ASYMPTOTICALLY P-PERIODIC SOLUTIONS OF A QUANTUM VOLTERRA INTEGRAL EQUATION

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.4 (27), No., (28), 59 7 DOI:.5644/SJM.4..6 ASYMPTOTICALLY P-PERIODIC SOLUTIONS OF A QUANTUM VOLTERRA INTEGRAL EQUATION MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER ABSTRACT. In this paper, we study the existence of an asymptotically periodic solution of a Volterra integral equation on the time scale q N, which we call a quantum Volterra integral equation. In the process, we study the existence of periodic solutions of an associated equation on the time scale q Z, which is an extension of q N. We employ Schauder s fixed point theorem in the analysis.. INTRODUCTION Qualitative analysis of dynamic equations has always been a primary research topic in applied mathematics. In particular, the periodicity of dynamic and integrodynamic equations on so called additively periodic time scale have been studied extensively in the last decade (see [2, 6,, 3, 7, 8, 9, 2], and the references therein). A time scale domain T is said to be additively periodic if there exists a P Tsuch that t± P Tfor all t T, (see [7]). A function f on an additively periodic domain T is said to be periodic if there exists a T [P, ) T with f(t± T)= f(t) for all t T. Examples of additively periodic domains include the set of reals R, the set of integers Z, and the set hz={ht : t Z}. Due to many applications, it has been found that the notion of periodicity on non-additively periodic time scales is very important. For example, to study periodicity of q-difference equations, which occur in the field of quantum calculus, one needs a non-additively periodic time scale domain such as q Z ={q n : n Z} {}, which is an extension of the so called q-time scale q N = {q n : n =,,2,...}, where q> is assumed. We refer the readers to [, 5, 5] for the periodicity notions, including definitions of periodic functions on the q-time scale q N. Article 2 Mathematics Subject Classification. 39A3, 39A2, 39A23, 39A2. Key words and phrases. dynamic equations, Volterra integral equations, time scales, periodic, asymptotically periodic.

2 6 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER [5] provides a comprehensive discussion on the two different periodicity definitions found in [5] and in []. Also, in [5] readers will find results on the existence of periodic solutions of a type of Volterra equation, which the authors called a q-volterra integral equation. In the present paper, we study a different type of periodicity, known as asymptotic periodicity, which we define later in the paper. In particular, we study the existence of asymptotically periodic solutions of a Volterra type nonlinear integral equation on the non-additively periodic time scale q N where the periodic part is extended to q Z. In the process, we study the existence of a periodic solution of an associated equation. The two articles that are somewhat related to our present paper are [, 4], in which authors studied existence of continuous asymptotically periodic solutions of Volterra integral equations. We refer the readers to [6] for a review of basic theory on quantum calculus, to [2] for a study of the existence of q-difference equations, and to [4] for some discussions on the equivalence between q-difference equations and differential equations. The first periodicity notion on q N was given by Bohner and Chieochan in [5]. Definition. ([5]). Let P N. A function f : q N R is said to be P-periodic if f(t)=q P f(q P t) for all t q N. (.) See [6, 7, 8, 9] for some applications. Afterwards, Adivar [] (see also [3]) introduced a more general periodicity notion on time scales that are not necessarily additively periodic. On q N, this is defined as follows. Definition.2 ([]). Let P N. A function f : q N R is said to be P-periodic if f(q P t)= f(t) for all t q N. In [5], it was shown that f is periodic with respect to Definition. if and only if f(t) = t f(t) is periodic with respect to Definition.2. Other relationships between these definitions involoving periodic solutions of q-difference and q-integral equations are also established in [5]. Notice each of these definitions can easily be extended to the time scale q Z. When we refer to P-periodic functions in this paper, we will be referring to Definition.2. We consider the nonlinear Volterra integral equation x(t)=a(t)+ C(t,s) f(s,x(s))d q s, t q N, (.2) where a : q N R is bounded, C : q N q N R, and f : q N R R is continuous in its second variable. Here q n n f(s)d q s :=(q ) q k f(q k ), q m k=m

