On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions

S. Carl; S. Heikkilä

Annales Polonici Mathematici (1999)

  • Volume: 71, Issue: 1, page 1-17
  • ISSN: 0066-2216

Abstract

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We consider the existence of extremal solutions to second order discontinuous implicit ordinary differential equations with discontinuous implicit boundary conditions in ordered Banach spaces. We also study the dependence of these solutions on the data, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to demonstrate the applicability of the method developed in this paper.

How to cite

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S. Carl, and S. Heikkilä. "On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions." Annales Polonici Mathematici 71.1 (1999): 1-17. <http://eudml.org/doc/262726>.

@article{S1999,
abstract = {We consider the existence of extremal solutions to second order discontinuous implicit ordinary differential equations with discontinuous implicit boundary conditions in ordered Banach spaces. We also study the dependence of these solutions on the data, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to demonstrate the applicability of the method developed in this paper.},
author = {S. Carl, S. Heikkilä},
journal = {Annales Polonici Mathematici},
keywords = {discontinuous implicit differential equations; discontinuous implicit boundary conditions; ordered Banach spaces; fixed point principles in partially ordered sets; existence; extremal solutions; second-order discontinuous implicit ordinary differential equations},
language = {eng},
number = {1},
pages = {1-17},
title = {On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions},
url = {http://eudml.org/doc/262726},
volume = {71},
year = {1999},
}

TY - JOUR
AU - S. Carl
AU - S. Heikkilä
TI - On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions
JO - Annales Polonici Mathematici
PY - 1999
VL - 71
IS - 1
SP - 1
EP - 17
AB - We consider the existence of extremal solutions to second order discontinuous implicit ordinary differential equations with discontinuous implicit boundary conditions in ordered Banach spaces. We also study the dependence of these solutions on the data, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to demonstrate the applicability of the method developed in this paper.
LA - eng
KW - discontinuous implicit differential equations; discontinuous implicit boundary conditions; ordered Banach spaces; fixed point principles in partially ordered sets; existence; extremal solutions; second-order discontinuous implicit ordinary differential equations
UR - http://eudml.org/doc/262726
ER -

References

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  7. [7] T. Kaczyński, Implicit differential equations which are not solvable for the highest derivative, in: Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990), S. Busenberg and M. Martelli (eds.), Lecture Notes in Math. 1475, Springer, Berlin, 1991, 218-224. Zbl0735.34002
  8. [8] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1961. 
  9. [9] S. A. Marano, On a boundary value problem for the differential equation f(t,x,x',x'') = 0, J. Math. Anal. Appl. 182 (1994), 309-319. Zbl0801.34031
  10. [10] S. A. Marano, Implicit elliptic differential equations, Set-Valued Anal. 2 (1994), 545-558. Zbl0824.35150
  11. [11] W. V. Petryshyn, Solvability of various boundary value problems for the equation x'' = f(t,x,x',x'') - y, Pacific J. Math. 122 (1986), 169-195. Zbl0585.34020
  12. [12] B. Ricceri, Applications de théorèmes de semi-continuité inférieure, C. R. Acad. Sci. Paris Sér. I 295 (1982), 75-78. Zbl0509.54008
  13. [13] S. Stanek, On a class of functional boundary value problems for the equation x'' = f(t,x,x',x'',λ), Ann. Polon. Math. 59 (1994), 225-237. Zbl0808.34025
  14. [14] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. I: Fixed-Point Theorems, Springer, Berlin, 1985. 

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