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1999 | 71 | 1 | 1-17
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On discontinuous implicit differential equations in ordered Banach spaces with discontinuous implicit boundary conditions

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Abstrakty
EN
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.
Twórcy
autor
  • Fachbereich Mathematik und Informatik, Institut für Analysis, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany
autor
  • Department of Mathematical Sciences, University of Oulu, 90570 Oulu, Finland
Bibliografia
  • [1] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Univ. Press, Cambridge, 1990.
  • [2] G. Bartuzel and A. Fryszkowski, Abstract differential inclusions with some applications to partial differential ones, Ann. Polon. Math. 53 (1991), 67-78.
  • [3] L. H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y'), Ann. Polon. Math. 54 (1991), 195-226.
  • [4] L. H. Erbe, W. Krawcewicz and T. Kaczyński, Solvability of two-point boundary value problems for systems of nonlinear differential equations of the form y''= g(t,y,y',y''), Rocky Mountain J. Math. 20 (1990), 899-907.
  • [5] M. Frigon and T. Kaczyński, Boundary value problems for systems of implicit differential equations, J. Math. Anal. Appl. 179 (1993), 317-326.
  • [6] S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, 1994.
  • [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.
  • [8] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1961.
  • [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.
  • [10] S. A. Marano, Implicit elliptic differential equations, Set-Valued Anal. 2 (1994), 545-558.
  • [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.
  • [12] B. Ricceri, Applications de théorèmes de semi-continuité inférieure, C. R. Acad. Sci. Paris Sér. I 295 (1982), 75-78.
  • [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.
  • [14] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. I: Fixed-Point Theorems, Springer, Berlin, 1985.
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