First-order impulsive ordinary differential equations with advanced arguments.(English)Zbl 1122.34042

Consider the boundary value problem $x'(t)= f(t, x(t), x(\alpha(t))\quad\text{for }[0,T]\setminus \{t_1,t_2,\dots, t_m\},$
$\Delta x(t_k)= I_k(x(t_k))\quad\text{for }k= 1,\dots, m,\tag{$$*$$}$
$0= g(x(0), x(T)),$ where $$f$$, $$\alpha$$, $$g$$ and $$I_k$$ $$(1\leq k\leq m)$$ are continuous functions, $$0\leq t\leq\alpha(t)\leq T$$, $$\Delta x(t_k)= x(t^+_k)- x(t^-_k)$$.
Using the method of lower and upper solutions in reversed order and the method of coupled lower and upper solutions, the author establishes conditions guaranteeing the existence of a solution and of a quasi-solution to $$(*)$$, respectively.

MSC:

 34K10 Boundary value problems for functional-differential equations 34K45 Functional-differential equations with impulses 34K07 Theoretical approximation of solutions to functional-differential equations
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References:

 [1] Agarwal, R. P.; Franco, D.; O’Regan, D., Singular boundary value problems for first and second order impulsive differential equations, Aequationes Math., 69, 83-96 (2005) · Zbl 1073.34025 [2] Ding, W.; Han, M.; Mi, J., Periodic boundary value problem for second-order impulsive functional differential equations, Comput. Math. Appl., 50, 491-507 (2005) · Zbl 1095.34042 [3] Ding, W.; Mi, J.; Han, M., Periodic boundary value problems for the first order impulsive functional differential equations, Appl. Math. Comput., 165, 433-446 (2005) · Zbl 1081.34081 [4] Franco, D.; Nieto, J. J., First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Anal., 42, 163-173 (2000) · Zbl 0966.34025 [5] Franco, D.; Nieto, J. J.; O’Regan, D., Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions, Appl. Math. Comput., 153, 793-802 (2004) · Zbl 1058.34015 [6] He, Z.; Yu, J., Periodic boundary value problem for first-order impulsive functional differential equations, J. Comput. Appl. Math., 138, 205-217 (2002) · Zbl 1004.34052 [7] Jankowski, T., On delay differential equations with nonlinear boundary conditions, Bound. Value Probl., 2005, 201-214 (2005) · Zbl 1148.34043 [8] Jankowski, T., Advanced differential equations with nonlinear boundary conditions, J. Math. Anal. Appl., 304, 490-503 (2005) · Zbl 1092.34032 [9] Jankowski, T., Solvability of three point boundary value problems for second order differential equations with deviating arguments, J. Math. Anal. Appl., 312, 620-636 (2005) · Zbl 1154.34367 [10] Jankowski, T., Boundary value problems for first order differential equations of mixed type, Nonlinear Anal., 64, 1984-1997 (2006) · Zbl 1095.45001 [11] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston · Zbl 0658.35003 [12] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Ordinary Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 [13] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003 [14] Zhang, F.; Ma, Z.; Yan, J., Boundary value problems for first order impulsive delay differential equations with a parameter, J. Math. Anal. Appl., 290, 213-223 (2004) · Zbl 1056.34041
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