A nonlinear impulsive integrodifference equation within the frame of qk-quantum calculus is investigated by applying using fixed point theorems. The conditions for existence and uniqueness of solutions are obtained.
1. Introduction
Recently, by introducing and applying the fractional difference operators to real world problems (see, e.g., [1–7] and the references therein) we revitalized the importance of the quantum calculus [8]. However the real world phenomena are usually described by complex model based involving different types of operators. In this way we hope to understand deeper the dynamics of complex or hypercomplex systems and to reveal their hidden aspects.
On this line of thought in this paper, we study the existence and uniqueness of solutions for nonlinear qk-integrodifference equation with nonlocal boundary condition and impulses:
(1)Dqku(t)=f(t,u(t))+Itkqkg(t,u(t)),0000000000<qk<1,t∈J′,Δu(tk)=Ik(u(tk)),k=1,2,…,m,u(0)=h(u)+u0,u0∈ℝ,
where Dqk, Itkqk are qk-derivatives and qk-integrals (k=0,1,2,…,m), respectively. f, g∈C(J×ℝ,ℝ), Ik, h∈C(ℝ,ℝ), J=[0,T](T>0), 0=t0<t1<⋯<tk<⋯<tm<tm+1=T, J′=J∖{t1,t2,…,tm}, Δu(tk)=u(tk+)-u(tk-), where u(tk+) and u(tk-) denote the right and the left limits of u(t) at t=tk(k=1,2,…,m), respectively.
2. Preliminaries
Let us set J0=[0,t1], J1=(t1,t2],…,Jm-1=(tm-1,tm], Jm=(tm,T] and introduce the space:
(2)PC(J,ℝ)={u:J⟶ℝ∣u∈C(Jk),k=0,1,…,m,0000andu(tk+)exist,k=1,2,…,m},
with the norm ∥u∥=supt∈J|u(t)|. Then, PC(J,ℝ) is a Banach space.
For convenience, let us recall some basic concepts of qk-calculus [9].
For 0<qk<1 and t∈Jk, we define the qk-derivatives of a real valued continuous function f as
(3)Dqkf(t)=f(t)-f(qkt+(1-qk)tk)(1-qk)(t-tk),Dqkf(tk)=limt→tkDqkf(t).
Higher order qk-derivatives are given by
(4)Dqk0f(t)=f(t),Dqknf(t)=DqkDqkn-1f(t),000000000000000000000000000n∈ℕ,t∈Jk.
The qk-integral of a function f is defined by
(5)tkIqkf(t)≔∫tktf(s)dqks=(1-qk)(t-tk)×∑n=0∞qknf(qknt+(1-qkn)tk),00000000000000000000t∈Jk,
provided the series converges. If a∈(tk,t) and f is defined on the interval (tk,t), then
(6)∫atf(s)dqks=∫tktf(s)dqks-∫tkaf(s)dqks.
Observe that
(7)Dqk(Itkqkf(t))=Dqk∫tktf(s)dqks=f(t),Itkqk(Dqkf(t))=∫tktDqkf(s)dqks=f(t),Iaqk(Dqkf(t))=∫atDqkf(s)dqks=f(t)-f(a),0000000000000000000000000000000a∈(tk,t).
For t∈Jk, the following reversing order of qk-integration holds
(8)∫tkt∫tksf(r)dqkrdqks=∫tkt∫qkr+(1-qk)tktf(r)dqksdqkr.
Note that if tk=0 and qk=q in (3) and (5), then Dqkf=Dqf, Itkqkf=I0qf, where Dq and I0q are the well-known q-derivative and q-integral of the function f(t) defined by
(9)Dqf(t)=f(t)-f(qt)(1-q)t,I0qf(t)=∫0tf(s)dqs=∑n=0∞t(1-q)qnf(tqn).
Lemma 1.
For given yqk∈C(J,ℝ), the function u∈PC(J,ℝ) is a solution of the impulsive qk-integrodifference equation
(10)Dqku(t)=yqk(t),0<qk<1,t∈J′,Δu(tk)=Ik(u(tk)),k=1,2,…,m,u(0)=h(u)+u0,u0∈ℝ,
if and only if u satisfies the qk-integral equation
(11)u(t)={∫0tyq0(s)dq0s+h(u)+u0,t∈J0;∫tktyqk(s)dqks+∑i=0k-1∫titi+1yqi(s)dqis+∑i=1kIi(u(ti))+h(u)+u0,t∈Jk.
