Separated boundary value problems for second-order impulsive q -integro-difference equations
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RESEARCH
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Separated boundary value problems for second-order impulsive q-integro-difference equations Chatthai Thaiprayoon1 , Jessada Tariboon1* and Sotiris K Ntouyas2 * Correspondence: [email protected] 1 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology, North Bangkok, Bangkok, Thailand Full list of author information is available at the end of the article
Abstract This article studies the existence and uniqueness of solutions for a boundary value problem of nonlinear second-order impulsive qk -integro-difference equations with separated boundary conditions. Several new results are obtained by applying a variety of fixed point theorems. Some examples are presented to illustrate the results. MSC: 26A33; 39A13; 34A37 Keywords: qk -derivative; qk -integral; impulsive qk -difference equation; existence; uniqueness; fixed point theorems
1 Introduction In this paper, we study the separated boundary value problem for impulsive qk -integrodifference equation of the following form: ⎧ Dqk x(t) = f (t, x(t), (Sqk x)(t)), t ∈ J := [, T], t = tk , ⎪ ⎪ ⎪ ⎨x(t ) = I (x(t )), k = , , . . . , m, k k k (.) + ∗ ⎪ D x(t ) – D x(t ) = I (x(t )), k = , , . . . , m, q q k k ⎪ k k k k– ⎪ ⎩ x() + Dq x() = , x(T) + Dqm x(T) = , where = t < t < t < · · · < tk < · · · < tm < tm+ = T, f : J × R → R, t (Sqk x)(t) = φ(t, s)x(s) dqk s, t ∈ (tk , tk+ ], k = , , , . . . , m,
(.)
tk
φ : J × J → [, ∞) is a continuous function, Ik , Ik∗ ∈ C(R, R), x(tk ) = x(tk+ ) – x(tk ) for k = , , . . . , m, x(tk+ ) = limh→ x(tk + h) and < qk < for k = , , , . . . , m. The notions of qk -derivative and qk -integral on finite intervals were introduced in []. For a fixed k ∈ N ∪ {} let Jk := [tk , tk+ ] ⊂ R be an interval and < qk < be a constant. We define qk -derivative of a function f : Jk → R at a point t ∈ Jk as follows. Definition . Assume f : Jk → R is a continuous function and let t ∈ Jk . Then the expression Dqk f (t) =
f (t) – f (qk t + ( – qk )tk ) , ( – qk )(t – tk )
t = tk ,
Dqk f (tk ) = lim Dqk f (t) t→tk
(.)
is called the qk -derivative of function f at t. ©2014 Thaiprayoon et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Thaiprayoon et al. Advances in Difference Equations 2014, 2014:88 http://www.advancesindifferenceequations.com/content/2014/1/88
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We say that f is qk -differentiable on Jk provided Dqk f (t) exists for all t ∈ Jk . Note that if tk = and qk = q in (.), then Dqk f = Dq f , where Dq is the well-known q-derivative of the function f (t) defined by Dq f (t) =
f (t) – f (qt) . ( – q)t
(.)
In addition, we should define the higher qk -derivative of functions. Definition . Let f : Jk → R be a continuous function, we call the second-order qk derivative Dq
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