FIRST- AND SECOND-ORDER DYNAMIC EQUATIONS WITH IMPULSE F. M. ATICI AND D. C. BILES Received 3 pdf

FIRST- AND SECOND-ORDER DYNAMIC EQUATIONS WITH IMPULSE F M ATICI AND D C BILES Received December 2004 and in revised form 11 February 2005 We present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions Introduction We first briefly survey the recent results for existence of solutions to first-order problems with fixed-time impulses Periodic boundary conditions using upper and lower solutions were considered in [19], using degree theory A nonlinear alternative of Leray-Schauder type was used in [15] for initial conditions or periodic boundary conditions The monotone iterative technique was employed in [14] for antiperiodic and nonlinear boundary conditions Lower and upper solutions and periodic boundary conditions were studied in [20] Semilinear damped initial value problems in a Banach space using fixed point theory were investigated in [6] In [9], existence of solutions for the differential equation u (t) = q(u(t))g(t,u(t)) subject to a general boundary condition is proven, in which g is Carath´ odory and q ∈ L∞ , and existence of lower and upper solutions is assumed e Schauder’s fixed point theorem was used there This generalized an earlier result found in [18] It appears that little has been done concerning dynamic equations with impulses on time scales (see [4, 5, 16] for earlier results) In Section 2, the present paper uses ideas from [9] to prove an existence result for discontinuous dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions The study of boundary value problems for nonlinear second-order differential equations with impulses has appeared in many papers (see [10, 11, 13] and the references therein) In Section 3, we use ideas from [12, 16] to prove an existence result for secondorder dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions Nonlinear boundary conditions cover, among others, the periodic and the Dirichlet conditions, and have been introduced for ordinary differential equations by Adje in [1] Assuming the existence of a lower and an upper solution, we prove that the solution of the boundary value problem stays between them In [2], it was shown that the upper and lower solution method will not work for first-order dynamic equations involving ∆-derivatives, unless restrictive assumptions are Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 119–132 DOI: 10.1155/ADE.2005.119 120 Dynamic equations with impulse made Hence, in Section 2, we work with the ∇-derivative In Section 3, we can use the more conventional ∆-derivative The monographs [17, 21] are good general references on impulsive differential equations—discussion of applications may be found in these books Applications of the results given in this paper could involve those typically modelled on time scales which are subjected to sudden major