3 ASYMPTOTICALLY P-PERIODIC SOLUTIONS... 6 where m= if the lower limit of integration is, and n= if the upper limit of integration is. Throughout the paper, we assume the following conditions on the functions a, C and f. We refer to these as the basic assumptions. We assume a(t)= b(t)+r(t), and C(t,s) f(s,x) = D(t,s)g(s,x)+Q(t,s)h(s,x), where h : q N R R and g : q N R R are continuous in their second variables. We also assume is bounded, b(t)=b(q P t) for all t q N, (.3) r(t) as t, (.4) g(t,x)=g(q P t,x) for all t q N, (.5) and D(t,s)=q P D(q P t,q P s) for all (t,s) q N q N. (.6) By using (.3), (.5) and (.6), b, g and D can be extended periodically so that b : q Z R, g : q Z R R and D : q Z q Z R. Notice that the assumptions on b and g relate to Definition.2 and the assumption on D relates to Definition.. Definition.3. A function x is asymptotically P-periodic if there exists a P-periodic function y and a function z such that x(t)=y(t)+z(t) with z(t) as t. In this article, we show that there exists an asymptotically periodic solution x of (.2). In the process, we show that there exists a periodic solution of u(t)=b(t)+ which we refer to as the associated equation of (.2). For a constant ρ>, let D(t,s)g(s,u(s))d q s, t q Z, (.7) 2. PRELIMINARIES B ρ ={x R: x ρ}. We assume (H) lim Q(t,s) d q s=; t (H2) there exists a d > such that D(t,s) d q s d < for all t q Z ; and

4 62 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER (H3) (i) (m f) ρ = sup{ f(t,x) : x B ρ, t q N }<, (ii) (mg) ρ = sup{ g(t,x) : x B ρ, t q Z }<, (iii) (mh) ρ = sup{ h(t,x) : x B ρ, t q N }<. Remark 2.. Assumption (H) implies that there exists a q such that sup t q N Remark 2.2. When D satisfies (H2), uniformly in t. lim k Q(t,s) d q s q <. D(q kp t,s) d q s=, Proof. Since D satisfies (H2), then for k N, D(t,s) d q s= By letting k, we have D(t,s) d q s= lim k D(t,s) d q s+ D(t,s) d q s. D(t,s) d q s+ D(t,s) d q s. This implies lim D(t,s) d q s=. k Since D(t,s)= q P D(q P t,q P s), D(t,s)= q kp D(q kp t,q kp s). So implying lim k D(t,s) d q s= lim k = lim k q kp D(q kp t,q kp s) d q s D(q kp t,s) d q s, lim D(q kp t,s) d q s=. k Theorem 2.3. In addition to the basic assumptions, let (H)-(H3) hold. Let x be an asymptotically P-periodic solution of (.2) with x(t) ρ for t q N. Then the function η(t)= C(t,s) f(s,x(s))d q s, t q N, is asymptotically P-periodic. Moreover, the P-periodic part of η(t) is ϕ(t)= D(t,s)g(s, π(s))d q s, t q N,

5 ASYMPTOTICALLY P-PERIODIC SOLUTIONS where x(t) = y(t)+z(t) with y(t)=y(q P t), z(t) as t, and where π is the P-periodic extension of y to q Z. Proof. First, notice for t q N, So ϕ is P-periodic. Now, for t q N, η(t) ϕ(t) = = = q P ϕ(q P t t)= = = = ϕ(t). D(q P t,s)g(q P s, π(s))d q s q P D(q P t,q P s)g(q P s, π(q P s))d q s D(t,s)g(s, π(s))d q s C(t,s) f(s,x(s))d q s D(t,s)g(s,x(s))d q s+ D(t,s)g(s, π(s))d q s + D(t,s)g(s, π(s))d q s Q(t,s)h(s,x(s))d q s D(t,s)g(s,x(s))d q s+ D(t,s)g(s,x(s))d q s Q(t,s)h(s,x(s))d q s D(t,s)g(s, π(s))d q s D(t,s)g(s, π(s))d q s. D(t,s) g(s,x(s)) d q s+ D(t,s) g(s, π(s)) d q s + + D(t,s) g(s,x(s)) g(s, π(s)) d q s Q(t,s) h(s,x(s)) d q s. Let x,y B ρ. By assumption (H3) and the fact that that D(t,s) d q s D(t,s) d q s, η(t) ϕ(t) 2(mg) p D(t,s) d q s

