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We obtain results of existence of weak solutions in the Hopf sense of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps A− g, where A is invertible and g is A-condensing, is used. Various problems for the Navier-Stokes equations describing the motion of the(More)
One of the most efficient methods for the study of boundary and periodic problems for nonlinear differential equations and inclusions, consists in the operator treatment of these problems in suitable functional spaces. However, for a number of problems of this sort, the maps constructed in functional spaces do not possess “nice” properties on the whole(More)
In this paper the concept of unbounded Fredholm operators on Hilbert C *-modules over an arbitrary C *-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C *-modules over C *-algebras of compact operators. In the framework of Hilbert C *-modules over C *-algebras of compact operators, the(More)
We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order(More)
Here v(t,x) = (v1, . . . ,vn) is a velocity of the medium at location x at time t, p(t,x) is a pressure, ρ, μ0, μ1, λ are positive constants, Div means a divergence of a matrix, the matrix (v) has coefficients i j(v)(t,x) = (1/2)(∂vi(t,x)/∂xj + ∂vj(t,x)/∂xi). In (1.1) and in the sequel repeating indexes in products assume their summation. The function z(τ;(More)
and Applied Analysis 3 holding for any t ∈ 0, T . Here ( a ij x e j k ) t : ∫ t 0 a ij x, t, τ · e j k x, τ dτ 2.4 are Volterra integral operators with kernels a ij x, t, τ ; a hk ij 0 x, t are instantaneous elastic coefficients out-of-integral terms and a ij x : a hk ij 0 x, t a ij x ; fi0 are components of a vector of external forces; φi x, t are(More)
Here u is the velocity vector, p is the pressure function, f is the body force, and σ is the deviator of the stress tensor (all of them depend on a point x of an arbitrary domain Ω in the space Rn, n= 2,3, and on a moment of time t). The gradient grad and the divergence div are taken with respect to the variable x. The divergence Divσ of a tensor σ is the(More)
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