influences, for example, an insect population sprayed with an insecticide or a financial market affected by a major terrorist attack For our purposes, we let T be a time scale (a closed subset of R), let [a,b] be the closed and bounded interval in T, that is, [a,b] := {t ∈ T : a ≤ t ≤ b} and a,b ∈ T For the readers’ convenience, we state a few basic definitions on a time scale T [7, 8] Obviously a time scale T may or may not be connected Therefore, we have the concept of forward and backward jump operators as follows: define σ,ρ : T → T by σ(t) = inf {s ∈ T : s > t }, ρ(t) = sup{s ∈ T : s < t } (1.1) If σ(t) = t, σ(t) > t, ρ(t) = t, ρ(t) < t, then t ∈ T is called right dense (rd), right scattered, left dense, left scattered, respectively We also define the graininess function µ : T → [0, ∞) as µ(t) = σ(t) − t The sets Tκ , Tκ which are derived from T are as follows: if T has a left-scattered maximum t1 , then Tκ = T − {t1 }, otherwise Tκ = T If T has a rightscattered minimum t2 , then Tκ = T − {t2 }, otherwise Tκ = T If f : T → R is a function, we define the functions f σ : Tκ → R by f σ (t) = f (σ(t)) for all t ∈ Tκ , f ρ : Tκ → R by f ρ (t) = f (ρ(t)) for all t ∈ Tκ and σ (t) = ρ0 (t) = t If f : T → R is a function and t ∈ Tκ , then the delta derivative of f at a point t is defined to be the number f ∆ (t) (provided it exists) with the property that, for each ε > 0, there is a neighborhood of U1 of t such that f σ(t) − f (s) − f ∆ (t) σ(t) − s ≤ ε σ(t) − s , (1.2) for all s ∈ U1 If t ∈ Tκ , then we define the nabla derivative of f at a point t to be the number f ∇ (t) (provided it exists) with the property that, for each ε > 0, there is a neighborhood of U2 of t such that f ρ(t) − f (s) − f ∇ (t) ρ(t) − s ≤ ε ρ(t) − s , (1.3) for all s ∈ U2 Remark 1.1 If T = R, then f ∆ (t) = f ∇ (t) = f (t), and if T = Z, then f ∆ (t) = ∆ f (t) = f (t + 1) − f (t) and f ∇ (t) = ∇ f (t) = f (t) − f (t − 1) A function F : T → R is called a ∆-antiderivative of f : T → R provided F ∆ (t) = f (t) holds for all t ∈ Tk Then the Cauchy ∆-integral from a to t of f is defined by t a f (s) s = F(t) − F(a) ∀t ∈ T (1.4) F M Atici and D C Biles 121 A function Φ : T → R is called a ∇-antiderivative of f : T → R provided Φ∇ (t) = f (t) for all t ∈ Tk We then define the Cauchy ∇-integral from a to t of f by t a f (s)∇s = Φ(t) − Φ(a) ∀t ∈ T (1.5) Note that, in the case T = R, we have b a f (t)∆t = b a f (t)∇t = b a f (t)dt, (1.6) and, in the case T = Z, we have b a b−1 f (t)∆t = b f (k), k=a a b f (t)∇t = f (k), (1.7) k=a+1 where a,b ∈ T with a ≤ b There are two types of impulse effects that are studied in the literature The first is the “fixed-time impulse”: a set of times < t1 < t2 < · · · < tn < T is specified, and the solution is required to satisfy + u tk = Ik u tk (1.8) for k = 1,2, ,n, where the functions Ik provide the “impulse.” Also studied are “variabletime