6 64 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER In the proof of Remark 2.2, it was shown that Thus + D(t,s) g(s,x(s)) g(s, π(s)) d q s +(mh) ρ Q(t,s) d q s. D(t,s) d q s= D(q kp t,s) d q s. η(t) ϕ(t) 2(mg) p D(q kp t,s) d q s + D(t,s) g(s,x(s)) g(s, π(s)) d q s +(mh) ρ Q(t,s) d q s. By Remark 2.2, for large enough t, there exists a k N, q kp < t, such that Thus, for this chosen k, D(q kp t,s) d q s< ε 6(mg) ρ. 2(mg) p D(q kp t,s) d q s< ε 3. Since π is a periodic extension of y, we can replace π(t) by y(t) for large t. So x(t) π(t) = x(t) y(t) as t, which, by the continuity of g in the second variable, implies g(t,x(t)) g(t, π(t)) as t. Thus there exists a T > q kp such that for t > T, D(t,s) g(s,x(s)) g(s, π(s)) d q s< ε 3. Finally, by assumption (H), Q(t,s) d qs as t. So there exists a T 2 such that for t > T 2, ε Q(t,s) d q s<. 3(mh) ρ Thus for t > T 2, (mh) ρ Q(t,s) d q s< ε 3. Let T = max{t,t 2 }. Then for t > T, Thus n is asymptotically P-periodic. η(t) ϕ(t) <ε.

7 ASYMPTOTICALLY P-PERIODIC SOLUTIONS Remark 2.4. When (H)-(H3) hold, then there exists a c such that sup t q N C(t,s) d q s c <. This follows when Remark 2., (H2) and (H3) are applied on C(t,s) f(s,x) = D(t,s)g(s,x)+Q(t,s)g(s,x). 3. EXISTENCE Let B ={x : q N R:xis bounded} be the Banach space of bounded functions on q N with norm x =sup t q N x(t). Theorem 3. (Schauder s Fixed Point Theorem). If S is a closed, bounded, convex subset of a Banach space X and H : S S is completely continuous, then H has a fixed point in S. A mapping H is completely continuous if it is continuous, and it maps a bounded set into a set whose closure is compact. Theorem 3.2. Suppose (H)-(H3) and the basis assumptions hold. Then (.2) has an asymptotically P-periodic solution. Proof. Let M = b +d (mg) ρ and N = q +c (m f) ρ + d (mg) ρ, where c and d are the constants from Remark 2.4 and (H2) respectfully, and (m f) ρ and (mg) ρ are the constants from (H3). Choose a ρ> such that M+ N < ρ. Define S ρ = { x B : x(t)=y(t)+z(t),y(t)=y(q P t),z(t) as t, y M, z N }. Notice S ρ is a closed and convex subset of B, and if x S ρ, x ρ. Define H on S ρ by Hx(t)=a(t)+ C(t,s) f(s,x(s))d q s. Since x S ρ, there exists y and z such that x(t) = y(t)+z(t), where y(t) = y(q P t) for all t q N, z(t) as t. From Theorem 2.3, η(t)= C(t,s) f(s,x(s))d q s is asymptotically P-periodic, and the P-periodic part of η(t) is given by ϕ(t)= where π is the periodic extension of y to q Z. D(t,s)g(s, π(s))d q s,

8 66 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER Let α(t) = η(t) ϕ(t). Then α(t) as t. By the basic assumptions, a(t) = b(t)+r(t), where b(t) = b(q P t) and r(t) as t. Thus H can be written as Hx(t)=b(t)+r(t)+ϕ(t)+α(t)=(b(t)+ϕ(t))+(r(t)+α(t))=u(t)+v(t), where u(t)=b(t)+ϕ(t) and v(t)=r(t)+α(t). Notice u(t)=b(t)+ϕ(t)=b(q P t)+ϕ(q P t)=u(q P t), so u is P-periodic. Also, since r(t) and α(t) as t, then v(t) as t. Now, notice that ϕ(t) which implies ϕ d (mg) ρ. Also, α(t) η(t) + ϕ(t) Thus α c (m f) ρ + d (mg) ρ. Therefore, D(t,s) g(s, π(s)) d q s d (mg) ρ, C(t,s) f(s,x(s)) d q s+ ϕ c (m f) ρ + d (mg) ρ. u b + ϕ b +d (mg) ρ + M, and v q + α q +c (m f) ρ + d (mg) ρ = N. Therefore, Hx ρ. Thus H : S ρ S ρ, and hence HS ρ is uniformly bounded. Next, we show H is continuous in x. For each x S ρ, define and (Ux)(t)= C(t,s)x(s)d q s (V x)(t)= f(t,x(t)), for all t q N. Since f is continuous in its second variable, V is continuous in x. Since U is linear in x, U is continuous in x. Thus, since Hx = a+(u V)x for x S ρ, H is continuous in x. Finally, we show HS ρ is compact. We will use the sequential criterion, which states that HS ρ is compact if and only if every sequence in HS ρ has a subsequence converging to an element in HS ρ. Let {(Hx) m } be a sequence in HS ρ. Since each (Hx) m HS ρ, there exists functions ϕ m and α m such that (Hx) m (t) = ϕ m (t)+ α m (t), where ϕ m is P-periodic and α m (t) as t. The sequence {ϕ m (q )} is uniformly bounded and therefore must have a subsequence {ϕ mk (q )} that converges. Then {α mk (q )} is uniformly bounded and also has a subsequence that converges. For notational purposes, label this subsequence {α mk }. Similarly, the sequence{ϕ mk (q )} is uniformly bounded and therefore must have a subsequence {ϕ mk j (q )} that converges. Then {α mk j (q )} is uniformly bounded and also has a subsequence that converges. For notational purposes, label this subsequence {α mk }. Continuing this process gives subsequences {ϕ m jk } and {α m jk } such that

