# Kamke's Uniqueness Theorem

@article{Ramankutty1980KamkesUT, title={Kamke's Uniqueness Theorem}, author={Padmaja Ramankutty}, journal={Journal of The London Mathematical Society-second Series}, year={1980}, pages={110-116} }

A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t) -> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on the boundary of the domain of/. Kamke's hypothesis that \\f(t,x)-f{t,y)\\ < <(>(\t-to\, ||x-,y||) is replaced by a weaker dissipative-type hypothesis formulated in terms of the duality map of X and a semi-inner… Expand

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#### References

SHOWING 1-10 OF 24 REFERENCES

Nonlinear operators in Banach spaces

- Mathematics
- 1966

From the point of view of its applications to nonlinear boundary value problems for partial differential equations (as well as to other problems in nonlinear analysis) the principal result of the… Expand

On restricted uniqueness for systems of ordinary differential equations

- Mathematics
- 1973

A uniqueness theorem is proved, for not necessarily Lipschitzian systems of ordinary differential equations, y'=f. This theorem compares with one of Okamura and Murakami, in that, here, at the… Expand

The generic property of existence of solutions of differential equations in Banach space

- Mathematics
- 1973

where f: 77 --P E is continuous. A solution z(*) of (I) will be said to be a solution through us = (to , x0) if (2) is satisfied. Examples of continuous f are known for which there is no solution x… Expand

On the nonexistence of solutions of differential equations in nonreflexive spaces

- Mathematics
- 1972

We consider the problem of the existence of solutions of ordinary differential equations in Banach spaces. Counterexamples in c0 by Dieudonné [2] and in l2 by Yorke [4] show that, in… Expand

An extension of the Nagumo uniqueness theorem

- Mathematics
- 1971

The classical Nagumo uniqueness theorem is a best possible result in the sense that if the Nagumo constant is replaced by a number greater than one then the result is false. This classical result… Expand

Two remarks on global solutions of ordinary differential equations in the real line

- Mathematics
- 1976

We prove two theorems concerning global solutions of initial and terminal value problems: a criteria for the existence of maximal and minimal solutions, as well as the connectedness of the set of… Expand

On nonlinear differential equations in Hilbert spaces

- Mathematics
- 1971

In a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains… Expand

Euler - Cauchy polygons and the local existence of solutions to abstract ordinary differential equations " , Funkcial

- Ekvac
- 1972