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Differential equations with "maxima"-differential equations that
contain the maximum of the unknown function over a previous
interval-adequately model real-world processes whose present state
significantly depends on the maximum value of the state on a past
time interval. More and more, these equations model and regulate
the behavior of various technical systems on which our
ever-advancing, high-tech world depends. Understanding and
manipulating the theoretical results and investigations of
differential equations with maxima opens the door to enormous
possibilities for applications to real-world processes and
phenomena. Presenting the qualitative theory and approximate
methods, Differential Equations with Maxima begins with an
introduction to the mathematical apparatus of integral inequalities
involving maxima of unknown functions. The authors solve various
types of linear and nonlinear integral inequalities, study both
cases of single and double integral inequalities, and illustrate
several direct applications of solved inequalities. They also
present general properties of solutions as well as existence
results for initial value and boundary value problems. Later
chapters offer stability results with definitions of different
types of stability with sufficient conditions and include
investigations based on appropriate modifications of the Razumikhin
technique by applying Lyapunov functions. The text covers the main
concepts of oscillation theory and methods applied to initial and
boundary value problems, combining the method of lower and upper
solutions with appropriate monotone methods and introducing
algorithms for constructing sequences of successive approximations.
The book concludes with a systematic development of the averaging
method for differential equations with maxima as applied to
first-order and neutral equations. It also explores different
schemes for averaging, partial averaging, partially additiv
Differential equations with "maxima"-differential equations that
contain the maximum of the unknown function over a previous
interval-adequately model real-world processes whose present state
significantly depends on the maximum value of the state on a past
time interval. More and more, these equations model and regulate
the behavior of various technical systems on which our
ever-advancing, high-tech world depends. Understanding and
manipulating the theoretical results and investigations of
differential equations with maxima opens the door to enormous
possibilities for applications to real-world processes and
phenomena. Presenting the qualitative theory and approximate
methods, Differential Equations with Maxima begins with an
introduction to the mathematical apparatus of integral inequalities
involving maxima of unknown functions. The authors solve various
types of linear and nonlinear integral inequalities, study both
cases of single and double integral inequalities, and illustrate
several direct applications of solved inequalities. They also
present general properties of solutions as well as existence
results for initial value and boundary value problems. Later
chapters offer stability results with definitions of different
types of stability with sufficient conditions and include
investigations based on appropriate modifications of the Razumikhin
technique by applying Lyapunov functions. The text covers the main
concepts of oscillation theory and methods applied to initial and
boundary value problems, combining the method of lower and upper
solutions with appropriate monotone methods and introducing
algorithms for constructing sequences of successive approximations.
The book concludes with a systematic development of the averaging
method for differential equations with maxima as applied to
first-order and neutral equations. It also explores different
schemes for averaging, partial averaging, partially additiv
This new and important book expands the idea of the wide spectre of
the theory of impulsive equations to the theoretical and applied
researchers in mathematics and provides them a tool and motivation
for deeper investigations and applications of these equations to
real world problems. It deals with the theory of impulsive
differential equations, which is a relatively new area and seems to
hold a considerable potential. This field started gaining interest
around 1980 and today we have several hundred researches working in
it and a couple of thousand publications. Many of the theoretical
results obtained in the field of impulsive differential equations
have already been used in the process of modelling and
investigating the properties of real life processes in the sciences
and technologies.
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