9 ASYMPTOTICALLY P-PERIODIC SOLUTIONS {ϕ m jk (q l )} and {α m j k (q l )} converge for each l =,..., j. Using a diagonalization technique, consider the subsequences {ϕ m j j } and {α m j j }. By construction, these subsequences must converge at each t q N. Denote lim j ϕ m j j (t) = ϕ (t) and lim j α m j j (t)=α (t). Then lim (Hx) m j j (t)=ϕ (t)+α (t) :=(Hx) (t). j Notice that since ϕ m (t) = ϕ j j m (qp j t) and ϕ j m (qp j t) ϕ (q P t), ϕ (t) = ϕ (q P t), j implying ϕ is P-periodic. Since lim t α m j (t)=, we have j lim t α (t)=. So(Hx) is asymptotically P-periodic. Finally, notice (Hx) ϕ + α. M αm N But ϕ m j j and j j for all j N, so ϕ M and α N. So (Hx) ρ. Thus, (Hx) HS ρ, which shows HS ρ is compact. Therefore, by Schauder s fixed point theorem, there exists a function x in HS ρ such that x = Hx; the function x is a solution of (.2). This concludes the proof of Theorem 3.2. Theorem 3.3. Suppose (H)-(H3) and the basic assumptions hold. Then (.7) has a P-periodic solution. Proof. It was shown in Theorem 3.2 that there exists an asymptotically periodic solution x of (.2). Then there exist functions y and z with y(t) = y(q P t) for all t q N and z(t) as t, such that x(t) = y(t)+z(t) for all t q N. Then from (.2), we can write y(t)+z(t)=b(t)+r(t)+ By Theorem 2.3, C(t,s) f(s,x(s))d q s= C(t,s) f(s,x(s))d q s, t q N. (3.) D(t,s)g(s, π(s))d q s+α(t), (3.2) where α(t) as t, and π is the P-periodic extension of y to q Z. Combining (3.) and (3.2) gives π(t)+z(t)=b(t)+ D(t,s)g(s, π(s))d q s+α(t)+r(t), t q Z. (3.3) Now equating the P-periodic parts of both sides of (3.3) yields π(t)=b(t)+ D(t,s)g(s, π(s))d q s, t q Z.

10 68 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER This shows that the function π, which is P-periodic, satisfies equation (.7). This concludes the proof that (.7) has a P-periodic solution. 4. AN EXAMPLE Example 4.. Consider the q-volterra integral equation ( ) ) lnt t (( ) x(t)=cos lnq π + e t lnq lnt s + t 2 + se t( +e t) x(s) d q s, t q N. (4.) ( lnt lnt Define b(t) = cos ), lnq π r(t) = e t ( ) lnq s, C(t,s) = t 2 + se t (+e t ) and f(t,x)=x. Let D(t,s)= s lnt for s t, g(t,x)=( ) lnq x, Q(t,s)=se t for t2 s t, and h(t,x) =(+e t )x. Then C(t,s) f(s,x) = D(t,s)g(s,x)+Q(t,s)h(s,x). Thus the basic assumptions, including (.3)-(.6), are satisfied ( with ) P = 2. lnt Extend b, g and D periodically to q Z by b(t) = cos lnq π for t q Z and b()=, g(t,x)=( ) lnq lnt x for t q Z and g(,x)=, and D(t,s)= s for <s t t2 and D(,)=. Let t = q n. To see (H), notice n Q(t,s) d q s=(q ) q k q k exp( q n ) as t. Also, k= =(q )exp( q n ) q2n q 2 = e t (t 2 ) q+ D(t,s) d q s=(q ) n k= = q q 2n q 2n q 2 = q+, q k qk q 2n implying (H2) holds. Finally, (H3) holds, since (m f) ρ = (mg) ρ = (mh) ρ = ρ. Therefore, by Theorem 3.2, (4.) has an asymptotically 2-periodic solution.

