EXISTENCE OF POSITIVE SOLUTIONS FOR DAMPING ELASTIC SYSTEMS IN BANACH SPACES

Haide Gou; Baolin Li

doi:10.11948/20250221

2026 Volume 16 Issue 3

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Haide Gou, Baolin Li. EXISTENCE OF POSITIVE SOLUTIONS FOR DAMPING ELASTIC SYSTEMS IN BANACH SPACES[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1140-1164. doi: 10.11948/20250221

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Haide Gou, Baolin Li. EXISTENCE OF POSITIVE SOLUTIONS FOR DAMPING ELASTIC SYSTEMS IN BANACH SPACES[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1140-1164. doi: 10.11948/20250221

EXISTENCE OF POSITIVE SOLUTIONS FOR DAMPING ELASTIC SYSTEMS IN BANACH SPACES

Haide Gou^1, ,,
Baolin Li^2,

1.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2.
Academic Affairs Office, Lanzhou Technology and Business College, Lanzhou 730101, China

Author Bio: Email: hsy_519@126.com(B. Li)
Corresponding author: Email: 842204214@qq.com(H. Gou)
Fund Project: Supported by the National Natural Science Foundation of China (12061062), Lanzhou Youth Science and Technology Talent Innovation Project (grant No. 2023-QN-106), and Project of NWNU-LKQN2023-02, and Project of Gansu Provincial National Natural Science Foundation (grant Nos. 24JRRA131, 24JRRA780)

Abstract

Abstract

The goal of this paper is to deal with the damped elastic systems with delay and nonlocal conditions in the framework of ordered Banach spaces. By combining a fixed point theorem for convex-power condensing operators with measures of noncompactness, we establish the existence of positive mild solutions for the aforementioned system. Our analysis assumes that the nonlinear function satisfies both measure conditions and order conditions. To demonstrate the practical applicability of our theoretical findings, we present a concrete example involving the vibration equation of a simply supported beam.
- Damped elastic systems /
- nonlocal conditions /
- measure of noncompactness /
- positive mild solutions /
- convex-power condensing operator
MSC: 34G20, 34K20, 34A08, 35B35, 47H08

