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Positive solutions to singular and delay higher-order differential equations on time scales. (English) Zbl 1181.34102

Summary: We are concerned with the singular three-point boundary value problem
\[ \begin{aligned} & (-1)^nu^{\Delta^{2n}}(t)=w(t)f(t,u(t-c)),\quad t\in[a,b],\\ & u(t)=\psi(t),\quad t\in[a-c,a),\\ & u^{\Delta^{2i}}(a)-\beta_{i+1}u^{\Delta^{2i+1}}(a)=a_{i+1}u^{\Delta^{2i}}(\varpi),\\ & \gamma_{i+1}u^{\Delta^{2i}}(\varpi)=u^{\Delta^{2i}}(b),\quad 0\leq i\leq n-1,\end{aligned}\tag{1.1} \]
where \(c\in[0,(b-a)/2]\), \(\varpi\in (a,b)\), \(\beta_i\geq 0\), \(1<\gamma_i<(b-a+\beta_i)/(\varpi-a+\beta_i)\), \(0\leq \alpha_i<(b-\gamma_i\varpi+(\gamma_i-1)(a-\beta_i))/(b-\varpi)\), \(i=1,2,\dots,n\) and \(\psi\in C([a-c,a])\). The functional \(w:(a,b)\to[0,+\infty)\) is continuous and \(f:[a,b]\times (0,+\infty)\to [0,+\infty)\) is continuous. Our nonlinearity \(w\) may have singularity at \(t=a\) and/or \(t=b\), and \(f\) may have singularity at \(u=0\).
Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K10 Boundary value problems for functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

[1] Atici FM, Guseinov GSh: On Green’s functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002, 141(1-2):75-99. 10.1016/S0377-0427(01)00437-X · Zbl 1007.34025 · doi:10.1016/S0377-0427(01)00437-X
[2] Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358. · Zbl 0978.39001 · doi:10.1007/978-1-4612-0201-1
[3] Agarwal, RP; Bohner, M.; Rehák, P., Half-linear dynamic equations, 1-57 (2003), Dordrecht, The Netherlands · Zbl 1056.34049 · doi:10.1007/978-94-010-0035-2
[4] Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(2):368-381. 10.1016/j.na.2006.05.014 · Zbl 1129.34015 · doi:10.1016/j.na.2006.05.014
[5] Agarwal, RP; Otero-Espinar, V.; Perera, K.; Vivero, DR, Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations, No. 2008, 13 (2008) · Zbl 1153.39006
[6] Ahmad, B.; Nieto, JJ, The monotone iterative technique for three-point second-order integrodifferential boundary value problems with [InlineEquation not available: see fulltext.]-Laplacian, No. 2007, 9 (2007)
[7] Anderson DR: Solutions to second-order three-point problems on time scales. Journal of Difference Equations and Applications 2002, 8(8):673-688. 10.1080/1023619021000000717 · Zbl 1021.34011 · doi:10.1080/1023619021000000717
[8] Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Computers & Mathematics with Applications 2008, 56(9):2429-2443. 10.1016/j.camwa.2008.05.018 · Zbl 1165.39300 · doi:10.1016/j.camwa.2008.05.018
[9] Anderson DR, Smyrlis G: Solvability for a third-order three-point BVP on time scales. Mathematical and Computer Modelling 2009, 49(9-10):1994-2001. 10.1016/j.mcm.2008.11.009 · Zbl 1171.34306 · doi:10.1016/j.mcm.2008.11.009
[10] Aulbach B, Neidhart L: Integration on measure chains. In Proceedings of the 6th International Conference on Difference Equations, 2004, Boca Raton, Fla, USA. CRC Press; 239-252. · Zbl 1083.26005
[11] Boey KL, Wong PJY: Positive solutions of two-point right focal boundary value problems on time scales. Computers & Mathematics with Applications 2006, 52(3-4):555-576. 10.1016/j.camwa.2006.08.025 · Zbl 1140.34328 · doi:10.1016/j.camwa.2006.08.025
[12] Cabada A, Cid JÁ: Existence of a solution for a singular differential equation with nonlinear functional boundary conditions. Glasgow Mathematical Journal 2007, 49(2):213-224. 10.1017/S0017089507003679 · Zbl 1128.34009 · doi:10.1017/S0017089507003679
[13] DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004, 295(2):378-391. 10.1016/j.jmaa.2004.02.049 · Zbl 1069.34012 · doi:10.1016/j.jmaa.2004.02.049
[14] Gatica JA, Oliker V, Waltman P: Singular nonlinear boundary value problems for second-order ordinary differential equations. Journal of Differential Equations 1989, 79(1):62-78. 10.1016/0022-0396(89)90113-7 · Zbl 0685.34017 · doi:10.1016/0022-0396(89)90113-7
[15] Henderson J, Tisdell CC, Yin WKC: Uniqueness implies existence for three-point boundary value problems for dynamic equations. Applied Mathematics Letters 2004, 17(12):1391-1395. 10.1016/j.am1.2003.08.015 · Zbl 1062.34012 · doi:10.1016/j.am1.2003.08.015
[16] Kaufmann ER, Raffoul YN: Positive solutions for a nonlinear functional dynamic equation on a time scale. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(7):1267-1276. 10.1016/j.na.2005.04.031 · Zbl 1090.34054 · doi:10.1016/j.na.2005.04.031
[17] Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales. Journal of Difference Equations and Applications 2008, 14(7):723-736. 10.1080/10236190701840906 · Zbl 1148.34007 · doi:10.1080/10236190701840906
[18] Liang J, Xiao T-J, Hao Z-C: Positive solutions of singular differential equations on measure chains. Computers & Mathematics with Applications 2005, 49(5-6):651-663. 10.1016/j.camwa.2004.12.001 · Zbl 1085.34019 · doi:10.1016/j.camwa.2004.12.001
[19] Yaslan İ: Multiple positive solutions for nonlinear three-point boundary value problems on time scales. Computers & Mathematics with Applications 2008, 55(8):1861-1869. 10.1016/j.camwa.2007.07.005 · Zbl 1159.34317 · doi:10.1016/j.camwa.2007.07.005
[20] Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045
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