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In this paper, we present a new alternative proof of the spectral radius inequality theorem for products of Hilbert space operators due to F. Kittaneh, in which the spectral radius of 𝐴 obtained by using the formula 𝑟 𝐴 = lim𝑛→∞ 𝐴 𝑛 1 𝑛 , where . represents any operator norm. Then, we use this alternative technique to prove some related results. Indeed, the spectral radius inequalities presented in this paper have diverse applications.

In this paper, we applied analytical-numerical method to approximate solutions of two-point boundary value problems of fourth-order Volterra integrodifferential equations based on the reproducing kernel theory . The solution is represented in the form of convergent series with easily computable components. The solution methodology is based on generating the orthogonal basis from the obtained kernel function in the space W5 2 [a, b]. The n-term approximation is obtained and proved to converge to the analytical solution. Moreover, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solutions and its all derivatives will be applicable. Numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate the method is simple and effective

This paper proposes an efficient numerical algorithm to obtain an approximate solution of first-order periodic boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. Its exact solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

In this paper, we study the oscillatory properties for asymptotic behaviors of solutions of a class of nonlinear second-order stochastic Itô equations. Meanwhile, we investigate existence of zeros of its solutions with probability 1. Sufficient conditions for the oscillation and nonoscillation of solutions are obtained on the half-line [t0,∞) for every t0 > 0.

In this paper we compute the number of spanning trees on the following decorated centred cubic lattices; base- centred cubic, side- centered cubic and edge- centred cubic lattices. For these lattices we also determine the asymptotic growth constant.

In this paper, we present a new alternative proof of the spectral radius inequality theorem for products of Hilbert space operators due to F. Kittaneh, in which the spectral radius of 𝐴 obtained by using the formula 𝑟 𝐴 = lim𝑛→∞ 𝐴 𝑛 1 𝑛 , where . represents any operator norm. Then, we use this alternative technique to prove some related results. Indeed, the spectral radius inequalities presented in this paper have diverse applications.

In this paper, we applied analytical-numerical method to approximate solutions of two-point boundary value problems of fourth-order Volterra integrodifferential equations based on the reproducing kernel theory . The solution is represented in the form of convergent series with easily computable components. The solution methodology is based on generating the orthogonal basis from the obtained kernel function in the space W5 2 [a, b]. The n-term approximation is obtained and proved to converge to the analytical solution. Moreover, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solutions and its all derivatives will be applicable. Numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate the method is simple and effective

This paper proposes an efficient numerical algorithm to obtain an approximate solution of first-order periodic boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. Its exact solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

In this paper, we study the oscillatory properties for asymptotic behaviors of solutions of a class of nonlinear second-order stochastic Itô equations. Meanwhile, we investigate existence of zeros of its solutions with probability 1. Sufficient conditions for the oscillation and nonoscillation of solutions are obtained on the half-line [t0,∞) for every t0 > 0.

In this paper we compute the number of spanning trees on the following decorated centred cubic lattices; base- centred cubic, side- centered cubic and edge- centred cubic lattices. For these lattices we also determine the asymptotic growth constant.