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Keywords:
third-order differential equation; multi-point and integral boundary conditions; Guo-Krasnosel'skii fixed point theorem in cone; positive solutions
Summary:
This paper concerns the following system of nonlinear third-order boundary value problem: \begin{equation*} u_{i}'''(t)+f_{i}(t,u_{1}(t),\dots ,u_{n}(t),u'_{1}(t),\dots ,u'_{n}(t))= 0, 0<t<1, i\in \{1,\dots ,n\} \end{equation*} with the following multi-point and integral boundary conditions: $$\begin{cases} u_{i}(0)=0 u_{i}'(0)=0 u_{i}'(1)= \sum^{p}_{j=1}\beta_{j,i}u_{i}'(\eta_{j,i}) + \int^{1}_{0}h_{i}(u_{1}(s),\dots ,u_{n}(s))\,ds \end{cases}$$ where $\beta_{j,i}>0$, $0< \eta_{1,i}<\dots <\eta_{p,i}<\frac{1}{2}$, $f_{i}:[0,1]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ and $h_{i}:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ are continuous functions for all $i\in \{1,\dots ,n\}$ and $j\in \{1,\dots ,p\}$. Using Guo-Krasnosel'skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.
References:
[1] Gregus M.: Third order linear differential equations. in Math. Appl., Reidel, Dordrecht, 1987. MR 0882545 | Zbl 0602.34005
[2] Yao Q., Feng Y.: The existence of solution for a third-order two-point boundary value problem. Appl. Math. Lett. 15 (2002), 227–232. DOI 10.1016/S0893-9659(01)00122-7 | MR 1880762 | Zbl 1008.34010
[3] Feng Y., Liu S.: Solvability of a third-order two-point boundary value problem. Appl. Math. Lett. 18 (2005), 1034–1040. DOI 10.1016/j.aml.2004.04.016 | MR 2156998 | Zbl 1094.34506
[4] Klaasen G.: Differential inequalities and existence theorems for second and third order boundary value problems. J. Differential Equations 10 (1971), 529–537. DOI 10.1016/0022-0396(71)90010-6 | MR 0288397 | Zbl 0218.34024
[5] Jackson L.K.: Existence and uniqueness of solutions of boundary value problems for third order differential equations. J. Differential Equations 13 (1973), 432–437. DOI 10.1016/0022-0396(73)90002-8 | MR 0335925 | Zbl 0256.34018
[6] O'Regan D.: Topological transversality: Application to third order boundary value problems. SIAM J. Math. Anal. 19 (1987), 630–641. DOI 10.1137/0518048 | MR 0883557
[7] Sun Y.: Positive solutions for third-order three-point nonhomogeneous boundary value problems. Appl. Math. Lett. 22 (2009) 45–51. DOI 10.1016/j.aml.2008.02.002 | MR 2483159 | Zbl 1163.34313
[8] Guo L.J., Sun J.P., Zhao Y.H.: Existence of positive solution for nonlinear third-order three-point boundary value problem. Nonlinear Anal. 68 (2008), 3151–3158. DOI 10.1016/j.na.2007.03.008 | MR 2404825
[9] Guo D., Lakshmikantham V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1988. MR 0959889 | Zbl 0661.47045
[10] Deimling K.: Nonlinear Functional Analysis. Springer, Berlin, 1985. MR 0787404 | Zbl 0559.47040
[11] Zhang X.G., Liu L.S., and Wu C.X.: Nontrivial solution of third-order nonlinear eigenvalue problems. Appl. Math. Comput. 176 (2006), 714–721. DOI 10.1016/j.amc.2005.10.017 | MR 2232063
[12] Chen H.: Positive solutions for the nonhomogeneous three-point boundary value problem of second order differential equations. Math. Comput. Modelling 45 (2007), 844–852. DOI 10.1016/j.mcm.2006.08.004 | MR 2297125 | Zbl 1137.34319
[13] Graef J.R., Yang B.: Multiple positive solution to a three-point third order boundary value problems. Discrete Contin. Dyn. Syst. 2005, suppl., pp. 1–8. MR 2192690
[14] Ma R.: Positive solutions for a second order three-point boundary value problems. Appl. Math. Lett. 14 (2001), 1–5. DOI 10.1016/S0893-9659(00)00102-6 | MR 1793693
[15] Yao Q.L.: The existence and multiplicity of positive solutions for third-order three-point boundary value problem. Acta Math. Appl. Sinica 19 (2003), 117–122. DOI 10.1007/s10255-003-0087-1 | MR 2053778
[16] Anderson D.: Multiple positive solutions for a three-point boundary value problem. Math. Comput. Modelling 27 (1998), 49–57. DOI 10.1016/S0895-7177(98)00028-4 | MR 1620897 | Zbl 0906.34014
[17] Goodrich C.S.: Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale. Comment. Math. Univ. Carolin. 54 (2013), no. 4, 509–525. MR 3125073
[18] Anderson D.: Green's function for a third-order generalized right focal problem. J. Math. Anal. Appl. 288 (2003), 1–14. DOI 10.1016/S0022-247X(03)00132-X | MR 2019740 | Zbl 1045.34008
[19] Boucherif A., Al-Malki N.: Nonlinear three-point third order boundary value problems. Appl. Math. Comput. 190 (2007), 1168–1177. DOI 10.1016/j.amc.2007.02.039 | MR 2339710 | Zbl 1134.34007
[20] Kong L., Kong Q.: Multi-point boundary value problems of second-order differential equations \rm (I). Nonlinear. Anal. 58 (2004), 909–931. DOI 10.1016/j.na.2004.03.033 | MR 2086064 | Zbl 1066.34012
[21] Kong L., Kong Q.: Multi-point boundary value problems of second-order differential equations \rm (II). Comm. Appl. Nonlinear. Anal. 14 (2007), 93–111. MR 2294496 | Zbl 1140.34008
[22] Henderson J., Ntouyas S.K., Purnaras I.K.: Positive solutions for systems of generalized three-point nonlinear boundary value problems. Comment. Math. Univ. Carolin. 49 (2008), no. 1, 79–91. MR 2433626 | Zbl 1212.34058
[23] Jebari R., Boukricha A.: Solvability and positive solution of a system of second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 2014, DOI: 10.1186/s13661-014-0262-8. DOI 10.1186/s13661-014-0262-8 | MR 3291555

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