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Some boundary value problems for systems of linear differential equat ions of hyperbol ic type
 Memoirs on Diff. Eq. and Math . Physics
, 1994
"... This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DMLCZ: The Czech Digital ..."
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This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DMLCZ: The Czech Digital
Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals
 Higher Order Logic Theorem Proving and its Applications (HUG `93
, 1993
"... In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from ..."
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Cited by 34 (4 self)
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In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and cryptosystems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...
Linear integral equations in the space of regulated functions
, 1997
"... In this paper, we investigate the existence of solutions to a wide class of systems of linear integral equations with solutions which can have in the closed interval [0,1] only discontinuities of the rst kind and are leftcontinuous on the corresponding open interval (0,1). The results cover, e.g., ..."
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Cited by 15 (0 self)
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In this paper, we investigate the existence of solutions to a wide class of systems of linear integral equations with solutions which can have in the closed interval [0,1] only discontinuities of the rst kind and are leftcontinuous on the corresponding open interval (0,1). The results cover, e.g., the results known for systems of linear generalized differential equations as well as systems of Stieltjes Integral equations. Some possible applications to functional differential equations are discussed as well.
On the correctness of linear boundary value problems for systems of ordinary differential equations
 Russian) Bull. Acad. Sci. Georgian SSR
"... Abstract. The sufficient conditions are established for the correctness of the linear boundary value problem dx(t) = dA(t) · x(t) + df(t), l(x) = c0, where A: [a, b] → R n×n and f: [a, b] → R n are matrix and vectorfunctions of bounded variation, c0 ∈ R n, and l is a linear continuous operator ..."
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Abstract. The sufficient conditions are established for the correctness of the linear boundary value problem dx(t) = dA(t) · x(t) + df(t), l(x) = c0, where A: [a, b] → R n×n and f: [a, b] → R n are matrix and vectorfunctions of bounded variation, c0 ∈ R n, and l is a linear continuous operator from the space of ndimentional vectorfunctions of bounded variation into R n. Let the matrix and vectorfunctions, A: [a, b] → R n×n and f: [a, b] → R n, respectively, be of bounded variation, c0 ∈ R n, and let l: BVn(a, b) → R n be a linear continuous operator such that the boundary value problem dx(t) = dA(t) · x(t) + df(t), (1) l(x) = c0 (2) has the unique solution x0. Consider the sequences of matrix and vectorfunctions of bounded variation Ak: [a, b] → R n×n (k = 1, 2,...) and fk: [a, b] → R n (k = 1, 2,...), respectively, the sequence of constant vectors ck ∈ R n (k = 1, 2,...) and the sequence of linear continuous operators lk: BVn(a, b) → R n (k = 1, 2,...). In this paper the sufficient conditions are given for the problem dx(t) = dAk(t) · x(t) + dfk(t), (3) lk(x) = ck (4) to have a unique solution xk for any sufficiently large k and lim k→+ ∞ xk(t) = x0(t) uniformly on [a, b]. (5)
Reasoning About the Reals: the marriage of HOL and Maple
, 1993
"... . Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In ..."
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. Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In this paper we try to get the best of both worlds by careful exploitation of a link between a theorem prover and a computer algebra system. 1 Motivation In the HOL theorem prover[5], a theory of real numbers has been developed, using a rigorous definition in terms of Dedekind cuts [8]. It is therefore possible to apply HOL to areas traditionally within the purview of Computer Algebra Systems (CASs). This offers two main benefits. Firstly, theorem provers are designed to manipulate proofs and theorems in a coherent and structured way, with all concepts clearly defined. By contrast, most CASs have no concept of `logic' as such  they usually take an algebraic expression and return another pur...
Tisdell: Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains
 Int. J. Difference Equ. Appl
"... This article introduces the basic qualitative and basic quantitative theory of Volterra integral equations on time scales and thus may be considered as a foundation for future advanced studies in the field. New sufficient conditions are introduced that guarantee: existence; uniqueness; approximation ..."
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This article introduces the basic qualitative and basic quantitative theory of Volterra integral equations on time scales and thus may be considered as a foundation for future advanced studies in the field. New sufficient conditions are introduced that guarantee: existence; uniqueness; approximation; boundedness and certain growth rates of solutions to both linear and nonlinear problems. The main techniques employed are contemporary components of nonlinear analysis, including: the fixedpoint theorems of Banach and Schäfer; Picard iterations; inequality theory on time scales; and a novel definition of measuring distance in metric spaces and normed spaces. As an application of the new findings, we present some results concerning nonlinear initial value problems for dynamic, differential and difference equations on unbounded domains. We also present some suggestions concerning open problems and possible directions for further work. AMS subject classification: 39A10, 39A12.
Continuous dependence of solutions of generalized linear differential equations on a parameter
 FUNCT. DIFFER. EQU
, 2009
"... This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have boun ..."
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Cited by 8 (1 self)
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This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bounded variation on [a, b]. The integrals are understood in the KurzweilStieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations on parameters and initial data. In particular, we generalize in several aspects the known result by Ashordia. Our main goal consists in a more general notion of a solution to the given system. In particular, neither g nor x need not be of bounded variation on [a, b] and, in general, they can be regulated functions.
ASYMPTOTIC SOLUTIONS OF FORCED NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS AND THEIR EXTENSIONS
, 2007
"... Abstract. Using a modified version of Schauder’s fixed point theorem, measures of noncompactness and classical techniques, we provide new general results on the asymptotic behavior and the nonoscillation of second order scalar nonlinear differential equations on a halfaxis. In addition, we extend ..."
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Abstract. Using a modified version of Schauder’s fixed point theorem, measures of noncompactness and classical techniques, we provide new general results on the asymptotic behavior and the nonoscillation of second order scalar nonlinear differential equations on a halfaxis. In addition, we extend the methods and present new similar results for integral equations and VolterraStieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of VolterraStieltjes integral equations is also presented. 1.
Approximated Solutions of Generalized Linear Differential Equations
 Institute of Mathematics, Acad. Sci. Czech Rep., Preprint 141/2008 [available as http://www.math.cas.cz/preprint/pre141.pdf
"... Abstract. This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g h ..."
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Abstract. This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bounded variation on [a, b]. The integrals are understood in the KurzweilStieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations on parameters and initial data. In particular, we generalize in several aspects the known result by Ashordia. Our main goal consists in a more general notion of a solution to the given system. In particular, neither g nor x need not be of bounded variation on [a, b] and, in general, they can be regulated functions. AMS Subject Classification. 34A37, 45A05, 34A30. 1