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With b belonging to a new B M O θ (ρ) space, L = − △ + V is a Schrödinger operator on R n with nonnegative potential V belonging to the reverse Hölder class R H n / 2 . The fractional integral operator associated with L is denoted by I β L . We investigate the boundedness of I β L and [ b , I β L ] , which are its commutators with b θ (ρ) on vanishing generalized mixed Morrey spaces V M p → , φ α , V related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V . The boundedness of the operator I β L is ensured by finding sufficient conditions on the pair (φ 1 , φ 2) , which goes from M p → , φ 1 α , V to M q → , φ 2 α , V , and from V M p → , φ 1 α , V to V M q → , φ 2 α , V , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β . When b belongs to B M O θ (ρ) and (φ 1 , φ 2) satisfies some conditions, we also show that the commutator operator [ b , I β L ] is bounded from M p → , φ 1 α , V to M q → , φ 2 α , V and from V M p → , φ 1 α , V to V M q → , φ 2 α , V . [ABSTRACT FROM AUTHOR]

Abstract With b belonging to a new B M O θ ( ρ ) $BMO_{\theta}(\rho )$ space, L = − △ + V $L=-\triangle +V$ is a Schrödinger operator on R n ${\mathbb{R}^{n}}$ with nonnegative potential V belonging to the reverse Hölder class R H n / 2 $RH_{n/2}$ . The fractional integral operator associated with L is denoted by I β L ${\mathcal{I}}_{\beta}^{L}$ . We investigate the boundedness of I β L ${\mathcal{I}}_{\beta}^{L}$ and [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ , which are its commutators with b θ ( ρ ) $b_{\theta}(\rho )$ on vanishing generalized mixed Morrey spaces V M p → , φ α , V $VM_{\vec{p},\varphi}^{\alpha ,V}$ related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V $M_{\vec{p},\varphi}^{\alpha ,V}$ . The boundedness of the operator I β L ${\mathcal{I}}_{\beta}^{L}$ is ensured by finding sufficient conditions on the pair ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ , which goes from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β $\sum \limits _{i=1}^{n}\frac{1}{p_{i}}-\sum \limits _{i=1}^{n}\frac{1}{q_{i}}=\beta $ . When b belongs to B M O θ ( ρ ) $BMO_{\theta}(\rho )$ and ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ satisfies some conditions, we also show that the commutator operator [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ is bounded from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ .

Abstract In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

Abstract In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form.

In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form. [ABSTRACT FROM AUTHOR]

Abstract Let L=−Δ+V $L=-\Delta+V$ be a Schrödinger operator on Rn $\mathbb{R}^{n}$, where n≥3 $n\ge3$ and the nonnegative potential V belongs to the reverse Hölder class RHq1 $RH_{q_{1}}$ for some q1>n/2 $q_{1} > n/2$. Let b belong to a new Campanato space Λνθ(ρ) $\Lambda_{\nu}^{\theta}(\rho)$ and IβL $\mathcal {I}_{\beta}^{L}$ be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL] $[b,\mathcal {I}_{\beta}^{L}]$ with b∈Λνθ(ρ) $b \in\Lambda_{\nu}^{\theta}(\rho)$ on local generalized Morrey spaces LMp,φα,V,{x0} $LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}$, generalized Morrey spaces Mp,φα,V $M_{p,\varphi}^{\alpha,V}$ and vanishing generalized Morrey spaces VMp,φα,V $VM_{p,\varphi}^{\alpha,V}$ associated with Schrödinger operator, respectively. When b belongs to Λνθ(ρ) $\Lambda_{\nu}^{\theta}(\rho)$ with θ>0 $\theta >0$, 0