11 ASYMPTOTICALLY P-PERIODIC SOLUTIONS Also, Theorem 3.3 gives the existence of a 2-periodic solution of ( ) lnt t u(t)= cos lnq π ( ) lnq lnt s + t 2 u(s) d q s, t q Z. REFERENCES [] M. Adivar, A new periodicity concept for time scales, Math. Slovaca 63 (23), No. 4, /s [2] M. Adivar, H. Can Koyuncuoğlu and Y. N. Raffoul, Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay, J. Diffrence Equ. Appl. 9 (23), No. 2, / [3] M. Adivar, H. Can Koyuncuoğlu and Y. N. Raffoul, Existence of periodic solutions in shifts δ ± for neutral nonlinear dynamic systems, Appl. Math. Comput. 242 (24), /j.amc [4] Y. André and L. Di Vizio, q-difference equations and p-adic local monodromy, Astérisque 296 (24), [5] M. Bohner and R. Chieochan, Floquet theory for q-difference equations, Sarajevo J. Math 8 (22), No. 2, /SJM [6] M. Bohner and R. Chieochan, Positive periodic solutions for higher-order functional q- difference equations, J. Appl. Funct. Anal. 8 (23), No., [7] M. Bohner and R. Chieochan, The Beverton-Holt q-difference equation with periodic growth rate, Springer Proc. Math. Stat. 5 (25), / [8] M. Bohner and J. G. Mesquita, Periodic averaging principle in quantum calculus, J. Math. Anal. Appl. 435 (26), No. 2, /j.jmaa [9] M. Bohner and S. Streipert, Optimal harvesting policy for the Beverton-Holt quantum difference model, Math. Morav. 2 (26), No. 2, [] T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations, Dover Publications, Inc., Mineola, NY, [] T. A. Burton and T. Furumochi, Periodic and asymptotically periodic solutions of Volterra integral equations, Funkcial. Ekvac. 39 (996), No., [2] M. El-Shahed and H. A. Hassan, Positive solutions of q-difference equation, Proc. Amer. Math. Soc. 38 (2), No. 5, /S [3] M. N. Islam, Three fixed point theorems: periodic solutions of a Volterra type integral equation with infinite heredity, Canad. Math. Bull. 56 (23), No., /CMB [4] M. N. Islam, Asymptotically periodic solutions of Volterra integral equations, Electron. J. Differential Equations 26 (26), No. 83, [5] M. N. Islam and J. T. Neugebauer, Existence of periodic solutions for a quantum Volterra equation, Adv. Dyn. Syst. Appl. (26), No., [6] V. Kac and P. Cheung, Quantum calculus, Universitext, Springer-Verlag, New York, / [7] E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Ana. Appl. 39 (26), No., /j.jmaa [8] Y. Li and L. Zhu, Existence of positive periodic solutions for difference equations with feedback control, Appl. Math. Lett. 8 (25), No., /j.aml [9] B. Lisena, Periodic solutions of logistic equations with time delay, Appl. Math. Lett. 2 (27), No., /j.aml.26..8

12 7 MUHAMMAD N. ISLAM AND JEFFREY T. NEUGEBAUER [2] W. Wang and Z. Luo, Positive periodic solutions for higher-order functional difference equations, Int. J. Difference Equ. 2 (27), No. 2, (Received: May 6, 27) (Revised: September 2, 27) Muhammad N. Islam Department of Mathematics University of Dayton Dayton, OH USA mislam@udayton.edu Jeffrey T. Neugebauer Eastern Kentucky University Department of Mathematics and Statistics Richmond, KY 4475 USA jeffrey.neugebauer@eku.edu