongly direct sum decomposition given by a strongly nil-clean endomorphism motivates us to introduce a class of new Drazin inverses, which is called strong Drazin inverse. In this paper, some basic properties of the strong Drazin inverse are obtained. We show the Cline’s Formula and Jacobson’s Lemma for the strong Drazin inverse. Further, some additive results for the strong Drazin inverse are presented under additional conditions that ab = 0 or ab = ba.

Solving linear systems is a classical problem of engineering and numerical analysis which has various applications in many sciences and engineering. In this paper, we study efficient iterative methods, based on the diagonal and off-diagonal splitting of the coefficient matrix A for solving linear system Ax = b, where A ? Cnxn is nonsingular and x,b ? Cnxm. The new method is a two-parameter two-step method that has some iterative methods as its special cases. Numerical examples are presented to illustrate the effectiveness of the new method.

In this paper, firstwe give some necessary and sufficient conditions for multiplication conditional type operators between two Lp-spaces to have closed range. Then we investigate Fredholm ones when the underlying measure space is non-atomic. Finally we give some examples.

In this paper, Cline's formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when $\left\{ \begin{aligned} acd=dbd \ \\ dba=aca. \\ \end{aligned} \right.$ Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

In this paper, we study Weyl type theorems for complex symmetric operator matrices. In particular, we give a necessary and sufficient condition for complex symmetric operator matrices to satisfy a-Weyl?s theorem. Moreover, we also provide the conditions for such operator matrices to satisfy generalized a-Weyl?s theorem and generalized a-Browder?s theorem, respectively. As some applications, we give various examples of such operator matrices which satisfy Weyl type theorems.

In this paper, we shall establish some Fej?r-type inequalities for L-Lipschitzian functions. These inequalities can connect with Fej?r inequality (1). Also, some applications to convex function, ?-th moment, mathematical expectation of a random variable and Euler?s Beta function are provided.