<abstract><p>We give a representation of solutions to linear nonhomogeneous $ \Psi $-fractional delayed differential equations with noncommutative matrices. We newly define $ \Psi $-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $ \Psi $-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.</p></abstract>

In the present paper, we construct a new class of operators based on new type Bézier bases with a shape parameter λ and positive parameter s. Our operators include some well-known operators, such as classical Bernstein, α-Bernstein, generalized blending type α-Bernstein and λ-Bernstein operators as special case. In this paper, we prove some approximation theorems for these operators. Approximation properties of our operators are illustrated on graphs for variables s, α, λ, and n. It should be mentioned that our operators for $\lambda =1$ λ = 1 have better approximation than Bernstein and α-Bernstein operators.

In this paper, we construct a new sequence of Riemann-Liouville type fractional Bernstein-Kantorovich operators K α n (f ; x) depending on a parameter α. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first and second order modulus of continuity of these operators. Moreover, we introduce a new operator that preserves affine functions from Riemann-Liouville type fractional Bernstein-Kantorovich operators. Further, we define the bivariate case of Riemann-Liouville type fractional Bernstein-Kantorovich operators and investigate the order of convergence. Some numerical results are given to illustrate the convergence of these operators and its comparison with the classical case of these operators.

In this paper, we introduce a family of blending type Bernstein operators L α,s n ( f ;x) which depends on two parameters, α and s . We prove a Korovkin type approximation theorem and obtain the rate of convergence of these operators. We also prove that these operators has monotonicity and convexity preserving properties for each α and s . So far, Lotosky matrices that generates blending type Bernstein operators were ignored. In this paper, we also introduce Lototsky matrices that generates these new family of blending type Bernstein operators.