Linear difference operator with multiple variable parameters and applications to second-order differential equations. (English) Zbl 1495.34099
Summary: In this article, we first investigate the linear difference operator \((Ax)(t):=x(t)-\sum_{i=1}^nc_i(t)x(t- \delta_i(t))\) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator \(A\) are explicitly pointed out. Afterwards, as applications of properties of the operator \(A\), we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 47B39 | Linear difference operators |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
difference operator; multiple variable parameters; periodic solution; second-order differential equationReferences:
| [1] | Burton, T., A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11, 85-88 (1998) · Zbl 1127.47318 · doi:10.1016/S0893-9659(97)00138-9 |
| [2] | Cheng, Z.; Bi, Z., Study on a kind of p-Laplacian neutral differential equation with multiple variable coefficients, J. Appl. Anal. Comput., 9, 501-525 (2019) · Zbl 1458.34075 |
| [3] | Cheng, Z.; Li, F., Positive periodic solutions for a kind of second-order neutral differential equations with variable coefficient and delay, Mediterr. J. Math., 15, 1-19 (2018) · Zbl 1395.34075 · doi:10.1007/s00009-018-1184-y |
| [4] | Cheng, Z.; Li, F., Weak and strong singularities fro second-order nonlinear differential equations with a linear difference operator, J. Fixed Point Theory Appl., 21, 1-23 (2019) · Zbl 1421.34046 · doi:10.1007/s11784-019-0687-x |
| [5] | Cheung, W.; Ren, J.; Han, W., Positive periodic solution of second-order neutral functional differential equations, Nonlinear Anal., 71, 3948-3955 (2009) · Zbl 1185.34093 · doi:10.1016/j.na.2009.02.064 |
| [6] | Danca, M.; Feckan, M.; Pospisil, M., Difference equations with impulses, Opusc. Math., 39, 5-22 (2019) · Zbl 1403.39012 · doi:10.7494/OpMath.2019.39.1.5 |
| [7] | Du, B., Anti-periodic solutions problems for inertial competitive neutral-type neutral networks via Wirtinger inequality, J. Inequal. Appl., 2019 (2019) · Zbl 1499.34357 · doi:10.1186/s13660-019-2136-1 |
| [8] | Du, B.; Guo, L.; Ge, W.; Lu, S., Periodic solutions for generalized Liénard neutral equation with variable parameter, Nonlinear Anal. TMA, 70, 2387-2394 (2009) · Zbl 1166.34325 · doi:10.1016/j.na.2008.03.021 |
| [9] | Ge, W.; Ren, J., An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p-Laplacian, Nonlinear Anal. TMA, 58, 477-488 (2004) · Zbl 1074.34014 · doi:10.1016/j.na.2004.01.007 |
| [10] | Hale, J., Theory of Functional Differential Equations (1977), Berlin: Springer, Berlin · Zbl 0352.34001 |
| [11] | Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), San Diego: Academic Press, San Diego · Zbl 0777.34002 |
| [12] | Lu, S.; Ge, W., Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. Math. Comput., 146, 195-209 (2003) · Zbl 1037.34065 |
| [13] | Lu, S.; Ge, W., Existence of periodic solutions for a kind of second order neutral functional differential equation, Appl. Math. Comput., 157, 433-448 (2004) · Zbl 1059.34043 |
| [14] | Luo, Y.; Wei, W.; Shen, J., Existence of positive periodic solutions for two kinds of neutral functional differential equations, Appl. Math. Lett., 21, 581-587 (2008) · Zbl 1149.34040 · doi:10.1016/j.aml.2007.07.009 |
| [15] | Lv, L.; Cheng, Z., Positive periodic solution to superlinear neutral differential equation with time-dependent parameter, Appl. Math. Lett., 98, 271-277 (2019) · Zbl 1426.34085 · doi:10.1016/j.aml.2019.06.024 |
| [16] | Manasevich, R.; Mawhin, J., Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differ. Equ., 145, 367-393 (1998) · Zbl 0910.34051 · doi:10.1006/jdeq.1998.3425 |
| [17] | Pinelas, S.; Dix, J., Oscillation of solutions to non-linear difference equations with several advanced arguments, Opusc. Math., 37, 889-898 (2017) · Zbl 1400.39012 |
| [18] | Radulescu, V.; Repovs, D., Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis (2015), Boca Raton: CRC Press, Boca Raton · Zbl 1343.35003 |
| [19] | Ren, J.; Cheng, Z.; Siegmund, S., Neutral operator and neutral differential equation, Abstr. Appl. Anal., 2011 (2011) · Zbl 1229.34120 · doi:10.1155/2011/969276 |
| [20] | Ren, J.; Zhu, D.; Wang, H., Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst., Ser. B, 24, 1843-1865 (2019) · Zbl 1411.35139 |
| [21] | Stevic, S., Solvability of a product-type system of difference equations with six parameters, Adv. Nonlinear Anal., 8, 29-51 (2019) · Zbl 1426.39008 · doi:10.1515/anona-2016-0145 |
| [22] | Wu, J.; Wang, Z., Two periodic solutions of second-order neutral functional differential equations, J. Math. Anal. Appl., 329, 677-689 (2007) · Zbl 1118.34063 · doi:10.1016/j.jmaa.2006.06.084 |
| [23] | Xu, Y.; Zhu, D.; Ren, J., On a reaction-diffusion-advection system: fixed boundary or free boundary, Electron. J. Qual. Theory Differ. Equ., 2018 (2018) · Zbl 1445.34096 · doi:10.1186/s13662-018-1474-5 |
| [24] | Yao, S.; Cheng, Z., The homotopy perturbation method for a nonlinear oscillator with a damping, J. Low Freq. Noise Vib. Act. Control, 38, 1110-1112 (2019) · doi:10.1177/1461348419836344 |
| [25] | Yao, S.; Ma, Z.; Cheng, Z., Pattern formation of a diffusive predator-prey model with strong Allee effect and nonconstant death rate, Physica A, 527, 1-11 (2019) · Zbl 07568343 · doi:10.1016/j.physa.2019.121350 |
| [26] | Zhang, M., Periodic solution of linear and quasilinear neutral functional differential equation, J. Math. Anal. Appl., 189, 378-392 (1995) · Zbl 0821.34070 · doi:10.1006/jmaa.1995.1025 |
| [27] | Zhou, T.; Du, B.; Du, H., Positive periodic solution for indefinite singular Liénard equation with p-Laplacian, Adv. Differ. Equ., 2019 (2019) · Zbl 1459.34081 · doi:10.1186/s13662-019-2100-x |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
