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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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[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 Boston · Zbl 0658.35003
[12] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive ordinary differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[13] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), 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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