Feckan, Michal

Zhao, Hou Yu

Miskolc Mathematical Notes

Hou Yu Zhao

Michal Feckan

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Pseudo almost periodic solutions of an iterative equation with variable coefficients

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 515–524 DOI: 10.18514/MMN.2017.2047PSEUDO ALMOST PERIODIC SOLUTIONS OF AN ITERATIVEEQUATION WITH VARIABLE COEFFICIENTSHOU YU ZHAO AND MICHAL FECˇKANReceived 09 June, 2016Abstract. In this paper, the exponential dichotomy, and Tikhonov and Banach fixed point the-orems are used to study the existence and uniqueness of pseudo almost periodic solutions ofnonhomogeneous iterative functional differential equations of the form x0.t/ D 1.t/x.t/C2.t/xŒ2.t/C : : :Cn.t/xŒn.t/Cf .t/.2010 Mathematics Subject Classification: 39B12; 39B82Keywords: iterative functional differential equation, pseudo almost periodic solutions, fixed pointtheorem1. INTRODUCTIONRecently, iterative functional differential equations of the formx0.t/DH.xŒ0.t/;xŒ1.t/;xŒ2.t/; : : : ;xŒn.t//have appeared in several papers, where xŒ0.t/D t , xŒi.t/D x.xŒi 1.t//, i  1. In[4], Cooke points out that it is highly desirable to establish the existence and stabilityproperties of periodic solutions for equations of the formx0.t/Cax.t  h.t;x.t///D F.t/;in which the lag h.t;x.t// implicitly involves x.t/. Stephan [14] studies the existenceof periodic solutions of equationx0.t/Cax.t   rCh.t;x.t///D F.t/:Eder [6] considers the iterative functional differential equationx0.t/D xŒ2.t/This work was partially supported by the National Natural Science Foundation of China (GrantNo. 11326120, 11501069), Foundation of Chongqing Municipal Education Commission (Grant No.KJ1400528), the Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0153/16 and by the Slovak Researchand Development Agency under the contract No. APVV-14-0378.c 2017 Miskolc University Press516 H. Y. ZHAO AND M. FECˇKANand obtains that every solution either vanishes identically or is strictly monotonic.FeLckan [7] studies the equationx0.t/D f .xŒ2.t//by obtaining an existence theorem for solutions satisfying x.0/D 0: Later, Wang andSi [17] studyx0.xŒr.t//D c0´C c1x.t/C c2xŒ2.t/C : : :CxŒn.t/;and show the existence theorem of analytic solutions. In particularly, Si and Cheng[12] discusses the smooth solutions of equation ofx0.t/D 1x.t/C2xŒ2.t/C : : :CnxŒn.t/Cf .t/:For some various properties of solutions for several iterative functional differentialequations, we refer the interested reader to [10, 11, 13, 15].On the other hand, the existence of pseudo almost periodic solutions is amongthe most attractive topics in qualitative theory of differential equations due to theirapplications, especially in biology, economics and physics [1,3,5,18,20]. As pointedout by Ait Das and Ezzinbi [1], it is an interesting thing to study the pseudo almostperiodic systems with delays. It is obviously that iterative functional differentialequations are special type state-dependent delay differential equations. In [9], Liupointed that the properties of the almost periodic functions do not always hold in theset of pseudo almost periodic functions and given an example: when x.t/ is a pseudoalmost periodic function, x.x.t// may not be a pseudo almost periodic function.To the best of our knowledge, there are few results about pseudo almost periodicsolutions