Non-linear factorization of linear operators

2009 ◽  
Vol 41 (4) ◽  
pp. 663-668 ◽  
Author(s):  
W. B. Johnson ◽  
B. Maurey ◽  
G. Schechtman
Keyword(s):  
Linear Operators ◽  
Non Linear ◽  
Linear Factorization

Complementary bounds for inner products associated with non-linear equations

1984 ◽  
Vol 96 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
R. J. Cole ◽  
J. Mika ◽  
D. C. Pack
Keyword(s):  
Real Hilbert Space ◽  
Linear Equations ◽  
Linear Operators ◽  
Inner Product ◽  
Upper And Lower Bounds ◽  
Non Linear Equation ◽  
Non Linear ◽  
Non Linear Operator ◽  
Non Linear Equations ◽  
Unknown Solution

SynopsisFunctionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.

scholarly journals Lipschitz Regularity of Solutions to Two-phase Free Boundary Problems

2017 ◽  
Vol 2019 (7) ◽  
pp. 2204-2222 ◽  
Author(s):  
D De Silva ◽  
O Savin
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Lipschitz Continuity ◽  
Free Boundary Problems ◽  
Linear Operators ◽  
Regularity Of Solutions ◽  
Two Phase ◽  
Boundary Problems ◽  
Lipschitz Regularity ◽  
Non Linear

AbstractWe prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully non-linear operators.

scholarly journals On the constructive approximation of non-linear operators in the modelling of dynamical systems

1997 ◽  
Vol 39 (1) ◽  
pp. 1-27 ◽  
Author(s):  
A. P. Torokhti ◽  
P. G. Howlett
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Input Signal ◽  
Linear Operators ◽  
Non Linear ◽  
Constructive Approximation ◽  
Modelling Process ◽  
Theoretical Procedure ◽  
Small Disturbances

AbstractIn this paper we propose a systematic theoretical procedure for the constructive approximation of non-linear operators and show how this procedure can be applied to the modelling of dynamical systems. We extend previous work to show that the model is stable to small disturbances in the input signal and we pay special attention to the role of real number parameters in the modelling process. The implications of computability are also discussed. A number of specific examples are presented for the particular purpose of illustrating the theoretical procedure.

Identification of a class of non-linear operators

1978 ◽  
Vol 18 (2) ◽  
pp. 7-15
Author(s):  
M.L. Agranovskii ◽  
R.D. Baglai ◽  
K.K. Smirnov
Keyword(s):  
Linear Operators ◽  
Non Linear

Non-linear operators on sets of measures

1976 ◽  
Vol 109 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Richard A. Alò ◽  
Charles A. Cheney ◽  
André de Korvin
Keyword(s):  
Linear Operators ◽  
Non Linear

scholarly journals Existence, Uniqueness and C −Differentiability of Solutions in a Non-linear Model of Cancerous Tumor

2018 ◽  
Vol 10 (6) ◽  
pp. 43
Author(s):  
Gossan D. Pascal Gershom ◽  
Yoro Gozo ◽  
Bailly Balé
Keyword(s):  
Mathematical Modeling ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Linear Operators ◽  
System Of Nonlinear Equations ◽  
Small Perturbations ◽  
Direct Mapping ◽  
Non Linear ◽  
Cancer Tumor ◽  
Differentiability Of Solutions

In this paper, we prove the existence and uniqueness of the weak solution of a system of nonlinear equations involved in the mathematical modeling of cancer tumor growth with a non homogeneous divergence condition. We also present  a new concept of generalized differentiation of non linear operators : C-differentiability. Through this notion, we also prove the uniqueness and the C-differentiability of the solution when the system is perturbed by a certain number of parameters. Two results have been established. In the first one, differentiability is according to Fréchet. The proof is given uses the theorem of reciprocal functions in Banach spaces. First of all, we give the proof of strict differentiability of a direct mapping, according to Fréchet. In the second result, differentiability is understood in a weaker sense than that of Fréchet. For the proof we use Hadamard's theorem of small perturbations of Banach isomorphism of spaces as well as the notion of strict differentiability.

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