Journal of Mathematical Extension
Volume:17 Issue: 10, Oct 2023

In this article, first we are going to review the concept of ordinary frames , in more general case in measure spaces, namely, gcframes. We try to develop the use of measure space in describing frames. Then by means of the gc-frames, we shall introduce gn-operators, which we shall show that each trace class operator has a vector-valued integral representation and vice-versa

In this parer, we give a necessary condition for Sasakian manifolds to be generalized weakly symmetric. We prove the odddimensional spheres are the only generalized weakly symmetric Sasakian manifolds. Then, we show that generalized weakly Ricci-symmetric Sasakian manifolds are Einstein. Thereafter, we define the sense of weakly parallel Riemannian submanifolds and show that every weakly parallel invariant submanifold of a Sasakian manifold is totally geodesic. Finally, we provide some examples which verify our main results.

In this paper, several inequalities involving the HilbertSchmidt numerical radius inequalities for 2 × 2 operator matrices operators are established. In particular, we obtain some generalizations and refinements of earlier inequalities. Some upper and lower bounds for the Hilbert-Schmidt numerical radius inequalities for 2 × 2 operator matrices operators is also given.

In this article, the current work suggests an efficient numerical approximation based on an operational matrix of Bernstein Polynomials to obtain numerical solution of high-order of integro-differential equations. At first, we present the integral and differential operator matrix of Bernstein Polynomials, and then apply this operator to the governing equation to transform it into an algebraic equations. Solving this system yields an approximate solution for the equation under study. Also, the convergence and error analysis for this method are study. To demonstrate the effectiveness of this scheme, we provide several numerical examples and compare the results with the exact solution and one of the other well-known method such as collocation Bernoulli method..

Let (A1, A2, A3) be a triple of nonempty convex subsets of a metric space Ω. In this paper, we determine optimal problems of the best proximity pair by proximal normal structure between two sets A1 and A2 with the help of a third set A3 and we find some necessary and sufficient conditions for existence these optimal problems. Also, we provide an example to illustrate the convergence behavior of our proposed results.

In this paper, we investigate the existence of asymptotically almost automorphic mild solution for a class of integro-differential equations. The existence results are established through the application of M¨onch’s fixed point theorem and the utilization of measures of non-compactness. Additionally, we present an illustrative example to showcase the obtained outcomes.