impulses,” in which a set of curves t = τ1 (x),t = τ2 (x), ,t = τn (x) is given, and the solution satisfies u(t + ) = Ik (u(t)) for t = τk (u(t)), k = 1,2, ,n Impulses of both types introduce discontinuities in the solution As mentioned in [17] and other works in the reference list, applications involving impulse effects can be found in biology, medicine, physics, economics, pharmacokinetics, and engineering In this paper, we consider fixedtime impulses Without loss of generality, we investigate systems with a single impulse First order Let 0,t1 ,T ∈ T with < t1 < T and t1 right dense Let J = [0,T] ∩ T, u∇ (t) = g t,u(t) , t ∈ J \ t1 , (2.1) + u t1 = I u t1 , (2.2) B u(0),u(T) = (2.3) Note that (2.3) covers as special cases many initial and boundary conditions found in b the literature Let J1 = [0,t1 ] ∩ J, J2 = (t1 ,T] ∩ J Define a y(s)∇s = (a,b] y(s)∇s, where the integrals in Section are with respect to the Lebesgue ∇-measure as defined by Atici and Guseinov in [3] 122 Dynamic equations with impulse Let ui be the restriction of u : J → R to Ji , i = 1,2, then Ꮿ J1 = u : J1 − R : u is continuous on J1 , → + Ꮿ J2 = u : J2 −→ R : u is continuous on J2 and u t1 exists , (2.4) A = u : J − R : u1 ∈ Ꮿ J1 and u2 ∈ Ꮿ J2 → For u ∈ A, let u = sup{|u(t)| : t ∈ J } (A, · ) is a Banach space For u,v ∈ A, [u,v] ≡ w ∈ A : u(t) ≤ w(t) ≤ v(t) ∀t ∈ J (2.5) Definition 2.1 u : T → R is a solution of (2.1)–(2.3) if (i) u ∈ A, (ii) u(t) = u(0) + u(t) = I u t1 t g s,u(s) ∇s, t + t1 t ∈ J1 , (2.6) g s,u(s) ∇s, t ∈ J2 , (iii) B(u(0),u(T)) = We call α : J → R a lower solution of (2.1)–(2.3) if (i) α ∈ A, b (ii) α(b) − α(a) ≤ a g(s,α(s))∇s for a ≤ b and a,b ∈ J1 , or a,b ∈ J2 , + (iii) α(t1 ) ≤ I(α(t1 )), (iv) B(α(0),α(T)) ≤ We call β : J → R an upper solution of (2.1)–(2.3) if it satisfies the same assumptions, but replace ≤ with ≥ Let p(t,x) = max{α(t),min{x,β(t)}} We have the following assumptions throughout this section: (1) for each x ∈ R, g(·,x) is Lebesgue ∇-measurable on J, (2) for a.e (∇) t ∈ J, g(t, ·) is continuous, T (3) there is an h : J → [0, ∞), h(s)∇s < ∞ such that |g(t, p(t,x))| ≤ h(t) a.e (∇) on J, for all x ∈ R, (4) I : R → R is continuous and nondecreasing, (5) B : R × R → R is continuous and for each x ∈ [α(0),β(0)], B(x, ·) is nonincreasing Note a.e (∇) denotes the Lebesgue ∇-measure Theorem 2.2 Assume that conditions (1)–(5) are satisfied and α, β are lower and upper solutions of (2.1)–(2.3) with α(t) ≤ β(t) for all t ∈ J Then, there exists a solution u to (2.1)– (2.3) such that u ∈ [α,β] F M Atici and D C Biles 123 Proof Our proof follows that of Cabada and Liz [9] Define an operator G : A → A by t Gu(t) = p 0,u(0) + g s, p s,u(s) ∇s, t Gu(t) = I p t1 ,u t1 + t1 t ∈ J1 , g s, p s,u(s) ∇s, t ∈ J2 , (2.7) (2.8) where u(0) ≡ u(0) − B(u(0),u(T)) Claim 2.3 If u is a fixed point of the operator G, then u is a solution of (2.1)–(2.3) such that u ∈ [α,β] Proof