for iterative functional differential equations except [9] and [16].In the present work, we propose an existence result for pseudo almost periodicsolutions of iterative functional differential equationsx0.t/D 1.t/x.t/C2.t/xŒ2.t/C : : :Cn.t/xŒn.t/Cf .t/: (1.1)by using Tikhonov fixed theorem. Uniqueness of the solution is achieved by Banachfixed point theorem.This paper is organized as follows. In Section 2 we give two theorems which areused in Section 3 to establish the main existence result. In Section 4, we show that(1.1) has a unique pseudo almost periodic solution under some suitable conditionsand present an example to illustrate the theory. Related problems are also studied in[2].PSEUDO ALMOST PERIODIC SOLUTIONS 5172. PRELIMINARIESLet C.R;R/ denote the set of all continuous functions from R to R endowed withthe usual metricd.f;g/D1XmD12 m kf  gkm1Ckf  gkm for kf  gkm D maxt2Œ m;m jf .t/ g.t/j;so the topology on C.R;R/ is the uniform convergence on each compact intervalsof R. We also consider the set BC.R;R/ of all bounded and continuous functionsfrom R to R with the norm kf k D supt2R jf .t/j, so the topology on BC.R;R/ is theuniform convergence on R. We denote by AP.R;R/ the set of all almost periodicfunctions from R to R. Define the setPAP0.R;R/Dng 2 BC.R;R/ˇˇˇlimT!C112TZ T Tjg.t/jdt D 0o:A function ' 2 BC.R;R/ is called pseudo almost periodic if it can be expressedas ' D hC g, where h 2 AP.R;R/ and g 2 PAP0.R;R/. The collection of suchfunctions will be denoted by PAP.R;R/. In particular, .PAP.R;R/;kk/ is a Banachspace [19]. For M;L > 0, defineBC .M;L/Dn' 2 C.R;R/ˇˇˇj'.t/j M; j'.t2/ '.t1/j  Ljt2  t1j;for all t; t1; t2 2 Ro;BS .M;L/D S.R;R/\BC .M;L/; S 2 fPAP;AP;PAP0g:From [9], it is easy to see that BS .M;L/, S 2 fPAP;AP;PAP0;C g are closed con-vex and bounded subsets of BC.R;R/ and C.R;R/, respectively. Furthermore, bythe Arzela´-Ascoli theorem, the subset BC .M;L/ is compact in C.R;R/.On the other hand, the subset BC .M;L/ is not precompact in BC.R;R/, sincea sequencenM sech L.tCn/Mo1nD1 2 BC .M;L/ has no a convergent subsequence inBC.R;R/.Furthermore, the subset BAP .M;L/ is not compact in C.R;R/, since we considerthe function from [8]g.t/D1XkD11ksin2t2k:From 1ksin2 t2k 1kt24kand that the seriesP1kD1 1kt24kconvergents, we see thatgn.t/DnXkD11ksin2t2k! g.t/518 H. Y. ZHAO AND M. FECˇKANin C.R;R/. Similarly we deriveg0n.t/DnXkD11k2k 1sint2kcost2k! g0.t/in BC.R;R/ and kg0nk  2, kg0k  2. Take'n.t/Dminf1;gn.t/g D 1Cgn.t/ j1 gn.t/j2:Clearly 'n 2BAP .1;2/ and 'n! ' Dminf1;gg in C.R;R/. Next we know from [8]that limsupt!1g.t/ D1 and liminft!1g.t/ D 0. So applying [8, Lemma 2.1],we know that ' …AP.R;R/. Hence the subset BAP .1;2/ is not complete in C.R;R/.By taking the sequencen'nL2Mto1nD1, we see that BAP .M;L/ is not complete inC.R;R/ for any M > 0 and L > 0. Since BAP .M;L/  BC .M;L/, BAP .M;L/ isjust precompact in C.R;R/. We have also shown thatBAP .M;L/ are not precompactin BC.R;R/, but they are complete.Furthermore, for any k 2N, we have 12kR 2k0 '.t/dt  12kR 2k0 sin2 t2D 12, so' … PAP0.R;R/. But we do not know ' … PAP.R;R/.Finally, taking n.t/D8ˆˆ<ˆ:ˆ0 for jt j  nC1;1 for jt j  n;tCnC1 for t j 2 Œ n 1; n;nC1  t for t  Œn;nC1;we see that  n 2 BPAP0.1;1/ and  n ! 