Degenerate Differential Equations in Banach Spaces


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Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

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The discussions of evolution equations include the usual initial-value problems as well as periodic or more general nonlocal constraints, history-value problems, those which may change type due to a possibly vanishing coefficient of the time derivative, and other implicit evolution equations or systems including hysteresis models.

The scalar conservation law and semilinear wave equations are briefly mentioned, and hyperbolic systems arising from vibrations of elastic-plastic rods are developed. The origins of a representative sample of such problems are given in the appendix. Advanced graduate students, research mathematicians, and engineers interested in numerical analysis, applied mathematics, control theory, or dynamical systems.

The book is extremely clear and well written, and so it is very readable. It contains a large variety of examples and applications and consequently it will prove quite useful not only to mathematicians, but also to engineers and physicists. The completeness and the way of presentation makes the text understandable to anybody who can be interested in existence and uniqueness theory for initial-boundary-value problems. AMS Homepage. Join our email list. Sign up. Ordering on the AMS Bookstore is limited to individuals for personal use only.

Advanced search. S ; surv Author s Product display : R. Abstract: The objectives of this monograph are to present some topics from the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realizations of those problems in appropriate function spaces.

Volume: Export Cancel. References E. Alvarez-Pardo and C. Lizama, Mild solutions for multi-term time-fractional differential equations with nonlocal initial conditions , Electr. Zentralblatt MATH: You have access to this content.

You have partial access to this content. The solution of the Cauchy problem for equation 2 is determined by the formula. The evolution operator has the properties. In the study of 2 the main attention has been focused on the behaviour of its solutions at infinity, in dependence on the behaviour of and. An important characteristic of the equation is the general or singular exponent.

Equations with periodic and almost-periodic coefficients have been studied in detail see Qualitative theory of differential equations in Banach spaces. Equation 2 can also be considered in the complex plane. If the functions and are holomorphic in a simply-connected domain containing the point , then the formulas 3 , 4 , 5 , 5prm remain valid if the integrals are understood to be integrals over a rectifiable arc joining and. A number of other questions arises in the case when the original linear equation is not solvable for the derivative.

If the operator is boundedly invertible everywhere except at one point, say , then in the space the equation reduces to the form.

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Here the main attention is focused on the study of the behaviour of the solutions in a neighbourhood of the origin, and the analytic and non-analytic cases are distinguished. If one looks for the operator in the form of a series. For this system to be solvable, that is, for 7 to be formally solvable, it is sufficient that the spectra of the operators and do not intersect cf.

1st Edition

Spectrum of an operator , or, equivalently, that there are no points differing by an integer in the spectrum of. Under this condition the series. Now, if there are finitely many integers representable as differences of points of the spectrum of , and each of them is an isolated point of the spectrum of the transformer. If the integer points of the spectrum of the transformer are poles of its resolvent, then the functions are polynomials. Under certain restrictions on it reduces by means of Laplace integrals to an equation with a regular singularity in the algebra of matrices with entries from.

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Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

In the finite-dimensional case a complete result has been obtained: If the equation has a formal solution in the form of a power series, then it has a solution that is infinitely differentiable on for which the formal series is the Taylor series at the point. In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions. Suppose that. If the spectrum of the operator does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element belonging to the invariant subspace of corresponding to the part of the spectrum of lying in the left half-plane.

Any solution that is continuous on appears in this family. If the whole spectrum of lies in the left half-plane, then there is only one infinitely-differentiable solution.

Degenerate Differential Equations Banach Spaces by Angelo Favini Atsushi Yagi

If there are no negative integers in the spectrum of , then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator , equations of the form 6 have been considered in which and have finite smoothness, and the solutions have the same smoothness.


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A rather different picture emerges when the differential equation is unsolvable for the derivative for all , for example when is a constant non-invertible operator. Suppose that in the equation. Suppose that the operator is continuously invertible for sufficiently small. Then there are decompositions into direct sums and such that and map into and into. The operator is invertible on and maps onto. The subspace is finite-dimensional. All solutions of 8 lie in the subspace and have the form , where is the restriction of to and.

For an inhomogeneous equation , a solution exists only if has a certain smoothness and under certain compatibility conditions for the values of and its derivatives with the initial data. The number of derivatives that certain components of must have and the number of compatibility conditions are equal to the maximal length of -adjoint chains of the operator.

Mod-01 Lec-05 Examples of Norms,Cauchy Sequence and Convergence, Introduction to Banach Spaces

If these conditions are satisfied, the solution of the Cauchy problem is unique. If the operator is non-invertible for all , then all solutions of 8 lie in a subspace that has, generally speaking, infinite deficiency cf. The solution of the Cauchy problem for it is not unique. For the function in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required. Suppose that is invertible for every , so that 1 can be solved for the derivative and takes the form. Even for the simplest equation with an unbounded operator, solutions of the Cauchy problem need not exist, they may be non-unique, and they may be non-extendable to the whole semi-axis, so the main investigations are devoted to the questions of existence and uniqueness of the solutions.

A solution of the equation on the interval is understood to be a function that takes values in , is differentiable on and satisfies the equation. Sometimes this definition is too rigid and one introduces the concept of a weak solution as a function that has the same properties on and is only continuous at. Suppose that the operator has a resolvent. If , then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of as. If for every there is a unique solution of the problem 10 that is continuously differentiable on , then this solution can be extended to the whole semi-axis and can be represented in the form , where is a strongly-continuous semi-group of bounded operators on , , for which the estimate holds.

For the equation to have this property it is necessary and sufficient that. These conditions are difficult to verify.


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Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces Degenerate Differential Equations in Banach Spaces

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