Positive solution for infinitely impulsive singular third-order $\phi$-Laplacian BVPs on the half line with first-order derivative dependence
Abdelhamid Benmezaï, Dhehbiya Belal, Kamal Bachouche
Received July 11, 2024. Published online January 17, 2025.
Abstract: We are concerned in this paper with the existence of positive solutions to the $\phi$-Laplacian third-order boundary value problem \begin{cases} -(\phi(u"))'(t)=f(t,u(t),u'(t))\text{ for a.e. }t\in J, u(0)=0, u'(0)=a\text{, }\lim\limits_{t\rightarrow\infty}u"(t)=0, \Delta u(t_k)=I_{1,k}(u(t_k),u'(t_k)), k=1,2,\ldots,\Delta u'(t_k)=I_{2,k}(u(t_k),u'(t_k)), k=1,2,\ldots, -\Delta\phi(u")(t_k)=I_{3,k}(u(t_k),u'(t_k)), k=1,2,\ldots,\end{cases} where $a\geq0,$ $J=(0,\infty)$, $0<t_1<t_2<\ldots<t_k\ldots$, $t_k\rightarrow\infty$ as $k\rightarrow\infty$, $\Delta u(t_k)=u(t_k^+)-u(t_k^-)$ and $J^{\ast}=J\backslash\{t_k k\geq1\}$. The function $\phi\colon\mathbb{R}\rightarrow\mathbb{R}$ is an increasing homeomorphism such that $\phi(0)=0$, $I_{i,k}\in C(I^2,[0,\infty))$ for $i=1,2,3$ and $k\geq1$, and the nonlinearity $f\colon J^3\rightarrow\mathbb{R}^+$ is a Caratheodory function. By means of a Guo-Krasnoselskii type fixed point theorem, we prove an existence result for at least one positive solution.
Keywords: BVP on infinite intervals; $\phi$-Laplacian; fixed point theory in cones
References: [1] R. P. Agarwal: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986). DOI 10.1142/0266 | MR 1021979 | Zbl 0619.34019
[2] R. P. Agarwal, D. O'Regan: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001). DOI 10.1007/978-94-010-0718-4 | MR 1845855 | Zbl 0988.34002
[3] Z. Benbaziz, S. Djebali: Positive solutions for nonlinear $\phi$-Laplacian third-order impulsive differential equations on infinite intervals. J. Nonlinear Funct. Anal. 2018 (2018), Article ID 32, 19 pages. DOI 10.23952/jnfa.2018.32
[4] M. Benchohra, J. Henderson, S. K. Ntouyas: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Application 2. Hindawi, New York (2006). DOI 10.1155/9789775945501 | MR 2322133 | Zbl 1130.34003
[5] F. Bernis, L. A. Peletier: Two problems from draining flows involving third-order ordinary differential equations. SIAM J. Math. Anal. 27 (1996), 515-527. DOI 10.1137/S0036141093260847 | MR 1377486 | Zbl 0845.34033
[6] C. Corduneanu: Integral Equations and Stability of Feedback Systems. Mathematics in Science and Engineering 104. Academic Press, New York (1973). DOI 10.1016/s0076-5392(08)x6099-0 | MR 0358245 | Zbl 0273.45001
[7] S. Djebali, O. Saifi: Third order BVPs with $\phi$-Laplacian operators on $[0,+\infty)$. Afr. Diaspora J. Math. 16 (2013), 1-17. MR 3091711 | Zbl 1283.34019
[8] S. Djebali, O. Saifi: Upper and lower solution for $\phi$-Laplacian third-order BVPs on the half-line. Cubo 16 (2014), 105-116. DOI 10.4067/S0719-06462014000100010 | MR 3185792 | Zbl 1319.34038
[9] S. Djebali, O. Saifi: Singular $\phi$-Laplacian third-order BVPs with derivative dependence. Arch. Math., Brno 52 (2016), 35-48. DOI 10.5817/AM2016-1-35 | MR 3475111 | Zbl 1374.34067
[10] M. Greguš: Third Order Linear Differential Equations. Mathematics and Its Applications 22. Reidel, Dordrecht (1987). DOI 10.1007/978-94-009-3715-4 | MR 0882545 | Zbl 0602.34005
[11] D. Guo: Existence of two positive solutions for a class of third-order impulsive singular integro-differential equations on the half-line in Banach spaces. Bound. Value Probl. 2016 (2016), Article ID 70, 31 pages. DOI 10.1186/s13661-016-0577-8 | MR 3479375 | Zbl 1383.45003
[12] J. Henderson, R. Luca: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations: Positive Solutions. Elsevier, Amsterdam (2016). DOI 10.1016/C2014-0-04797-1 | MR 3430818 | Zbl 1353.34002
[13] S. Liang, J. Zhang: Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. Acta Appl. Math. 111 (2010), 27-43. DOI 10.1007/s10440-009-9528-z | MR 2653048 | Zbl 1203.34038
[14] Y. Liu: Boundary value problem for second order differential equations on unbounded domains. Acta Anal. Funct. Appl. 4 (2002), 211-216. (In Chinese.) MR 1956716 | Zbl 1038.34030
[15] V. D. Mil'man, A. D. Myškis: On the stability of motion in the presence of impulses. Sib. Math. Zh. 1 (1960), 233-237. (In Russian.) MR 0126028 | Zbl 1358.34022
[16] F. M. Minhós, H. Carrasco: Higher Order Boundary Value Problems on Unbounded Domains: Types of Solutions, Functional Problems and Applications. Trends in Abstract and Applied Analysis 5. World Scientific, Hackensack (2018). DOI 10.1142/10448 | MR 3752165 | Zbl 1380.34002
[17] S. Padhi, S. Pati: Theory of Third-Order Differential Equations. Springer, New Delhi (2014). DOI 10.1007/978-81-322-1614-8 | MR 3136420 | Zbl 1308.34002
[18] J. Wang, Z. Zhang: A boundary value problem from draining and coating flows involving a third-order differential equation. Z. Angew. Math. Phys. 49 (1998), 506-513. DOI 10.1007/s000000050104 | MR 1629561 | Zbl 0907.34016
[19] X. Yang, Y. Liu: Existence of unbounded solutions of boundary value problems for singular differential systems on whole line. Bound. Value Probl. 2015 (2015), Article ID 42, 39 pages. DOI 10.1186/s13661-015-0300-1 | MR 3315714 | Zbl 1317.34034
[20] F. Yoruk Deren, N. Aykut Hamal: Second-order boundary-value problems with integral boundary conditions on the real line. Electron. J. Differ. Equ. 2014 (2014), Article ID 19, 13 pages. MR 3159428 | Zbl 1292.34017
[21] X. Zhang: Computation of positive solutions for nonlinear impulsive integral boundary value problems with $p$-Laplacian on infinite intervals. Abstr. Appl. Anal. 2013 (2013), Article ID 708281, 13 pages. DOI 10.1155/2013/708281 | MR 3035186 | Zbl 1266.65135
Affiliations: Abdelhamid Benmezaï (corresponding author), National High School of Mathematics, P.O.Box 75, Mahelma 16093, Algiers, Algeria, e-mail: aehbenmezai@gmail.com; Dhehbiya Belal, Faculty of Sciences, Adrar University, Rue nationale 6, Adrar, Algeria, e-mail: belal.dhehbiya@gmail.com; Kamal Bachouche, Faculty of Sciences, Algiers University 1, Algiers, Algeria, e-mail: kbachouche@gmail.com