Proof.
Let u be a solution of qk-difference equation (10). For t∈J0, applying the operator I0q0 on both sides of Dq0u(t)=yq0(t), we have
(12)u(t)=u(0)+I0q0yq0(t)=u(0)+∫0tyq0(s)dq0s.
Thus,
(13)u(t1-)=u(0)+∫0t1yq0(s)dq0s.
Similarly, for t∈J1, applying the operator It1+q1 on both sides of Dq1u(t)=yq1(t), then
(14)u(t)=u(t1+)+∫t1tyq1(s)dq1s.
In view of Δu(t1)=u(t1+)-u(t1-)=I1(u(t1)), it holds
(15)u(t)=u(0)+∫t1tyq1(s)dq1s+∫0t1yq0(s)dq0s+I1(u(t1)),∀t∈J1.
Repeating the above process, we can get
(16)u(t)=u(0)+∫tktyqk(s)dqks+∑i=0k-1∫titi+1yqi(s)dqis+∑i=1kIi(u(ti)),t∈Jk.
Using the boundary value condition given in (10), it follows
(17)u(t)=∫tktyqk(s)dqks+∑i=0k-1∫titi+1yqi(s)dqis+∑i=1kIi(u(ti))+h(u)+u0,t∈Jk.
Conversely, assume that u satisfies the impulsive qk-integral equation (11); applying Dqk on both sides of (11) and substituting t=0 in (11), then (10) holds. This completes the proof.
3. Main Results
Letting yqk(t)=f(t,u(t))+Itkqkg(t,u(t)), in view of Lemma 1, we introduce an operator Q:PC(J,ℝ)→PC(J,ℝ) as
(18)(Qu)(t)=∫tkt[f(s,u(s))+∫tksg(r,u(r))dqkr]dqks+∑i=0k-1∫titi+1[f(s,u(s))+∫tisg(r,u(r))dqir]0000000000×(s)dqis+∑i=1kIi(u(ti))+h(u)+u0.
By reversing the order of integration, we obtain
(19)(Qu)(t)=∫tkt[f(s,u(s))+[(t-tk)-qk(s-tk)]000000×g(s,u(s))]dqks+∑i=0k-1∫titi+1[f(s,u(s))000000000000+[(ti+1-ti)-qi(s-ti)]000000000000×g(s,u(s))]dqis+∑i=1kIi(u(ti))+h(u)+u0.
Then, the impulsive qk-integrodifference equation (1) has a solution if and only if the operator equation u=Qu has a fixed point.
In order to prove the existence of solutions for (1), we need the following known result [10].
Theorem 2.
Let E be a Banach space. Assume that T:E→E is a completely continuous operator and the set V={x∈E∣x=μTx,0<μ<1} is bounded. Then T has a fixed point in E.
Theorem 3.
Assume the following.
There exist nonnegative bounded functions Mi(t)(i=1,2,3,4) such that
(20)|f(t,u)|≤M1(t)+M2(t)|u|,|g(t,u)|≤M3(t)+M4(t)|u|,
for any t∈J, u∈ℝ.
There exist positive constants L¯,L~ such that
(21)|Ik(u)|≤L¯,|h(u)|≤L~,
for any u∈ℝ, k=1,2,…,m.
Then problem (1) has at least one solution provided
(22)supt∈J[TM2(t)+M4(t)∑i=0m(ti+1-ti)21+qi]<1.
Proof.