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References

[1]	M. S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah and F. Jarad, Existence of positive solutions for weighted fractional order differential equations, Chaos, Solitons & Fractals, 2020, 141, 110341. Google Scholar
[2]	S. Aizicovici and V. Staicu, Multivalued evolution equations with nonlocal initial conditions in Banach spaces, Nonlinear Differ. Equ. Appl., 2007, 14(3–4), 361–376. Google Scholar
[3]	J. Banaś and K. Goebel, Measure of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980. Google Scholar
[4]	J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, 2014. Google Scholar
[5]	A. Boucherif, Semilinear evolution inclusions with nonlocal conditions, Appl. Math. Lett., 2009, 22(8), 1145–1149. Google Scholar
[6]	L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 1991, 162, 494–505. Google Scholar
[7]	L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 1998, 33, 2413–2426. Google Scholar
[8]	L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 1991, 40, 11–19. Google Scholar
[9]	G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 1982, 39(1982), 433–454. Google Scholar
[10]	S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The systems: The case $\frac{1}{2}\leq\alpha\leq1$, Pacific Journal of Mathematics, 1989, 136(1), 15–55. $\frac{1}{2}\leq\alpha\leq1$" target="_blank">Google Scholar
[11]	S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: The case $0 <\alpha< \frac{1}{2}$, Proceedings of the American Mathmematical Society, 1990, 110(2), 401–415. $0 <\alpha< \frac{1}{2}$" target="_blank">Google Scholar
[12]	K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. Google Scholar
[13]	K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 1993, 179, 630–637. Google Scholar
[14]	T. Diagana, Well-posedness for some damped elastic systems in Banach spaces, Applied Mathematics Letters, 2017, 71, 74–80. Google Scholar
[15]	E. Emmrich and D. Siska, Full discretisation of second-order nonlinear evolution equations: Strong convergence and applications, Comput. Methods Appl. Math., 2011, 11(4), 441–459. Google Scholar
[16]	E. M. Emmrich and M. Thalhammer, Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math., 2010, 10, 171–190. Google Scholar
[17]	K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. Google Scholar
[18]	K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 2007, 67, 1613–1622. $\alpha$-norm for some neutral partial differential equations with nonlocal conditions" target="_blank">Google Scholar
[19]	H. Fan and F. Gao, Asymptotic stability of solutions to elastic systems with structural damping, Electron. J. Differential Equations, 2014, 245, 9. Google Scholar
[20]	H. Fan and Y. Li, Analyticity and exponential stability of semigroup for elastic systems with structural damping in Banach spaces, J. Math. Anal. Appl., 2014, 410, 316–322. Google Scholar
[21]	H. Fan and Y. Li, Monotone iterative technique for the elastic systems with structural damping in Banach spaces, Comput. Math. Appl., 2014, 68, 384–391. Google Scholar
[22]	H. Fan, Y. Li and P. Chen, Existence of mild solutions for the elastic systems with structural damping in Banach spaces, Abstract and Applied Analysis, 2013, 746893, 1–6. Google Scholar
[23]	H. Fan and X. Zhang, Asymptotic stability for stochastic elastic systems with damping, Appl. Anal., 2024, 1974–1988. DOI: 10.1080/00036811.2024.2441237. CrossRef Google Scholar
[24]	P. J. Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, 2014, 88(2), 333–365. Google Scholar
[25]	D. Guo, Nonlinear Functional Analysis, Shandong Science and Technology, Jinan (Chinese), 1985. Google Scholar
[26]	D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, Orlando, 1988. Google Scholar
[27]	H. Gou and Y. Li, A study on damped elastic systems in Banach spaces, Numer. Func. Anal. Opt., 2020, 41, 542–570. Google Scholar
[28]	H. Gou and Y. Li, Mixed monotone iterative technique for damped elastic systems in Banach spaces, J. Pseudo-Differ. Oper. Appl., 2020, 11, 917–933. Google Scholar
[29]	H. Gou snd Y. Li, Existence and approximate controllability of semilinear measure driven systems with nonlocal conditions, Bulletin of the Iranian Mathematical Society, 2022, 48, 769–789. Google Scholar
[30]	D. Guo and J. Sun, Ordinary Differential Equations in Abstract Spaces, Shandong Science and Technology, Jinan (Chinese), 1989. Google Scholar
[31]	H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-value functions, Nonlinear Anal., 1983, 7, 1351–1371. Google Scholar
[32]	E. M. Hern$\acute{a}$ndez and S. M. Tanaka, Global solutions for abstract functional differential equations with nonlocal conditions, Electr. J. Qualti., 2009, 50, 1–8. Google Scholar
[33]	F. Huang, A problem for linear elastic systems with structural damping, Acta Mathematica Scientia, 1986, 6(1), 101–107. (Chinese). Google Scholar
[34]	F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM Journal on Control and Optimization, 1988, 26(3), 714–724. Google Scholar
[35]	F. L. Huang, On the holomorphic property of the semigroup associated with linear elastic systems with structural damping, Acta Mathematica Scientia, 1985, 5(3), 271–277. Google Scholar
[36]	F. L. Huang, Y. Z. Huang and F. M. Guo, Holomorphic and differentiable properties of the $C_0$-semigroup associated with the Euler-Bernoulli beam equations with structural damping, Science in China A, 1992, 35(5), 547–560. $C_0$-semigroup associated with the Euler-Bernoulli beam equations with structural damping" target="_blank">Google Scholar
[37]	F. L. Huang and K. S. Liu, Holomorphic property and exponential stability of the semigroup associated with linear elastic systems with damping, Annals of Differential Equations, 1988, 4(4), 411–424. Google Scholar
[38]	F. L. Huang, K. S. Liu and G. Chen, Differentiability of the semigroup associated with a structural damping model, in Proceedings of the 28th IEEE Conference on Decision and Control, (Tampa, FL, 1989), IEEE, New York, 1989, 2034–2038. Google Scholar
[39]	R. Kasinathan, R. Kasinathan, D. Chalishajar and D. Kasinathan, Various controllability results for Fredholm-Volterra type stochastic elastic damped integro-differential systems with applications, International Journal of Dynamics and Control, 2025, 13, 117. Google Scholar
[40]	Y. J. Kim, Stieltjes derivatives and its applications to integral inequalities of Stieltjes type, Journal of the Korean Society of Mathematical Education Series B-Pure and Applied Mathematics, 2011, 18, 63–78. Google Scholar
[41]	F. Li and H. Wang, $S-$asymptotically $w-$periodic mild solutions of neutral fractional differential equations with finite delay in Banach space, Mediterr. J. Math., 2017, 14, 57. $S-$asymptotically $w-$periodic mild solutions of neutral fractional differential equations with finite delay in Banach space" target="_blank">Google Scholar
[42]	Q. Li and Y. Li, Positive periodic solutions for abstract evolution equations with delay, Positivity, 2021, 25, 379–397. Google Scholar
[43]	Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sin., 1996, 39(5), 666–672. (in Chinese). Google Scholar
[44]	Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 2011, 261, 1309–1324. Google Scholar
[45]	Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sin., 1996, 39, 666–672. (Chinese). Google Scholar
[46]	K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces, J. Differ. Equations, 1997, 141, 340–355. Google Scholar
[47]	L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, Journal of Mathematical Analysis and Applications, 2005, 309, 638-–649. Google Scholar
[48]	Z. Liu and Q. Zhang, A note on the polynomial stability of a weakly damped elastic abstract system, Z. Angew. Math. Phys., 2015, 66(4), 1799–1804. Google Scholar
[49]	V. T. Luong and N. T. Tung, Decay mild solutions for elastic systems with structural damping involving nonlocal conditions, Vestnik St. Petersburg University, Mathematics, 2017, 62(1), 87–103. Google Scholar
[50]	V. T. Luong and N. T. Tung, Exponential decay for elastic systems with structural damping and infinite delay, Appl. Anal., 2020, 99(1), 13–28. Google Scholar
[51]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Google Scholar
[52]	J. Sun and X. Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Mathematica Sinica, Chinese Series, 2005, 48, 439–446. Google Scholar
[53]	R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl., 1975, 52, 383–403. Google Scholar
[54]	I. I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal., 2012, 262(4), 1363–1391. Google Scholar
[55]	R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 2013, 85, 180–191. Google Scholar
[56]	M. Wei and Y. Li, Monotone iterative technique for nonlocal problems of damped elastic systems with delay, Dyn. Syst., 2022, 37(3), 444–465. Google Scholar
[57]	M. Wei, Y. Li and Q. Li, Positive mild solutions for damped elastic systems with delay and nonlocal conditions in ordered Banach spaces, Qualitative Theory of Dynamical Systems, 2022, 21, 128. Google Scholar
[58]	S. Wei, Global existence of mild solutions for the elastic system with structural damping, Ann. Appl. Math., 2019, 35, 180–188. Google Scholar
[59]	T. Xiao and J. Liang, Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal. : Theory Methods Appl., 2005, 63, e225–e232. Google Scholar