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1
Content available remote Inequalities and means from a cyclic differential equation
Herzog G., Kunstmann P.
Demonstratio Mathematica
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2014
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Vol. 47, nr 2
300--309
EN
We prove that the solution of the cyclic initial value problem (…) is convergent to an equilibrium (…) , and study the properties of the function (…) and its relation to Shapiro’s inequality.
2
Content available remote On coupled first and second order boundary value problems in ordered Banach spaces
Herzog G., Lemmert R.
Demonstratio Mathematica
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2010
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Vol. 43, nr 3
561-574
EN
By means of a result on coupled first and second order differential inequalities and an intermediate value theorem in ordered Banach spaces, we obtain the existence of extremal solutions of boundary value problems of the form u܉ = f(t, u1, u2), u𔈀 + g(t, u1, u2) = 0, u1(a) = xa, u2(a) = ya, u2(b) = yb, between lower and upper solutions.
3
Content available remote A note Gateaux differentials of hermitian elements in Banach algebras
Herzog G., Schmoeger C.
Demonstratio Mathematica
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2008
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Vol. 41, nr 4
909-913
EN
Let A be a complex unital Banach algebra with unit 1. If a is an element of A is hermitian then we show that [...] and we give a proof of an inequality due to J. Nieto.
4
Content available remote Second order differential inequalities via aggregation
Herzog G., Lemmert R.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
347-364
EN
Let E be a Banach space ordered by a solid and normal cone. We introduce a polynorm with respect to a given selection of positive pairwise disjoint vectors p1, . . . , pm, and derive monotonicity properties of solutions of second order differential inequalities under one-sided matrix Lipschitz conditions.
5
Content available remote On bounded solutions of second order systems
Herzog G.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
793-801
EN
We prove existence and uniqueness of bounded solutions of u"+f(t, u) = 0, u(0) = x on [0,infinity) under quasimonotonicity and one-sided Lipschitz conditions on f.
6
Content available remote Banach algebra homomorphisms via orthogonal projections
Herzog G., Schmoeger C.
Demonstratio Mathematica
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2006
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Vol. 39, nr 1
137-148
EN
Let A be a unital complex Banach algebra with unit e, and p1,... ,pn a collection of orthogonal projections with sum e. The aim of this note is to investigate the close connections of properties of a is an element of A and of (piapj) ia an element of Mn(A), where Mn(A) denotes the matrix algebra of all n x n matrices with entries in A.
7
Content available remote A class of hypercyclic Volterra composition operators
Herzog G., Weber A.
Demonstratio Mathematica
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2006
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Vol. 39, nr 2
465-468
EN
We prove that certain Volterra composition operators are hypercyclic on the Frechet space of all continuous functions u : [0,1) ->- R or C with u(0) = 0.
8
Content available remote On-sided estimates for quasimonotone systems of boundary value problem
Herzog G.
Demonstratio Mathematica
|
2004
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Vol. 37, nr 4
847--856
EN
We prove existence and uniqueness theorems for Dirichlet boundary value problems of the form u" + f(t,u) = 0, u(0) = uo, u(1) = ui in ordered finite dimensional Banach spaces, involving one-sided estimates and quasimonotonicity.
9
Content available remote On quasimonotone homeomorphisms in ordered Banach spaces
Herzog G.
Demonstratio Mathematica
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2003
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Vol. 36, nr 3
747--755
10
Content available remote Semicontinuous solutions of systems of functional equations
Herzog G.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2003
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[Z] 43(2)
207-219
EN
We use the method of upper and lower solutions to prove the existence of upper and lower semicontinuous solutions of functional equations of the form F(w,u(w),u(g_1(w)),...,u(g_m(w)) = ) in R^n under monotonicity and quasimonotonicity assumptions on F, and for w from a metrizable topological spaces.
11
Content available remote Universal elements for families of discontinuous mappings
Herzog G., Lemmert R.
Demonstratio Mathematica
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2002
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Vol. 35, nr 1
199-204
EN
We extend a result of K.-G. Grofie-Erdmann on residuality of universal elements for families of continuous mappings to families of quasicontinuous mappings in the sense of Kempisty.
12
Content available remote On Riccati equation in ordered Banach algebras
Herzog G., Lemmert R.
Demonstratio Mathematica
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2002
|
Vol. 35, nr 4
783-790
EN
We consider Riccati differential equations in ordered Banach algebras A, and prove invariance and comparison theorems for the case that the right hand side of a Riccati equation is quasimonotone increasing on the set of quasipositive elements (which are the quasimonotone increasing linear mappings in case that A is the operator algebra of an ordered Banach space).
13
Content available remote Stability for families of positive semigroups and partial deifferential equations via one-sided estimates
Herzog G., Kunstmann P.
Demonstratio Mathematica
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2001
|
Vol. 34, nr 1
77-82
EN
Let a Banach space E be ordered by a normal solid cone and A be the generator of a positive strongly continuous semigroup (Tt).
14
Content available remote On positive solutions of linear second order equations in Banach spaces
Herzog G.
Demonstratio Mathematica
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2001
|
Vol. 34, nr 3
587-591
EN
In an ordered Banach space we consider initial value problems of the form u"= C(t)u(t)+f(t), u(0) , u'(0)=u1. Involving quasimonotonicity methods we give conditions which imply that u(t) > 0.
15
Content available remote Nonlinear fundamental systems for linear differential equations in Frechet spaces
Herzog G., Lemmert R.
Demonstratio Mathematica
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2000
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Vol. 33, nr 2
313-318
EN
Let E be a Frechet space. We prove that ex (E) = ex1 (E), that is that the IVP u' = Au + f, u(0) = uo is always solvable if the homogeneous problem u' = Au, u(0) = uo is always solvable (even if this solution is not unique). Moreover we prove that there is a continuous, in general nonlinear selection of solutions, which can be applied to prove an existence theorem for u = Au u+ g(',u), u(0) = uo.
16
Content available remote On maximal and minimal solutions for x'(t) = F(t,x(t),x(h(t))), x(0) = x_0
Herzog G., Lemmert R.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2000
|
[Z] 40
93-102
EN
We prove existence of maximal and minimal solutions for initial value problems for certain functional-differential equations of the form x'(t) = F(t,x(t),x(h(t))). Moreover we give conditions for these problems to be well posed. Under our conditions several forms of the case h(t) > t are included.
17
Content available remote On row-finite systems of differential equations in Banach spaces
Herzog G.
Demonstratio Mathematica
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1998
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Vol. 31, nr 4
835-840
EN
We prove an existence theorem for initial value problems in Banach spaces, including a wide class of row-finite systems of ordinary differential equations.
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