1 in C.R;R/, so BPAP0.M;L/ are notcomplete in C.R;R/. We have also shown that BPAP0.M;L/ are not precompact inBC.R;R/, but they are complete.Summarizing, we arrive at the following result.Theorem 1. The subsets BS .M;L/, S 2 fPAP;AP;PAP0;C g are not precom-pact in BC.R;R/. They are precompact in C.R;R/. Furthermore, BAP .M;L/and BPAP0.M;L/ are just precompact in C.R;R/, while BC .M;L/ is compact inC.R;R/.So (non-)compactness of BPAP .M;L/ in C.R;R/ is still open.Finally, we recall Tikhonov fixed point theorem.Theorem 2. (Tikhonov) Let˝ be a non-empty compact convex subset of a locallyconvex topological vector space X . Then any continuous function A W˝!˝ has afixed point.PSEUDO ALMOST PERIODIC SOLUTIONS 5193. EXISTENCE RESULTIn this section, the existence of pseudo almost periodic solutions of equation (1.1)will be studied. Throughout this paper, it will be assumed that 1 W R! . 1;0/ isan almost periodic function, i W R! R are pseudo almost periodic functions, andC1 D supt2R1.t/ < 0;Ci D supt2Rji .t/j;where i D 2; : : : ;n.Now we apply Theorem 2. Let X be either PAP.R;R/ or BC.R;R/ but fixed. If' 2X , from Corollary 5.4 in [19], it follows 'Œ2 2X . Consider an auxiliary equationx0.t/D 1.t/x.t/C2.t/'Œ2.t/C : : :Cn.t/'Œn.t/Cf .t/; (3.1)where f 2 PAP.R;R/ and 1.t/ < 0. It is easy to see that the linear equationx0.t/D 1.t/x.t/admits an exponential dichotomy on R, by Theorem 2.3 in [18], we know that (3.1)has exactly one solutionx'.t/DZ t 1eR ts 1.u/du nXiD2i .s/'Œi.s/Cf .s/dsin X .Now take f 2 BPAP .M 0;L0/ for constants M 0 and L0. Let A W BC .M;L/!BC.R;R/ be defined by.A'/.t/DZ t 1eR ts 1.u/du nXiD2i .s/'Œi.s/Cf .s/ds; t 2 R: (3.2)Note A W BPAP .M;L/! PAP.R;R/.Lemma 1. For any '; 2 BC .M;L/; t1; t2 2 R, the following inequality hold,jj'Œn  Œnjj n 1XjD0Lj k'  k; nD 1;2; : : : : (3.3)Proof. It can be obtained by direct calculation by the definition of BC .M;L/. Lemma 2. Operator A W BC .M;L/! C.R;R/ is continuous.Proof. Let 'j ! '0 as j !1 for 'j 2 BC .M;L/, j 2N0 DN[f0g uniformlyon any compact interval Œ m;m, m 2N of R. Sethj .s/DnXiD2i .s/'Œij .s/Cf .s/; j 2N0:520 H. Y. ZHAO AND M. FECˇKANThen hj ! h0 uniformly on Œ m;m. Next we haveˇˇˇeR ms 1.u/duhj .s/ˇˇˇMnXiD2Ci Ckf ke C1 .sCm/; s 2 Œ 1; m:Since Z  m 1e C1 .sCm/ds D  1C1<1;we can apply the Lebesgue dominated convergence theorem to obtain .A'j /. m/!.A'0/. m/. From (3.1),x0j .t/ x00.t/D 1.t/.xj .t/ x0.t//Chj .t/ h0.t/ (3.4)for xj .t/D .A'j /.t/, k 2N0. Integrating the both sides of (3.4) from  m to t , wehavexj .t/ x0.t/D xj . m/ x0. m/CZ t m1.s/.xj .s/ x0.s//dsCZ t m.hj .s/ h0.s//dsandjxj .t/ x0.t/j  jxj . m/ x0. m/jC2mkhj  h0km C1Z t mjxj .s/ x0.s/jdsfor any t 2 Œ m;m. Then Gronwall’s inequality implieskxj  x0km  jxj . m/ x0. m/jC2mkhj  h0kme 2C1 m;which meanskA'j  A'0km  jA'j . m/ A'0. m/jC2mkhj  h0kme 2C1 m:Hence A'j .t/! A'0.t/ uniformly on t 2 Œ m;m. Since m 2N is arbitrarily, wegetA'j !A'0 in C.R;R/, i.e.,A WBC .M;L/!C.R;R/ is continuous. This provesthe continuity of A. Of course, then A W BPAP .M;L/! C.R;R/ is continuous as well.Theorem 3. Suppose C1  nXiD2Ci > 0: (3.5)Then for any f 2 BPAP .M 0;L0/, Eq. (1.1) has a solution' 2 BPAP .M;L/ C.R;R/PSEUDO ALMOST PERIODIC SOLUTIONS 521for the constants M and L given byM D  M0C1 CPniD2Ci; LD 2M0C1C1 CPniD2Ci: (3.6)Proof. For any '; 2 BC .M;L/; t; t1; t2 2 R, from (3.6) we haveˇˇˇ.A'/.t/ˇˇˇnXiD2CiˇˇˇˇˇZ t 1eR ts 1.u/du'Œi.s/dsˇˇˇˇˇCˇˇˇˇˇZ t 1eR ts 1.u/duf .s/dsˇˇˇˇˇ  MC1nXiD2Ci  M 0C1DM:Without loss of generality, assume t2  t1, thenˇˇˇ.A'/.t2/  .A /.t1/ˇˇˇnXiD2CiˇˇˇˇˇZ t2 1eR t2s 1.u/du'Œi.s/ds Z t1 1eR t1s 1.u/du'Œi.s/dsˇˇˇˇˇCˇˇˇˇˇZ t2 1eR t2s 1.u/duf .s/ds Z t1 1eR t1s 1.u/duf .s/dsˇˇˇˇˇnXiD2Ci"ˇˇˇZ t1 1eR t1s 1.u/dueR t2t11.u/du 1'Œi.s/dsˇˇˇCˇˇˇ Z t2t1eR t2s 1.u/du'Œi.s/dsˇˇˇ#C"ˇˇˇZ t1 1eR t1s 1.u/dueR t2t11.u/du 1f .s/dsˇˇˇCˇˇˇ Z t2t1eR t2s 1.u/duf .s/dsˇˇˇ#MnXiD2Ci"jC1 jjt2  t1jˇˇˇ Z t1 1eC1 .t1 s/dsˇˇˇCˇˇˇ Z t2t1eC1 .t2 s/dsˇˇˇ#CM 0"jC1 jjt2  t1jˇˇˇ Z t1 1eC1 .t1 s/dsˇˇˇCˇˇˇ Z t2t1eC1 .t2 s/dsˇˇˇ# 2.MnXiD2Ci CM 0/jt2  t1j D Ljt2  t1j:This shows that A' 2 BC .M;L/. Hence A W BC .M;L/! BC .M;L/ and also A WBPAP .M;L/!BPAP .M;L/. By Lemma 2 we getA WBPAP .M;L/!BPAP .M;L/.So all conditions of Tikhonov fixed theorem are satisfied forAwith˝DBPAP .M;L/and X D C.R;R/. Thus there exists a fixed point ' in BPAP .M;L/ of A. This is522 H. Y. ZHAO AND M. FECˇKANequivalent to say that ' is a solution of (1.1) in BPAP .M;L/. This completes theproof. The solution ' of Theorem 3 is a limit in C.R;R/ of a sequence of pseudo al-most periodic functions in BPAP .M;L/, i.e., ' is a uniform approximation on eachcompact intervals of R of a sequence of pseudo almost periodic functions. If f 2BAP .M0;L0/ then we know by Theorem 1 only that in general, the solution ' ofTheorem 3 belongs just to f 2 BAP .M;L/.4. UNIQUENESS RESULTIn this section, uniqueness of (1.1) will be proved, moreover, an example will beshowed.Theorem 4. Suppose (3.5). Then for any f 2 BPAP .M 0;L0/, Eq. (1.1) has theunique solution ' 2 BPAP .M;L/ provided that it holds  D  1C1nXiD2i 1XjD0Ci Lj < 1 (4.1)for the constants M and L given by (3.6).Proof. We already known that A W BPAP .M;L/! BPAP .M;L/ where we con-sider BPAP .M;L/ in BC.R;R/. For '; 2 BPAP .M;L/, by (3.2) and (3.3)j'.t/  .t/j Dˇˇˇ.A'/.t/  .A /.t/ˇˇˇnXiD2CiZ t 1eR ts 1.u/duˇˇˇ'Œi.s/  Œi.s/ˇˇˇds   1C1nXiD2i 1XjD0Ci Lj k'  k D   k'  k;thusk'  k    k'  k:From (4.1), we know   <1 andBPAP .M;L/ is closed in the Banach spaceBC.R;R/.The Banach fixed point theorem gives a unique fixed point of A in BPAP .M;L/. Theproof is finished. Example 1. Now, we apply Theorem 4. Consider the following equation:x0.t/ D  .jsin tC sinp3t jC6/x.t/C 15jcos tC cosp5t jC8x.x.t//C17sin tC sinp2tC sech t; (4.2)where 1.t/D .jsin tCsinp3t jC6/, 2.t/D 15jcos tCcosp5t jC8and f .t/D17sin tC sinp2tC sech t. Take M 0 D 37and L0 D 47. A simple calculation yieldsPSEUDO ALMOST PERIODIC SOLUTIONS 523C1 D  6 and C2 D 2, so (3.5) holds:  C1  C2 D 4 > 0. Then by (3.6) we deriveM D 328and LD 97. Thus (4.1) gives   D 1621< 1. Hence by Theorem 4, Eq. (4.2)has a unique solution in BPAP 328; 97.REFERENCES[1] E. Ait Dads and K. Ezzinbi, “Pseudo almost periodic solutions of some delay differentialequations.” J. Math. Anal. Appl., vol. 201, no. 3, pp. 840–850, art. no. 0287, 1996, doi:10.1006/jmaa.1996.0287.[2] J. Andres and A. M. Bersani, “Almost-periodicity problem as a fixed-point problem for evolutioninclusions.” Topol. Methods Nonlinear Anal., vol. 18, no. 2, pp. 337–349, 2001.[3] F. Che´rif, “Existence and global exponential stability of pseudo almost periodic solution forSICNNs with mixed delays.” J. Appl. Math. Comput., vol. 39, no. 1-2, pp. 235–251, 2012, doi:10.1007/s12190-011-0520-1.[4] K. Cooke, “Functional-differential equations: Some models and perturbation problems.” Differ.Equations dynam. Systems, Proc. Int. Sympos. Puerto Rico 1965, 167-183 (1967)., 1967.