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Existence of solutions to some differential delay equations. (English) Zbl 0849.34050

This paper deals with two problems concerning differential delay equations. First, the following initial value problem is examined: \(\psi (t) y'(t) = f(t,y(t), y(t - r))\), \(0 \leq t \leq T\), \(y(t) = 0\), \(t \in [-r,0]\), where \(r > 0\) is a fixed constant, \(\psi : [0,T] \to [0, \infty )\) and \(f:[0,T] \times \mathbb{R}^2 \to \mathbb{R}\) are continuous functions with \(\psi > 0\) on \((0,T)\). Sufficient conditions under which this problem has a solution \(y \in C [-r,T] \cap C^1 (0,T)\) are given. These conditions are specified on interesting examples from which the relation between the zero set of \(f\) and the width of the interval of existence of solutions can be deduced.
Secondly, the following boundary value problem is studied: \(\psi (t) y''(t) = f(t,y(t), y(t - r_1)\), \(y'(t), y'(t - r_2))\), \(0 \leq t \leq 1\), \(y(t) = 0\), \(t \in [-r,0]\), \(r = \max \{r_1, r_2\}\), \(y(1) = 0\), where \(r_1 > 0\), \(r_2 > 0\) are fixed constants, \(\psi : [0,T] \to [0, \infty)\) and \(f : [0,1] \times \mathbb{R}^4 \to \mathbb{R}\) are continuous functions, with \(\psi > 0\) on \((0,T)\). For this problem sufficient conditions for the existence of solutions \(y\) such that \(y \in C [-r,1] \cap C^1 [- r,0] \cap C^1 [0,1]\), \(y'\) is absolutely continuous on \([0,1]\) and \(y''\) exists a.e. on \((0,1)\) are also given. These conditions are illustrated by appropriate examples.
The proofs of the existence theorems in this paper are based on the well-known topological transversality method, which is applied in the same way as in a paper of J. W. Lee and the author [J. Differ. Equations 102, No. 2, 342-359 (1993; Zbl 0782.34070)].

MSC:

34K05 General theory of functional-differential equations
34K10 Boundary value problems for functional-differential equations

Citations:

Zbl 0782.34070
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References:

[1] Driver, R. D., Ordinary and delay differential equations, Applied Mathematical Sciences, Vol. 20 (1977), Springer: Springer Berlin · Zbl 0374.34001
[2] Hale, J., Functional Differential Equations (1971), Springer: Springer New York · Zbl 0222.34003
[3] Lee J. W. & O’Regan D., Existence results for differential delay equations I, J. diff. Eqns; Lee J. W. & O’Regan D., Existence results for differential delay equations I, J. diff. Eqns
[4] Granas, A.; Guenther, R. B.; Lee, J. W., Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. math., 10, 35-58 (1980) · Zbl 0476.34017
[5] O’Farrell, A. G.; O’Regan, D., Existence results for initial and boundary value problems, Proc. Am. math. Soc., 110, 661-673 (1990) · Zbl 0713.34024
[6] O’Regan, D., Solvability of some second and higher order boundary value problems, Nonlinear Analysis, 16, 507-516 (1991) · Zbl 0732.34015
[7] Rodriguez, A.; Tineo, A., Existence theorems for the Dirichlet problem without growth restrictions, J. math. Analysis Applic., 135, 1-7 (1988) · Zbl 0674.34016
[8] Agarwal, R. P.; Chow, Y. M., Finite difference methods for boundary value problems for differential equations with deviating arguments, Comp. Math. Appls., 12A, 11, 1143-1153 (1986) · Zbl 0617.65092
[9] Lee, J. W.; O’Regan, D., Existence results for differential delay equations II, Nonlinear Analysis, 17, 683-702 (1991) · Zbl 0782.34071
[10] Schmitt, K., Comparison theorems for second-order delay differential equations, Rocky Mountain J. math., 1, 3, 459-467 (1971) · Zbl 0226.34063
[11] Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963), The Rand Corporation: The Rand Corporation Santa Monica, California · Zbl 0118.08201
[12] Bobisud, L. E.; O’Regan, D., Existence of solutions to some singular initial value problems, J. math. Analysis Applic., 133, 214-230 (1988) · Zbl 0646.34003
[13] Devasahayam, M. P., Existence of monotone solutions for functional differential equations, J. math. Analysis Applic., 118, 487-495 (1986) · Zbl 0611.34065
[14] Dugundji, J.; Granas, A., Fixed Point Theory, Monographie Mathematyczne, Vol. I (1982), PNN: PNN Warsaw · Zbl 0483.47038
[15] Lee, J. W.; O’Regan, D., Existence of solutions to some initial value, two-point and multipoint boundary value problems with discontinuous nonlinearities, Applicable Analysis, 33, 57-77 (1989) · Zbl 0643.34018
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