Firstly, we prove the operator Q:PC(J,ℝ)→PC(J,ℝ) is completely continuous. Clearly, continuity of the operator Q follows from the continuity of f, g, Ik, and h. Let Ω⊂PC(J,ℝ) be bounded. Then ∀t∈J, u∈Ω; there exist positive constants Li(i=1,2,3,4) such that |f(t,u)|≤L1, |g(t,u)|≤L2, |Ik(u)|≤L3, |h(u)|≤L4. Thus
(23)|(Qu)(t)|≤∫tkt[|f(s,u(s))|+[(t-tk)-qk(s-tk)]00000×|g(s,u(s))|]dqks+∑i=0k-1∫titi+1[|f(s,u(s))|00000000000+[(ti+1-ti)-qi(s-ti)]00000000000×|g(s,u(s))|]dqis+∑i=1k|Ii(u(ti))|+|h(u)|+|u0|≤∫tkt[L1+[(t-tk)-qk(s-tk)]L2]dqks+∑i=0k-1∫titi+1[L1+[(ti+1-ti)000000000000000-qi(s-ti)]L2]dqis+∑i=1kL3+L4+|u0|≤L1(t-tk)+L2(t-tk)21+qk+∑i=0k-1[L1(ti+1-ti)+L2(ti+1-ti)21+qi]+mL3+L4+|u0|≤TL1+L2∑i=0m(ti+1-ti)21+qi+mL3+L4+|u0|≔ℒ(constant).
This implies ∥Qu∥≤ℒ.
Furthermore, for any t′, t′′∈Jk(k=0,1,2,…,m) satisfying t′<t′′, we have
(24)|(Qu)(t′′)-(Qu)(t′)|≤|∫tkt′′[|f(s,u(s))|0000000+[(t′′-tk)-qk(s-tk)]0000000×|g(s,u(s))|]dqks000-∫tkt′[|f(s,u(s))|000000+[(t′-tk)-qk(s-tk)]000000×|g(s,u(s))|]dqks∫tkt′′|≤∫t′t′′|f(s,u(s))00000+[(t′′-tk)-qk(s-tk)]0000000×g(s,u(s))|dqks0000+∫tkt′(t′′-t′)00000000000×|g(s,u(s))|dqks≤∫t′t′′[|f(s,u(s))|+(t′′-tk)000000×|g(s,u(s))||f(s,u(s))|+(t′′-tk)]dqks+L2(t′-tk)(t′′-t′)≤[L1+L2(t′′-tk)](t′′-t′)+L2(t′-tk)(t′′-t′).
As t′→t′′, the right hand side of the above inequality tends to zero. Thus, Q(Ω) is relatively compact. As a consequence of Arzela Ascoli's theorem, Q is a compact operator. Therefore, Q is a completely continuous operator.
Define the set 𝒲={u∈PC(J,ℝ)∣u=λQu,0<λ<1}.
Next, we show 𝒲 is bounded. Let u∈𝒲; then u=λQu, 0<λ<1. For any t∈J, by conditions (H1) and (H2), we have
(25)|u(t)|=λ|(Qu)(t)|≤∫tkt[|f(s,u(s))|+[(t-tk)-qk(s-tk)]000000×|g(s,u(s))|]dqks+∑i=0k-1∫titi+1[|f(s,u(s))|000000000000+[(ti+1-ti)-qi(s-ti)]000000000000×|g(s,u(s))|]dqis+∑i=1k|Ii(u(ti))|+|h(u)|+|u0|≤∫tkt[[M3(s)+M4(s)|u(s)|]M1(s)+M2(s)|u(s)|000000+[(t-tk)-qk(s-tk)]000000×[M3(s)+M4(s)|u(s)|]]dqks+∑i=0k-1∫titi+1[M1(s)+M2(s)|u(s)|000000000000+[(ti+1-ti)-qi(s-ti)]000000000000×[M3(s)+M4(s)|u(s)|]]dqis+∑i=1kL¯+L~+|u0|≤(M1+M2∥u∥)(t-tk)+(M3+M4∥u∥)×(t-tk)21+qk+∑i=0k-1[(ti+1-ti)21+qi(M1+M2∥u∥)(ti+1-ti)×(t-tk)21+qk+∑i=0k-100+(M3+M4∥u∥)(ti+1-ti)21+qi]+mL¯+L~+|u0|≤(M1+M2∥u∥)T+(M3+M4∥u∥)×∑i=0m(ti+1-ti)21+qi+mL¯+L~+|u0|≤M1T+M3∑i=0m(ti+1-ti)21+qi+mL¯+L~+|u0|+[M2T+M4∑i=0m(ti+1-ti)21+qi]∥u∥,
which implies
(26)∥u∥≤M1T+M3∑i=0m((ti+1-ti)2/(1+qi))+mL¯+L~+|u0|1-[M2T+M4∑i=0m((ti+1-ti)2/(1+qi))]≔constant.
