Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. (English) Zbl 1129.34015

In this work, the existence and multiplicity of positive solutions are studied for the Dirichlet boundary value problem \[ -u^{\triangle \triangle}(t) = f(\sigma(t),u(\sigma(t))), \quad t \in (a,\rho(b)) \cap \mathbb{T}, \]
\[ u(0) = u(b) = 0, \] where \(\mathbb{T}\) is a time scale with \(a = \min \mathbb{T}\) and \(b = \max \mathbb{T}\). The inhomogeneous term \(f(\sigma(t),z)\) (\(\triangle\)-measurable in \(\mathbb{T}\)) possesses a singularity in the dependent variable at \(z=0\) and also is bounded away from zero by a constant proportional to the the size of every interval near zero, where such lower bound is assumed to be valid. It also is dominated by a \(\triangle\)-Lebesgue integrable function on an interval of \(z\) bounded away from zero. In addition, the \(\triangle\)-antiderivative, \(F(\sigma(t),z)\), of \(f\), with respect to the dependent variable, is a \(\triangle\)-measurable function of the independent variable in all of \(\mathbb{T}\) for each nonnegative value of \(z\). There is a crucial condition imposed on the time scale, namely, that there exists a \(\xi \in (0,1)\) such that
\[ \int_{[a,\rho(b)) \cap\mathbb{T}}(b-\sigma(s))^{-\xi} \, \triangle s < +\infty. \]
The constant \(\xi\) plays an additional role by involvement in control of the singular behavior of \(f\) near \(z=0\). The above assumptions allow the authors to introduce a suitable regularization of \(f\). Using variational methods and under additional (growth-type) assumptions on \(f\), several existence results are obtained for at least one solution in the resonant and non-resonant scenarios. Multiplicity results are also obtained. This work will definitely aid those interested in applications of the critical point theory to boundary value problems on time scales.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Agarwal, R.P.; O’Regan, D., Singular differential and integral equations with applications, (2003), Kluwer Academic Publishers Dordrecht · Zbl 1027.34014
[2] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0923.39002
[3] Agarwal, R.P.; Otero-Espinar, V.; Perera, K.; Vivero, D.R., Basic properties of Sobolev spaces on time scales, Adv. difference equ., 2006, (2006), 14 pages. Article ID 38121 · Zbl 1139.39022
[4] R.P. Agarwal, V. Otero-Espinar, K. Perera, D.R. Vivero, Wirtinger’s inequalities on time scales, Canad. Math. Bull. (in press) · Zbl 1148.26020
[5] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear anal., 58, 69-73, (2004) · Zbl 1070.39005
[6] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular problems by variational methods, Proc. amer. math. soc., 134, 3, 817-824, (2006) · Zbl 1094.34013
[7] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston, Inc. Boston, MA · Zbl 1025.34001
[8] Cabada, A.; Vivero, D.R., Expression of the Lebesgue \(\operatorname{\Delta}\)-integral on time scales as a usual Lebesgue integral. application to the calculus of \(\operatorname{\Delta}\)-antiderivatives, Math. comput. modelling, 43, 194-207, (2006) · Zbl 1092.39017
[9] Cabada, A.; Vivero, D.R., Criterions for absolutely continuity on time scales, J. difference equ. appl., 11, 11, 1013-1028, (2005) · Zbl 1081.39011
[10] Cerami, G., An existence criterion for the critical points on unbounded manifolds, Istit. lombardo accad. sci. lett. rend. A, 112, 2, 332-336, (1979), 1978 · Zbl 0436.58006
[11] Guseinov, G.Sh., Integration on time scales, J. math. anal. appl., 285, 1, 107-127, (2003) · Zbl 1039.26007
[12] Hewitt, E.; Stromberg, K.R., Real and abstract analysis, (1975), Springer-Verlag New York
[13] Rabinowitz, P.H., Minimax methods in critical point theory with apinimax methods in critical point theory with applications to differential equations, ()
[14] Rudin, W., Real and complex analysis, (1987), McGraw-Hill Book Co. New York, (First ed. McGraw-Hill Book Co., 1966) · Zbl 0925.00005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.