of Claim 2.3 We assume that u ∈ A satisfies u(t) = p 0,u(0) + t g s, p s,u(s) ∇s, t u(t) = I p t1 ,u(t1 + t1 g s, p s,u(s) ∇s, t ∈ J1 , t ∈ J2 (2.9) (2.10) Subclaim u(t) ∈ [α(t),β(t)], for all t ∈ J Note that, by letting t = in the right-hand side of (2.9), we have ∅ g(s, p(s,u(s)))∇s = and hence u(0) = p(0,u(0)) which is in [α(0),β(0)] by the definition of p Suppose there exists a t1 ∈ (0,t1 ] ∩ J such that α(t1 ) > u(t1 ) Since α(0) ≤ u(0), there exists a t2 ∈ [0,t1 ) ∩ J such that α(t2 ) ≤ u(t2 ) and α(t) > u(t) on (t2 ,t1 ] ∩ J Then, g(t, p(t,u(t))) = g(t,α(t)) for all t ∈ (t2 ,t1 ] ∩ J We then have, for any t ∈ (t2 ,t1 ] ∩ J, t g s, p s,u(s) ∇s = u(t) − p 0,u(0) = u t2 − p 0,u(0) + u(t) − u t2 , t2 g s, p s,u(s) ∇s + u(t) − u t2 = u(t) − u t2 = ⇒ = t t2 t g s, p s,u(s) ∇s − t2 g s, p s,u(s) ∇s = t2 (2.11) g s, p s,u(s) ∇s g s,α(s) ∇s From assumption (ii) of the definition of lower solution, we have α(t) − α t2 ≤ t g s,α(s) ∇s (2.12) g s,α(s) ∇s ≥ α(t) − α t2 (2.13) t2 We then have u(t) − u t2 = t t2 and recalling that α(t2 ) ≤ u(t2 ) and u(t) < α(t), this is a contradiction Hence, α ≤ u on J1 Similarly, u ≤ β on J1 124 Dynamic equations with impulse We then have α t1 ≤ u t1 ≤ β t1 , (2.14) and using the fact that I is nondecreasing, we have + α t1 ≤ I α t1 ≤ I u t1 ≤ I β t1 + ≤ β t1 (2.15) + We also have I(u(t1 )) = I(p(t1 ,u(t1 )) = u(t1 ) and hence from (2.15) we conclude + + + α t1 ≤ u t1 ≤ β t1 (2.16) We may now proceed as before to get α ≤ u ≤ β on J2 , establishing Subclaim We may apply Subclaim to (2.9) to verify that u satisfies the first equation in property (ii) of a solution to (2.1)–(2.3), and apply Subclaim to (2.10) to verify that u satisfies the second equation in property (ii) Subclaim u(0) ∈ [α(0),β(0)] Suppose that α(0) > u(0) = u(0) − B(u(0),u(T)) Thus, u(0) = p(0,u(0)) = α(0) and hence B(u(0),u(T)) > Using assumption (5), we have B(α(0),α(T)) ≥ B(α(0),u(T)) > 0, which contradicts α being a lower solution of (2.1)–(2.3) We then have α(0) ≤ u(0) and, similarly, u(0) ≤ β(0), establishing Subclaim As a result of Subclaim 2, we have u(0) = p(0,u(0)) = u(0) = u(0) − B(u(0),u(T)) and hence B(u(0),u(T)) = 0, establishing Claim 2.3 Claim 2.4 G : A → A has a fixed point Proof of Claim 2.4 We will apply Schauder’s fixed point theorem t Let K = α + β Define w : J → R by w(t) = K + h(s)∇s Let + S = u ∈ A : u(0) ≤ K, u t1 ≤ K, u(b) − u(a) ≤ w(b) − w(a) on ≤ a ≤ b ≤ t1 or t1 < a ≤ b ≤ T, where a,b ∈ J (2.17) It can be shown that S is a convex and compact subset of (A, · ) Subclaim G(S) ⊆ S Let u ∈ S and consider Gu Let t = in (2.7) to obtain Gu(0) = p 0,u(0) ≤ max α(0) , β(0) ≤ K (2.18) Note that α(t1 ) ≤ p(t1 ,u(t1 )) ≤ β(t1 ), hence + α t1 ≤ I α t1 I p t1 ,u t1 ≤ I p t1 ,u t1 ≤ max ≤ I β t1 + + α t1 , β t1 + ≤ β t1 , ≤ K (2.19) Let t ↓ t1 in (2.8) to obtain + Gu t1 = I p t1 ,u t1 ≤ K (2.20) F M Atici and D C Biles 125 Let