Engineering topics

Hale, J. K.

Publications and source records attributed to Hale, J. K..

At least 19 records

NASA NTRS1974

Behavior near constant solutions of functional differential equations

Techniques have been developed to determine in a systematic way the local behavior near constant solutions. Local integral manifolds play a very important role in this development, as they have also for ordinary differential equations. An attempt is made to indicate a few more applications of these methods to some problems in bifurcation in the spirit of Sotomayor (to appear) and to a growth model of Cooke and Yorke (to appear). It is also shown how to prove a theorem on stability under constantly acting disturbances using these methods.

Hale, J. K.↗

NASA NTRS1974

Coincidence degree and periodic solutions of neutral equations

A study is made of the problem of existence of periodic solutions for some nonautonomous neutral functional differential equations. It is essentially an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations recently obtained by Hale (1973) and of a generalized Leray-Schauder theory developed by Mawhin (1972, and to appear). Although their proofs are surprisingly simple, the obtained results are nontrivial extensions to the neutral case of a number of recent existence theorems for periodic solutions of functional differential equations.

Hale, J. K.↗

NASA NTRS1973

Coincidence degree and periodic solutions of neutral equations

The problem of existence of periodic solutions for some nonautonomous neutral functional differential equations is examined. It is an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations and of a generalized Leray-Schauder theory. Although proofs are simple, the results are nontrivial extensions to the neutral case of existence theorems for periodic solutions of functional differential equations.

Hale, J. K.↗

NASA NTRS1973

Fixed point theorems and dissipative processes.

Operators of the type considered by Hale et al. (1972) are used to show that under certain conditions there is a fixed point in a dissipative map within a Banach space. The conditions required for the existence of this fixed point are discussed in detail. Several fixed point theorems are formulated and proved.

Hale, J. K.↗

NASA NTRS1973

Continuous dependence of fixed points of condensing maps

Many problems in analysis are concerned with the dependence upon parameters of fixed points of maps. For contraction mappings, criteria are relatively easy to obtain and have been known for some time. In the study of solutions of functional differential equations, more general results were needed. It is the purpose of this paper to give a rather general fixed-point theorem for condensing maps depending on a parameter, to prove continuous dependence and to indicate how many of the previous results are special cases.

Hale, J. K.↗

NASA NTRS1972

Smoothing properties of neutral equations

A neutral functional differential equation is defined as d/dt(Dx sub t) = f(x sub t), with D linear, continuous, atomic at zero. The solution generally is no smoother than the initial data after any finite number of steps. A more restrictive class of D-operators for which some smoothing takes place after an infinite number of steps is defined. This result indicates that a solution can be in an omega-limit set only if it corresponds to initial data which are smooth. A space which can be considered as a Banach space with the topology of uniform convergence is also defined.

Hale, J. K.↗

NASA NTRS1972

Theory of a general class of dissipative processes.

Development of a theory of periodic processes that is of sufficient generality for being applied to systems defined by partial differential equations (distributed parameter systems) and functional differential equations of the retarded and neutral type (hereditary systems), as well as to systems arising in the theory of elasticity. In particular, the attempt is made to develop a meaningful general theory of dissipative periodic systems with a wide range of applications.

Hale, J. K.↗

NASA NTRS1972

Oscillations in neutral functional differential equations

Fredholm alternative for the existence of periodic solutions of a nonhomogeneous linear neutral functional differential equation was announced. Boundary value problems were used to solve the theorems.

Hale, J. K.↗

NASA NTRS1972

Local behavior of autonomous neutral functional differential equations.

Basic problems for a special class of neutral functional differential equations (NFDE) are formulated, and some contributions to a general qualitative theory in the neighborhood of an equilibrium point are indicated. The properties of a NFDE (G,f) are examined to determine in what sense these properties are insensitive to small changes in (G,f) in the topology G x F. The special class of equations that is introduced includes retarded functional differential equations and difference equations.

Hale, J. K.↗

NASA NTRS1972

Fixed point theorems and dissipative processes

The deficiencies of the theories that characterize the maximal compact invariant set of T as asymptotically stable, and that some iterate of T has a fixed point are discussed. It is shown that this fixed point condition is always satisfied for condensing and local dissipative T. Applications are given to a class of neutral functional differential equations.

Hale, J. K.↗

NASA NTRS1971