Existence and uniqueness of solutions for nonlinear impulsive differential equations with nonlocal boundary conditions
In this paper, we aim to study differential equations х(t) = f (f,х(t)), t ∈ [0,T],t ≠ ti, і = 1,2,...,p, with nonlocal boundary conditions Ax(0)=∫ n(t)x(t)dt=B and subject to impulsive conditions x(ti+)-x(ti)=Ii (x(ti)), i = 1,2,..., p, where 0 = t0 < t1 < ... < tp < tp+1 = T, A ∈ Rn×n, n(t) ∈ Rn×n known matrices such that det N ≠ 0 ,N = A + ∫n(t)dt; f: [0,T]× Rn → Rn and Ii: Rn → Rn are given functions; ∆x(ti) = х(tі+) - х(tі-), where х(tі+) = lim х(ti + h), х(tі-) = lim х(ti - h) = х(/і) are right- and left-hand limits of х(t) at t = ti, respectively. The Green function is constructed and the considered problem is reduced to an equivalent integral equation. The existence and uniqueness of the solutions for the given problem are analyzed using the Banach contraction principle. The Schaefer fixed point theorem is then used to prove the existence of the solutions. The continuous dependence of the solutions on the right side of the boundary conditions is also established. AMS Mathematical Subject Classification: 34B10, 34A37, 34B37.
Keywords
existence and uniqueness of solutions,
continuous dependence of the solutions,
nonlocal boundary conditions,
impulsive systems,
непрерывная зависимость решений,
существование и единственность решений,
нелокальные краевые условия,
Дифференциальные уравнения с импульсными воздействиямиAuthors
| Mardanov Misir J. | Institute Mathematics and Mechanics of ANAS | misirmardanov@yahoo.com |
| Sharifov Yagub A. | Baku State University | farahzeynalli@rambler.ru |
| Zeynally Farah M. | Ganja State University | sharifov22@rambler.ru |
Всего: 3
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