a,b ∈ J with ≤ a ≤ b ≤ t1 , Gu(b) − Gu(a) = b a g s, p s,u(s) ∇s ≤ b a h(s)∇s = w(b) − w(a) (2.21) Similar results hold for t1 < a ≤ b ≤ T Subclaim G : S → S is continuous ∞ Let un n=1 ⊆ S which converges to u ∈ S in the space (A, · ) Note that un → u uniformly on compact subsets of J Let n ∈ N and t ∈ J1 , then Gu(t) − Gun (t) = Gu(t) − p 0,u(0) − Gun (t) − p 0,un (0) t = g s, p s,u(s) ∇s − t + p 0,u(0) − p 0,un (0) (2.22) g s, p s,un (s) ∇s + p 0,u(0) − p 0,un (0) and hence Gu(t) − Gun (t) ≤ t1 g s, p s,u(s) − g s, p s,un (s) ∇s + p 0,u(0) − p 0,un (0) (2.23) Now take limn→∞ and apply the Lebesgue dominated convergence theorem and the continuity of g in its second variable and of p to conclude lim Gun (t) − Gu(t) = (2.24) n→∞ Note that (2.23) does not involve t in its right-hand side, so we can conclude that the convergence is uniform on J1 A similar argument shows that Gun → G uniformly on compact subsets of J2 Hence, by Subclaims and 4, Schauder’s fixed point theorem applies to G, finishing the proof of Claim 2.4 Claims 2.3 and 2.4 yield the desired result Example 2.5 Let T = [0,1] ∪ [2,3], t1 = 2, g(t,x) = t + x2 , I(x) = x + 1, u(0) = (Note that I is not bounded, as required in [4].) α(t) = is a lower solution To construct β on [0,1], we solve β = + β2 (≥ t + β2 ), β(0) = Then this implies that β(t) = tant By considering boundary conditions, we have β(2) = β(0) + β(2) = tan1 + β∇ (s)∇s = β(0) + β (s)ds + β∇ (2), β(2) − β(1) = β(2) = β(2) = β(2) is arbitrary, ⇒ ⇒ 2−1 β(2+ ) = I β(2) = let β(2) = 1, (2.25) 126 Dynamic equations with impulse To construct β on [2,3], we solve β = + β2 (≥ t + β2 ), β(2) = 2, then we have + 3t − β = 3tan tan−1 (2.26) Applying Theorem 2.2, we know there exists a solution u such that ≤ u(t) ≤ β(t) for t ∈ T Second order In this section, we are concerned with second-order dynamic equations with functional boundary conditions and impulse: y (t) = f t, y σ (t) , t ∈ Tκ ≡ [a,b] \ t1 , L1 y(a), y ∆ (a), y σ (b) , y ∆ σ b) L2 y(a), y σ b) y y ∆ + t1 −y + t1 ∆ = 0, = 0, (3.2) (3.3) − − y t1 = r1 , − (3.1) (3.4) ∆ − t1 = I y t1 , y t1 , (3.5) where t1 ∈ T with a < t1 < b and t1 right-dense, r1 ∈ R, I is a real-valued function and + − J = [a,b] We set y ∆ (t1 ) = y ∆ (t1 ) if t1 is left-scattered, and y ∆ (t1 ) = y ∆ (t1 ) if t1 is leftdense We note that these impulses are different from those studied in [16] (3.2) and (3.3) cover many conditions found in the literature such as separated and nonseparated boundary conditions, respectively, L1 (x, y,z,w) = x, L1 (x, y,z,w) = y − z, L2 (x, y) = y, L2 (x, y) = y − x, (3.6) as in [7, Chapter 4] Let J1 = [a,t1 ], J2 = (t1 ,b] We define the following spaces of functions Let yi be the restriction of y : J → R to Ji , i = 1,2, then Ꮿ J1 = y : J1 −→ R : y and y ∆ are continuous on J1 , + + Ꮿ J2 = y : J2 −→ R : y and y ∆ are continuous on J2 and y t1 and y ∆ t1 exist , A = y : J −→ R : y1 ∈ Ꮿ J1 and y2 ∈ Ꮿ J2 (3.7) For y ∈ A, let y = sup{| y(t)| : t ∈ J } (A, · ) is a Banach space For x, y ∈ A, [x, y] ≡ z ∈ A : x(t) ≤ z(t) ≤ y(t) ∀t ∈ J (3.8) Now we introduce the concept of lower and upper solutions of problem (3.1)–(3.5) as follows F M Atici and D C Biles 127 Definition 3.1 The functions α and β are, respectively, a lower and an upper solution of problem (3.1)–(3.5) if the following properties hold: (i) α, β ∈ A; (ii) α∆∆ (t) ≥ f t,ασ (t) ∆ on t ∈ [a,b] \ t1 , L1 α(a),α (a),α σ (b) ,α∆ σ(b) 2 = 0, L2 α(a),α σ (b) α ∆ + t1 −α L2 α(a), · is injective, (3.9) − + t1 α ∆ ≥ 0, − α t1 = r1 , − t1 ≥ I α(t1 ,α∆ t1 ) ; − (iii) β∆∆ (t) ≤ f t,βσ (t) ∆ on t ∈ [a,b] \ t1 , L1 β(a),β (a),β σ (b) ,β∆ σ(b) L2 β(a),β σ (b) β + t1 L2 β(a), · is injective, = 0, −β ∆ (3.10) − + t1 β ∆ ≤ 0, − β t1 = r1 , − t1 ≤ I β t1 ,β∆ t1 − We assume the following conditions are satisfied for the functions f , L1 and L2 , and I (F) The function f : [a,b] × R → R is continuous (L) L1 ∈ C(R4 , R) is nondecreasing in the second variable, nonincreasing in the fourth Moreover, L2 : R2 → R is a continuous function and it is nonincreasing with respect to its first variable (I) I is continuous and strictly increasing with respect to the first variable and nonincreasing in the second variable We consider the following modified truncated problem: y (t) − y σ (t) = f t, p σ(t), y σ (t) − p σ(t), y σ (t) , t ∈ Tκ ≡ [a,b] \ t1 , (3.11) y(a) = L∗ y(a), y ∆ (a), y σ (b) , y ∆ σ(b) , ∗ 2 (3.12) y σ (b) = L2 y(a), y σ (b) , + − r1 = y t1 − y t1 , y (3.13) (3.14) ∆ + t1 −y ∆ − ∆ − t1 = I y t1 , y t1 , (3.15) where p(t, y) = min{max{α(t), y },β(t)}, L∗ (x, y,z,w) = p a,x + L1 (x, y,z,w) L∗ (x, y) = p σ (b), y − L2 (x, y) ∀(x, y,z,w) ∈ R4 , ∀(x, y) ∈ R2 (3.16) Theorem 3.2 Assume that conditions (F) and (L) are satisfied If there exist a lower solution α and an upper solution β of (3.1)–(3.5) such that α ≤ β on T, then the BVP (3.11)– (3.15) has a solution 128 Dynamic equations with impulse Proof It is not difficult to verify that the problem y ∆∆ − y σ = 0, t ∈ [a,b], (3.17) y(a) = y σ (b) = 0, has only the trivial solution By using [7, Theorem 4.67 and Corollary 4.74], we have that for every h ∈ Crd [a,b] and A,B ∈ R, the problem y ∆∆ − y σ = h(t), t ∈ [a,b], y(a) = A, (3.18) y σ (b) = B, has a solution y(t) if and only if the operator Qy(t) = Ay1 (t) + B y2 (t) + σ(b) a G(t,s)h(s)∆s (3.19) has a fixed point Here y1 (t), y2 (t) are the solutions of the linear homogeneous equation y ∆∆ − y σ = 0, t ∈ [a,b] and satisfy the boundary conditions y1 (a) = 1, y1 (σ (b)) = and y2 (a) = 0, y2 (σ (b)) = G is called the Green’s function of the Dirichlet problem One can verify that (see [7, page 169]) it is continuous in [a,σ (b)] × [a,σ (b)] and G∆ (·,s) is continuous at t = s = σ(s) and bounded in [a,σ (b)] Define Qy(t) = L∗ y(a), y ∆ (a), y σ (b) , y ∆ σ(b) y1 (t) + L∗ y(a), y