So, the set 𝒲 is bounded. Thus, Theorem 2 ensures the impulsive qk-integrodifference equation (1) has at least one solution.
Corollary 4.
Assume the following.
There exist nonnegative constants Li(i=1,2,3,4) such that
(27)|f(t,u)|≤L1,|g(t,u)|≤L2,|Ik(u)|≤L3,|h(u)|≤L4,
for any t∈J, u∈ℝ, k=1,2,…,m.
Then problem (1) has at least one solution.
Theorem 5.
Assume the following.
There exist nonnegative bounded functions M(t) and N(t) such that
(28)|f(t,u)-f(t,v)|≤M(t)|u-v|,|g(t,u)-g(t,v)|≤N(t)|u-v|,
for t∈J,u,v∈ℝ.
There exist positive constants K,G such that
(29)|Ik(u)-Ik(v)|≤K|u-v|,|h(u)-h(v)|≤G|u-v|,
Denote supt∈J|M(t)|=M, supt∈J|N(t)|=N. For ∀u,v∈PC(J,ℝ), by (H4) and (H5), we have
(30)|(Qu)(t)-(Qv)(t)|≤∫tkt[|f(s,u(s))-f(s,v(s))|00000000+[(t-tk)-qk(s-tk)]00000000×|g(s,u(s))-g(s,v(s))|]dqks+∑i=0k-1∫titi+1[|f(s,u(s))-f(s,v(s))|0000000000000+[(ti+1-ti)-qi(s-ti)]0000000000000×|g(s,u(s))-g(s,v(s))|]dqis+∑i=1k|Ii(u(ti))-Ii(v(ti))|+|h(u)-h(v)|≤∫tkt[M(s)+[(t-tk)-qk(s-tk)]N(s)]000000×|(u-v)(s)|dqks+∑i=0k-1∫titi+1[M(s)+[(ti+1-ti)-qi(s-ti)]000000000000000×N(s)]|(u-v)(s)|dqis+∑i=1kK|(u-v)(ti)|+G|u-v|≤[M(t-tk)+N(t-tk)21+qk00000+∑i=0k-1[M(ti+1-ti)+N(ti+1-ti)21+qi]00000+mK+G(t-tk)21+qk]∥u-v∥≤[Mt+mK+G+N∑i=0m(ti+1-ti)21+qi]∥u-v∥≤𝒦∥u-v∥.
As 𝒦<1 by (H6), then ∥Qu-Qv∥<∥u-v∥. Therefore, Q is a contractive map. Thus, the conclusion of the Theorem 5 follows by Banach contraction mapping principle.
4. Example
Consider the following nonlinear qk-integrodifference equation with impulses
(31)D1/(2+k)u(t)=8+3t+ln(1+5t3+t25|u(t)|)+∫1/(1+2k)t[10s+s33sinu(s)]d1/(2+k)s,00000000000000000t∈[0,1],t≠11+2k,Δu(11+2k)=cos(u(11+2k)),k=1,2,…,6,u(0)=5+e-u2(1/2).
Obviously, qk=1/(2+k)(k=0,1,2,…,6), tk=1/(1+2k)(k=1,2,…,6), f(t,u)=8+3t + ln(1+5t3+(t2/5)|u|), g(t,u)=10t+(t3/3)sinu, Ik(u)=cosu, and h(u)=e-u2.
By a simple calculation, we can get
(32)|f(t,u)|≤8+3t+5t3+t25|u|,|g(t,u)|≤10t+t33|u|,|Ik(u)|≤1,|h(u)|≤1.
Take M1(t)=8+3t+5t3, M2(t)=t2/5, M3(t)=10t, M4(t)=t3/3, and L¯=L~=1. Then all conditions of Theorem 3 hold. By Theorem 3, nonlinear impulsive qk-integrodifference (31) has at least one solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (2012021002-3).
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