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1

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.

2

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.

3

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.

4

On Riccati equation in ordered Banach algebras

Herzog G., Lemmert R.

Demonstratio Mathematica

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2002

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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).

5

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.

6

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

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2000

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[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.