σ (b) y2 (t) σ(b) + a G(t,s) f s, p σ(s), y σ (s) − p σ(s), y σ(s) ∆s + L t, y(t) , (3.20) where   y2 (t) −   y1 t1 I y t1 , y ∆ t1  W  y1 (t)  −  y2 t1 I y t1 , y ∆ t1 W L(t, y) =  − r1 y1 t1 , a ≤ t ≤ t1 , (3.21) − r1 y2 t1 , t1 ≤ t ≤ σ (b), where W = y2 (t1 )y1 ∆ (t1 ) − y1 (t1 )y2 ∆ (t1 ) One can easily observe that Q has a fixed point y if and only if y is a solution of (3.11)– (3.15) Since L1 , L2 , p, and G are bounded and continuous, it can be shown that there exists R > such that the compact operator Q : S → S where S = { y ∈ A : y ≤ R} Since S is a closed, bounded, and convex set, in view of the Tychonoff-Schauder fixed point theorem, there is at least one fixed point of Q Theorem 3.3 Assume that conditions (F) and (I) are satisfied Let α and β be a lower and upper solution, respectively, of the problem (3.1)–(3.5) such that α ≤ β on T Then every solution of the BVP (3.11)–(3.15) belongs to the sector [α,β] F M Atici and D C Biles 129 Proof Let y be a solution of (3.11)–(3.15), by definition of L∗ and L∗ , we know that y(a) ∈ [α(a),β(a)] and y(σ (b)) ∈ [α(σ (b)),β(σ (b))] We will prove that y ∈ [α,β] for t ∈ (a,σ (b)) Consider z(t) = y(t) − β(t) By definition of β, z is continuous on [a,σ (b)] Suppose, to the contrary, there is a t ∗ ∈ (a,t1 ) ∪ (t1 ,σ (b)) such that (y − β)(t ∗ ) = maxt∈T { y(t) − β(t)} > Suppose that t ∗ is left scattered In this case, we have that y ∆ t ∗ ≤ β∆ t ∗ , y ∆∆ ρ t ∗ ≤ β∆∆ ρ t ∗ (3.22) Consequently, by using condition F, we arrive at the following contradiction: > y ∆∆ ρ t ∗ − β∆∆ ρ t ∗ − y(t ∗ ) − β(t ∗ ) (3.23) ≥ f ρ(t ∗ ),β(t ∗ ) − f ρ(t ∗ ),β(t ∗ ) = When t ∗ is left dense the contradiction holds in a similar way Now suppose t ∗ = t1 Case t1 is left scattered + + Then we have z∆ (t1 ) ≤ and z∆ (t1 ) = z∆ (t1 ) Consequently, by using condition (I), we arrive at the following contradiction: + − = z∆ t1 − z∆ t1 + − + − = y ∆ t1 − y ∆ t1 − β∆ t1 − β∆ t1 ≥ I y(t1 , y ∆ t1 > I β t1 , y ∆ t1 − I β t1 ,β∆ t1 (3.24) − I β t1 ,β∆ t1 ≥ Case t1 is left dense For sufficiently small > 0, we have z∆ (s) ≥ for s ∈ t1 − ,t1 , z∆ s∗ ≤ for s∗ ∈ t1 ,t1 + (3.25) Then the contradiction holds in a similar way Analogously, the fact that α(t) ≤ y(t) for all t ∈ T can be shown Theorem 3.4 Assume that (L) holds If y ∈ [α,β] is a solution of (3.11)–(3.15), then y satisfies equalities (3.1)–(3.5) Proof If y(σ (b)) − L2 (y(a), y(σ (b))) < α(σ (b)), the definition of L∗ gives us that y(σ (b)) = α(σ (b)) Now using (L), we obtain a contradiction: α σ (b) > y σ (b) − L2 y(a), y σ (b) ≥ α σ (b) − L2 α(a),α σ (b) = α σ (b) (3.26) Analogously, we arrive at α(σ (b)) ≤ y(σ (b)) − L2 (y(a), y(σ (b))) ≤ β(σ (b)) and (3.13) implies that L2 (y(a), y(σ (b))) = 130 Dynamic equations with impulse To prove that L1 (y(a), y ∆ (a), y(σ (b)), y ∆ (σ(b))) = 0, it is enough using (3.12) to show α(a) ≤ y(a) + L1 y(a), y ∆ (a), y σ (b) , y ∆ σ(b) ≤ β(a) (3.27) If y(a) + L1 (y(a), y ∆ (a), y(σ (b)), y ∆ (σ(b))) < α(a), then y(a) = α(a) implies that = L2 (y(a), y(σ (b))) = L2 (α(a),α(σ (b))) Since L2 is injective with respect to the second variable, we have y(σ (b)) = α(σ (b)) Using the definition of L1 , we obtain a contradiction: α(a) > y(a) + L1 y(a), y ∆ (a), y σ (b) , y ∆ σ(b) ≥ α(a) + L1 α(a),α∆ (a),α σ (b) ,α∆ σ(b) ≥ α(a) (3.28) Here we used the fact that y ∈ [α,β], and α(a) = y(a), α(σ (b)) = y(σ (b)), consequently, it follows that y ∆ (a) ≥ α∆ (a) and y ∆ (σ(b)) ≤ α∆ (σ(b)) The other inequality holds similarly Example 3.5 Let T be any time scale and let the point 1/2 be a right-dense point in T ∩ [0,1] We define f , L1 , and L2 in the following way: f (t, y) = y sinh (y − 1)2 , L1 (x, y,z,w) = − x, L2 (x, y) = − y, I(x, y) = x − (3.29) Next we consider the following boundary value problem: y (t) = f t, y σ (t) , t ∈ [0,1]κ \ , (3.30) y(0) = 1, y(1) = 0, y y∆ + (3.31) (3.32) −y − = −1, (3.33) 1+ 1− 1− − y∆ , y∆ =I y 2 2 (3.34) One can easily verify that   0,   if t ∈ 0, , α(t) =   2(t − 1), if t ∈ ,1 ,  (3.35) is a lower solution and   2t + 1,   if t ∈ 0,  −2(t − 1),  if t ∈ β(t) =  is an upper solution of the problem (3.30)–(3.34) , ,1 , (3.36) F M Atici and D C Biles 131 Theorem 3.3 assures that there exists a solution y(t) of the problem (3.30)–(3.34) such that y ∈ [α,β] We note that   1,   y(t) =   0,  if t ∈ 0, if t ∈ , ,1 , (3.37) is one such solution References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] 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11, 1191–1198 C Pierson-Gorez, Impulsive differential equations of first order with periodic boundary conditions, Differential Equations Dynam Systems (1993), no 3, 185–196 J Qi and K Wang, Upper and lower solutions for impulsive differential equations with application to ODE, Northeast Math J 18 (2002), no 3, 189–196 A M Samo˘lenko and N A Perestyuk, Impulsive Differential Equations, World Scientific Series ı on Nonlinear Science Series A: Monographs and Treatises, vol 14, World Scientific, New Jersey, 1995 F M Atici: Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA E-mail address: ferhan.atici@wku.edu D C Biles: Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA E-mail address: daniel.biles@wku.edu ... upper solution of the problem (3. 30)– (3. 34) , ,1 , (3. 36) F M Atici and D C Biles 131 Theorem 3. 3 assures that there exists a solution y(t) of the problem (3. 30)– (3. 34) such that y ∈ [α,β] We note... I(x, y) = x − (3. 29) Next we consider the following boundary value problem: y (t) = f t, y σ (t) , t ∈ [0,1]κ \ , (3. 30) y(0) = 1, y(1) = 0, y y∆ + (3. 31) (3. 32) −y − = −1, (3. 33) 1+ 1− 1− − y∆... (3. 8) Now we introduce the concept of lower and upper solutions of problem (3. 1)– (3. 5) as follows F M Atici and D C Biles 127 Definition 3. 1 The functions α and β are, respectively, a lower and