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This paper studies the Stokes system -Δ⁢𝐮+∇⁡ρ=𝐟{-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}}, ∇⋅𝐮=χ{\nabla\cdot{\mathbf{u}}=\chi} in Ω with three boundary conditions: 𝐧⋅𝐮=𝐧⋅𝐠,\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}},𝐧×(∇×𝐮)=𝐧×𝐡\displaystyle{\mathbf{n}}\times(\nabla\times{\mathbf{u}})={\mathbf{n}}\times{% \mathbf{h}}on ⁢∂⁡Ω,\displaystyle\phantom{}\text{on }\partial\Omega,𝐧⋅𝐮=𝐧⋅𝐠,\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g},𝝉⋅[∂⁡𝐮∂⁡𝐧-ρ⁢𝐧+b⁢𝐮]=𝐡⋅τ\displaystyle{\boldsymbol{\tau}}\cdot\bigg{[}\frac{\partial{\mathbf{u}}}{% \partial{\mathbf{n}}}-\rho{\mathbf{n}}+b{\mathbf{u}}\bigg{]}={\mathbf{h}}\cdot\tauon ⁢∂⁡Ω,\displaystyle\phantom{}\text{on }\partial\Omega,𝐧⋅𝐮=𝐧⋅𝐠,\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}},[T⁢(𝐮,ρ)⁢𝐧+b⁢𝐮]⋅τ=𝐡⋅τ\displaystyle[T({\mathbf{u}},\rho){\mathbf{n}}+b{\mathbf{u}}]\cdot\tau={% \mathbf{h}}\cdot\tauon ⁢∂⁡Ω.\displaystyle\phantom{}\text{on }\partial\Omega. Here Ω is a bounded simply connected planar domain. We find a necessary and sufficient condition for the existence of a solution in Sobolev spaces Ws,q⁢(Ω;ℝ2)×Ws-1,q⁢(Ω){W^{s,q}(\Omega;{\mathbb{R}}^{2})\times W^{s-1,q}(\Omega)}, with 1+1/q<s<∞{1+1/q<s<\infty}, in Besov spaces Bsq,r⁢(Ω;ℝ2)×Bs-1q,r⁢(Ω){B_{s}^{q,r}(\Omega;{\mathbb{R}}^{2})\times B_{s-1}^{q,r}(\Omega)}, with 1+1/q<s<∞{1+1/q<s<\infty}, and classical solutions in 𝒞k,α⁢(Ω¯,ℝ2)×𝒞k-1,α⁢(Ω¯){{\mathcal{C}}^{k,\alpha}(\overline{\Omega},{\mathbb{R}}^{2})\times{\mathcal{C% }}^{k-1,\alpha}(\overline{\Omega})}, with 0<α<1{0<\alpha<1}, k∈ℕ{k\in{\mathbb{N}}}.

In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of (f,g){(f,g)}-generalized k -strictly pseudononspreading mappings in a CAT⁢(0){\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete CAT⁢(0){\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

In the present work, the interior spectral data is used to investigate the inverse problem for a diffusion operator with an impulse on the half line. We show that the potential functions q0⁢(x){q_{0}(x)} and q1⁢(x){q_{1}(x)} can be uniquely established by taking a set of values of the eigenfunctions at some internal point and one spectrum.

In this paper, we study the stability of k -Hessian overdetermined problems under small perturbations in R2{R^{2}}. Additionally, we give a proof for k -Hessian partially overdetermined problems and the related stability problems in R2{R^{2}}. The proof mainly replies to choosing a suitable auxiliary f .