A quasilinear Keller-Segel model with saturated discontinuous advection

Generalities and local theory.- The global existence problem.- Theory of monotone operators and applications.- Smoothing effect for some nonlinear evolution equations.- Schrodinger and wave equations with a logarithmic nonlinearity.- The linear case: Hilbertian theory and applications.- Some nonlinear monotone cases.- Some nonlinear, non monotone cases.- Autonomous dissipative systems.- General results for quasi-autonomous periodic systems.- More on asymptotic behavior for solutions of the nonlinear dissipative forced wave equation.- Boundedness of trajectories for quasi-autonomous dissipative systems.- Almost-periodic quasi-autonomous dissipative systems in a Hilbert space.

In this paper, sufficient conditions for trajectory controllability of nonlinear fractional integro‐differential systems involving Caputo fractional derivative of order α∈(1,2] in finite and as well as in infinite dimensional Hilbert spaces are obtained. Our tools of study include set‐valued functions, theory of monotone operators and α‐order cosine family of operators. The main results are well illustrated with the aid of examples.

Remarks on perturbation theory of maximal monotone and m-accretive operators in Banach spaces

The theory of monotone operators captured the attention of mathematicians not only because of the fineness of the results but also because of the large number of applications, especially in fields like nonlinear analysis, variational inequalities and partial differential equations (see for instance [88, 130]). Different attempts to establish links to the convex analysis have been made (see [90,91,119]), but the most fruitful ones turned out to be the ones based on the so-called Fitzpatrick function discovered by Simons Fitzpatrick in [70]. Neglected for many years until re-popularized in [7, 8, 52, 98, 104–106, 121], this class of functions along with its extensions have given rise to a great number of publications which rediscovered and extended the important results of the theory of monotone operators by using tools from the convex analysis. The investigations we make in this chapter are to be seen belonging to this class of results, whereby, we concentrate ourselves on results based on the conjugate duality theory.

In this chapter, we introduce the reader to the theory of monotone operators, \(\phi \)-accretive operators and their generalizations. The concept of monotone operator was first introduced by Minty in his paper of 1962, Minty, Duke Math J 29:341–346, 1962, [400], wherein he gave a surjectivity theorem for such operators. Since then, this theory is widely developed and has found useful applications in the investigation of the solvability of nonlinear operator equations and in particular of partial differential equations and integral equations. Our purpose is to give a systematic treatment (with historical development) of various topics in the theory of monotone operators needed for such an investigation.

In this chapter we investigate the abstract Hammerstein equation (1) , with the nonlinear operator F: X* → X and the linear operator K: X → X*, with the aid of the theory of monotone operators and the fixed-point index. The applications relate to Hammerstein integral equations and boundary value problems for semilinear elliptic partial differential equations. Whereas, in Chapter 7, we made use of B-spaces of smooth functions and monotone increasing operators in ordered B-spaces, we now work in L p (G)-spaces and apply the theory of monotone operators.

Using the Leray-Schauder fixed point theorem and the theory of dissipative operators, we first consider the solution of the linear Burgers-KdV equation. Second we find the control function h by using some inequalities, the theory of monotone operators and division integral theory. Finally we prove the exact controllability for the Burgers-KdV equation, that is to say, for suitable T > 0, for any given initial state u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> ∈ H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> and terminal state u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> ∈ H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> , we can find an appropriate control function h which drives the solution u for Burgers-KdV equation to satisfy u (x, 0) = u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> and u (x, T) = u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> .

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.

The AC power flow equations underlie all operational aspects of power systems. They are solved routinely in operational practice using the Newton-Raphson method and its variants. These methods work well given a good initial “guess” for the solution, which is always available in normal system operations. However, with the increase in levels of intermittent generation, the assumption of a good initial guess always being available is no longer valid. In this paper, we solve this problem using the theory of monotone operators. We show that it is possible to compute (using an offline optimization) a “monotonicity domain” in the space of voltage phasors. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or provably certify that no solutions exist in it. We validate the approach on several IEEE test cases and demonstrate that the offline optimization can be performed tractably and the computed “monotonicity domain” includes all practically relevant power flow solutions.

The AC power flow equations underlie all operational aspects of power systems. In this paper, Here we solve this problem using the theory of monotone operators. We show that it is possible to characterize a “contractivity domain” in the power flow variables (voltage magnitudes and phases). The construction of these domains depends on the specific representation chosen for the power flow equations, so that different representations lead to different domains. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or certify that no solutions exist in it. We validate the approach on several IEEE test cases.

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.

The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p-Laplace operator.

Nonlinear Oscillations in the Frame of Alternative Methods

SynopsisA non-linear spectral theory is developed which includes the spectral theory of linear operators and the theory of (maximal) monotone operators. In this nonlinear theory certain polytone operators will play the role of the linear or monotone operators. The concept of λ-polytonicity allows the characterization of regular points in terms of maximality. Furthermore, properties of the spectrum of non-linear operators are discussed in terms of the corresponding properties of their linearizations and vice versa.

We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson–Cowan model with a smooth firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson–Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes.