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Stability of Solutions to Some Evolution Problems
, 2011
"... Abstract. Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: ˙u = A(t)u + F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+: = [0, ∞), A(t) is a linear dt dissipative operator: Re(A( ..."
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Abstract. Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: ˙u = A(t)u + F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+: = [0, ∞), A(t) is a linear dt dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), γ(t) ≥ 0, F (t, u) is a nonlinear operator, ‖F (t, u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p are constants, ‖b(t) ‖ ≤ β(t), β(t) ≥ 0 is a continuous function. Sufficient conditions are given for the solution u(t) to problem (*) to exist for all t ≥ 0, to be bounded uniformly on R+, and a bound on ‖u(t) ‖ is given. This bound implies the relation limt→ ∞ ‖u(t) ‖ = 0 under suitable conditions on γ(t) and β(t). The basic technical tool in this work is the following nonlinear inequality: ˙g(t) ≤ −γ(t)g(t) + α(t, g(t)) + β(t), t ≥ 0; g(0) = g0, which holds on any interval [0, T) on which g(t) ≥ 0 exists and has bounded derivative from the right, ˙g(t): = lims→+0 g(t+s)−g(t) s. It is assumed that γ(t), and β(t) are realvalued, continuous functions of t, defined on R+: = [0, ∞), the function α(t, g) is defined for all t ∈ R+, locally Lipschitz with respect to g uniformly with respect to t on any compact subsets [0, T], T < ∞.If there exists a function µ(t)> 0, µ(t) ∈ C 1 (R+), such that α t, 1 + β(t) ≤ µ(t) 1 γ(t) − µ(t)
DSM for general nonlinear equations
 APPL. MATH. LETT., (2012)
"... If F: H → H is a map in a Hilbert space H, F ∈ C2 loc, and there exists a solution y, possibly nonunique, such that F (y) = 0, F ′(y) ̸ = 0, then equation F (u) = 0 can be solved by a DSM (Dynamical Systems Method) and the rate of convergence of the DSM is given provided that a sourcetype assump ..."
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If F: H → H is a map in a Hilbert space H, F ∈ C2 loc, and there exists a solution y, possibly nonunique, such that F (y) = 0, F ′(y) ̸ = 0, then equation F (u) = 0 can be solved by a DSM (Dynamical Systems Method) and the rate of convergence of the DSM is given provided that a sourcetype assumption holds. A discrete version of the DSM yields also a convergent iterative method for finding y. This method converges at the rate of a geometric series. Stable approximation to a solution of the equation F (u) = f is constructed by a DSM when f is unknown but the noisy data fδ are known, where fδ − f  ≤ δ.
On the DSM Newtontype method
"... A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Ex ..."
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A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Existence and uniqueness of the solution to evolution equation ˙u(t) = −[F ′ (u(t))] −1 (F (u(t)) − h), u(0) = u0, is proved without assuming that F ′ (u) satisfies the Lipschitz condition. The method of the inverse function theorem.
Stability of Solutions to Evolution Problems
, 2013
"... Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, ..."
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Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, in particular, sufficient conditions for this limit to be zero. The evolution problem is: ˙u = A(t)u + F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+:= dt [0, ∞), A(t) is a linear dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), where F (t, u) is a nonlinear operator, ‖F (t, u) ‖ ≤ c0‖u‖p, p> 1, c0 and p are positive constants, ‖b(t) ‖ ≤ β(t), and β(t) ≥ 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The nonclassical case γ(t) ≤ 0 is also treated.
Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces
"... Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ..."
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Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ‖A −1 a (u) ‖ ≤ c1 a  b, a > 0, b> 0, a ∈ L. Here a may be a complex number, and L is a smooth path on the complex aplane, joining the origin and some point on the complex a−plane, 0 < a  < ɛ0, where ɛ0> 0 is a small fixed number, such that for any a ∈ L estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) ˙u(t) = −A −1 a(t) (u(t))[F (u(t)) + a(t)u(t) − f], du u(0) = u0, ˙u = dt, converges to y as t → +∞, where a(t) ∈ L, F (y) = f, r(t): = a(t), and r(t) = c4(t + c2) −c3, where cj> 0 are some suitably chosen constants, j = 2, 3, 4. Existence of a solution y to the equation F (u) = f is assumed. It is also assumed that the equation F (wa) + awa − f = 0 is uniquely solvable for any f ∈ X, a ∈ L, and lima→0,a∈L ‖wa − y ‖ = 0.