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Qualitative problems in the
theory of hyperbolic differential equations
Nicolaie Lungu
In this monograph we present new
results of the author related to study the qualitative behavior of the solutions
of certain partial differential equations of hyperbolic type by operatorial
method. In this context also are includes results on Picard and weakly Picard
operators, introduced before by I. A. Rus. An important tool useful for the
study of various classes of differential equations are differential and integral
inequalities.
In this monograph we discuss
Gronwall-Wendorff-type integral inequalities by method of operatorial
inequalities. In this case "The Abstract Gronwall Lemma" plays an
important role, and we present some operatorial inequalities for weakly Picard
operators, established by I. A. Rus ([2]-[8]). The principal results are based
on the operatorial inequality for WPO (Rus [4]), and we presents certain
considerations on some lemmas of Gronwall-Bellman-Bihari-Wendorff-type, which
are resulted from AGL for Picard operators.
By this method certain
generalization for hyperbolic differential inequalities of Gronwall-Wendorff's
classical inequalities are presented. Snow [1] and Young [1]-[3] have given
Gronwall-type integral inequalities involving two or many independent variables
for scalar vector functions by using the notion of a Riemann function.
Carte listată
în Zentralblatt:
http://www.zentralblatt-math.org/zmath/en/search?q=an:1097.35001&format=complete
Zbl 1097.35001
Lungu, Nicolaie
Qualitative problems in the theory of hyperbolic differential equations.
(English)
[B] Cluj-Napoca: Digital Data Publishing. 105~p. (2006). ISBN 973-7768-19-1/pbk
Integral inequalities with explicit estimates play a very important role in the
study of qualitative behavior of solutions of various types of differential
equations. \par The present monograph deals with some results of the author and
his co-workers related to the study of qualitative behavior of solutions of
certain partial differential equations of hyperbolic type by operatorial method.
The well known Gronwall-Wendroff type integral inequalities are discussed by the
method of operatorial inequalities. The results on Picard and weakly Picard
operators and the abstract Gronwall lemma are given. It also deals with the
problems of existence, uniqueness, boundedness and comparison results.
Differential and operatorial inequalities, Volterra integral equations in higher
dimensions and applications to homogeneous economic growth models are also
discussed. The monograph will be of interest to those, whose work involves the
qualitative problems in the theory of hyperbolic differential equations.
[B. G. Pachpatte (Aurangabad)]
Cuprins
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PREFACE.
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1
Prof.
Dr. Nicolaie Lungu