[5] L. Duan and L. Huang, “Pseudo almost periodic dynamics of delay Nicholson’s blowflies modelwith a linear harvesting term.” Math. Methods Appl. Sci., vol. 38, no. 6, pp. 1178–1189, 2015, doi:10.1002/mma.3138.[6] E. Eder, “The functional differential equation x0.t/D x.x.t//.” J. Differ. Equations, vol. 54, pp.390–400, 1984, doi: 10.1016/0022-0396(84)90150-5.[7] M. Fecˇkan, “On a certain type of functional differential equations.” Math. Slovaca, vol. 43, no. 1,pp. 39–43, 1993.[8] A. Haraux and P. Souplet, “An example of uniformly recurrent function which is not almost peri-odic.” J. Fourier Anal. Appl., vol. 10, no. 2, pp. 217–220, 2004, doi: 10.1007/s00041-004-8012-4.[9] B. Liu and C. Tunc¸, “Pseudo almost periodic solutions for a class of first order differential iterativeequations.” Appl. Math. Lett., vol. 40, pp. 29–34, 2015, doi: 10.1016/j.aml.2014.08.019.[10] S. Minsker, “On splitting the area under curves.” J. Differ. Equations, vol. 26, pp. 443–457, 1977,doi: 10.1016/0022-0396(77)90090-0.[11] S. Minsker, “On splitting the area under curves. II.” J. Differ. Equations, vol. 45, pp. 182–190,1982, doi: 10.1016/0022-0396(82)90064-X.[12] J. G. Si and S. S. Cheng, “Smooth solutions of a nonhomogeneous iterative functional-differentialequation.” Proc. R. Soc. Edinb., Sect. A, Math., vol. 128, no. 4, pp. 821–831, 1998, doi:10.1017/S0308210500021806.[13] S. Staneˇk, “On global properties of solutions of functional differential equation x0.t/D x.x.t//Cx.t/.” Dyn. Syst. Appl., vol. 4, no. 2, pp. 263–278, 1995.[14] B. H. Stephan, “On the existence of periodic solutions of ´0.t/D a´.t rCuk.t;´.t///CF.t/.”J. Differ. Equations, vol. 6, pp. 408–419, 1969, doi: 10.1016/0022-0396(69)90002-3.[15] K. Wang, “On the equation x0.t/ D f .x.x.t///.” Funkc. Ekvacioj, Ser. Int., vol. 33, no. 3, pp.405–425, 1990.[16] W. Wang, “Positive pseudo almost periodic solutions for a class of differential iterativeequations with biological background.” Appl. Math. Lett., vol. 46, pp. 106–110, 2015, doi:10.1016/j.aml.2015.02.015.[17] X. P. Wang and J. G. Si, “Analytic solutions of an iterative functional differential equation.” J.Math. Anal. Appl., vol. 262, no. 2, pp. 490–498, 2001, doi: 10.1006/jmaa.2001.7527.[18] C. Zhang, “Pseudo almost periodic solutions of some differential equations. II.” J. Math. Anal.Appl., vol. 192, no. 2, pp. 543–561, 1995, doi: 10.1006/jmaa.1995.1189.[19] C. Zhang, Almost periodic type functions and ergodicity. Dordrecht: Kluwer Academic Publish-ers; Beijing: Science Press, 2003.524 H. Y. ZHAO AND M. FECˇKAN[20] H. Zhang, “New results on the positive pseudo almost periodic solutions for a generalized modelof hematopoiesis.” Electron. J. Qual. Theory Differ. Equ., vol. 2014, p. 10, 2014.Authors’ addressesHou Yu ZhaoSchool of Mathematics, Chongqing Normal University, Chongqing 401331, P.R.ChinaDepartment of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, CanadaE-mail address: houyu19@gmail.comMichal FecˇkanDepartment of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratis-lava, Mlynska´ dolina, 842 48 Bratislava, SlovakiaMathematical Institute of Slovak Academy of Sciences, Sˇtefa´nikova 49, 814 73 Bratislava, Slov-akiaE-mail address: Michal.Feckan@fmph.uniba.sk

Pseudo almost periodic solutions of an iterative equation with variable coefficients

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