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Many authors have considered and investigated generalized fractional differential operators. The main object of this present paper is to define a new generalized fractional differential operator $mathfrak{T}^{beta,tau,gamma},$ which generalized the Srivastava-Owa operators. Moreover, we investigate of the geometric properties such as univalency, starlikeness, convexity for their normalization. Further, boundedness and compactness in some well known spaces, such as Bloch space for last mention operator also are considered. Our tool is based on the generalized hypergeometric function.
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Let $Sinmathcal{M}_d(mathbb{C})^+$ be a positive semidefinite $dtimes d$ complex matrix and let $mathbf a=(a_i)_{iinmathbb{I}_k}in mathbb{R}_{>0}^k$, indexed by $mathbb{I}_k={1,ldots,k}$, be a $k$-tuple of positive numbers. Let $mathbb T_{d}(mathbf a )$ denote the set of families $mathcal G={g_i}_{iinmathbb{I}_k}in (mathbb{C}^d)^k$ such that $|g_i|^2=a_i$, for $iinmathbb{I}_k$; thus, $mathbb T_{d}(mathbf a )$ is the product of spheres in $mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $mathcal{M}_d(mathbb{C})$, we consider the generalized frame operator distance function $Theta_{( N , , , S, , , mathbf a)}$ defined on $mathbb T_{d}(mathbf a )$, given by $$ Theta_{( N , , , S, , , mathbf a)}(mathcal G) =N(S-S_{mathcal G }) quad text{where} quad S_{mathcal G}=sum_{iinmathbb{I}_k} g_i,g_i^*inmathcal{M}_d(mathbb{C})^+,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $mathcal G_0inmathbb T_{d}(mathbf a )$ of $Theta_{( N , , , S, , , mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.
The aim of this paper is twofold. First, we obtain the explicit exact formal solutions of differential equations of different types in the form with Dyson chronological operator exponents. This allows us to deal directly with the solutions to the equations rather than the equations themselves. Second, we consider in detail the algebraic properties of chronological operators, yielding an extensive family of operator identities. The main advantage of the approach is to handle the formal solutions at least as well as ordinary functions. We examine from a general standpoint linear and non-linear ODEs of any order, systems of ODEs, linear operator ODEs, linear PDEs and systems of linear PDEs for one unknown function. The methods and techniques involved are demonstrated on examples from important differential equations of mathematical physics.
Given a sequence of elements $xi={xi_n}_{nin mathbb{N}}$ of a Hilbert space, an operator $T_xi$ is defined as the operator associated to a sesquilinear form determined by $xi$. This operator is in general different to the classical frame operator but possesses some remarkable properties. For instance, $T_xi$ is self-adjoint (in an specific space), unconditionally defined and, when $xi$ is a lower semi-frame, $T_xi$ gives a simple expression of a dual of $xi$. The operator $T_xi$ and lower semi-frames are studied in the context of sequences of integer translates.
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions.
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