On the solvability of a~singular boundary-value problem for the equation $f(t,x,x',x'')=0$

M. K. Grammatikopulos ; P. S. Kelevedzhiev ; N. I. Popivanov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 2, Tome 16 (2006), pp. 10-21.

Résumé

In this work we consider boundary value problems of the form \begin{gather*} f(t,x,x',x'')=0,\quad 01,\\ x(0)=0,\quad x'(1)=b,\quad b>0, \end{gather*} where the the scalar function $f(t,x,p,q)$ may be singular at $x=0$. As far as we know, the solvability of the singular boundary value problems of this form has not been treated yet. Here we try to fill in this gap. Examples, illustrating our main result, are included.

Bibliographie

[1] Agarwal R. P., O'Regan D., “Boundary value problems with sign changing nonlinearities for second order singular ordinary differential equations”, Appl. Anal., 81 (2002), 1329–1346 | DOI | MR | Zbl

[2] Agarwal R. P., O'Regan D., Lakshmikantham V., Leela S., “An upper and lower solution theory for singular Emden—Fowler equations”, Nonlinear Anal. Real World Appl., 3 (2002), 275–291 | DOI | MR | Zbl

[3] Agarwal R. P., O'Regan D., Stanek S., “Singular Lidstone boundary value problem with given maximal values for solutions”, Nonlinear Anal., 55 (2003), 859–881 | DOI | MR | Zbl

[4] Agarwal R. P., O'Regan D., Wong P. J. Y., Positive solutions of differential, difference and integral equations, Kluwer Acad. Publ., Dordrecht, 1998 | MR

[5] Bobisud L. E., Lee Y. S., “Existence of monotone or positive solutions of second-order sublinear differential equations”, J. Math. Anal. Appl., 159 (1991), 449–468 | DOI | MR | Zbl

[6] Fitzpatrick P. M., “Existence results for equations involving noncompact perturbation of Fredholm mappings with applications to differential equations”, J. Math. Anal. Appl., 66 (1978), 151–177 | DOI | MR | Zbl

[7] Ge W., Mawhin J., “Positive solutions to boundary value problems for second order ordinary differential equations with singular nonlinearities”, Results Math., 34 (1998), 108–119 | MR | Zbl

[8] Grammatikopoulos M. K., Kelevedjiev P. S., Popivanov N. I., On the solvability of a Neumann boundary value problem, Techn. Report 11, University of Loania, June 2004, 2004, 93–101

[9] Granas A., Guenther R. B., Lee J. W., Nonlinear boundary value problems for ordinary differential equations, Dissnes Math., Warszawa, 1985 | MR | Zbl

[10] Guo Y., Gao Y., Zhang G., “Existence of positive solutions for singular second order boundary value problems”, Appl. Math. E-Notes, 2 (2002), 125–131 | MR | Zbl

[11] Huang Q., Li Y., “Nagumo theorems of nonlinear singular boundary value problems”, Nonlinear Anal., 29 (1997), 1365–1372 | DOI | MR | Zbl

[12] Jiang D., Pang P. Y. H., Agarwal R. P., “Nonresonant singular boundary value problems for the one-dimensional $p$-Laplacian”, Dynam. Systems Appl., 11 (2002), 449–457 | MR | Zbl

[13] Kelevedjiev P., “Existence of positive solutions to a singular second order boundary value problems”, Nonlinear Anal., 50 (2002), 1107–1118 | DOI | MR | Zbl

[14] Kelevedjiev P., Popivanov N., “Existence of solutions of boundary value problems for the equation $f(t,x,x',x'')=0$ with fully nonlinear boundary conditions”, Annuaire Univ. Sofia Fac. Math. Inform., 94 (2000), 65–77 | MR

[15] Maagli H., Masmoudi S., “Existence theorems of nonlinear singular boundary value problem”, Nonlinear Anal., 46 (2001), 465–473 | DOI | MR | Zbl

[16] Mao Y., Lee J., “Two-point boundary value problems for nonlinear differential equations”, Rocky Mountain J. Math., 26 (1996), 1499–1515 | DOI | MR | Zbl

[17] Marano S. A., “On a boundary value problem for the differential equation $f(t,x,x',x'')=0$”, J. Math. Anal. Appl., 1994, 309–319 | DOI | MR | Zbl

[18] Ntouyas S. K., Palamides P. K., “The existence of positive solutions of nonlinear singular second-order boundary value problems”, Math. Comput. Modelling, 34 (2001), 641–656 | DOI | MR | Zbl

[19] O'Regan D., Theory of singular boundary value problems, World Scientific, Singapore, 1994 | MR

[20] Palamides P. K., “Boundary-value problems for shallow elastic membrane caps”, IMA J. Appl. Math., 67 (2002), 281–299 | DOI | MR | Zbl

[21] Rach.{u}nková I., “Singular Dirichlet second-order BVPs with impulses”, J. Differential Equations, 193 (2003), 435–459 | DOI | MR | Zbl

[22] Rach.{u}nková I., Stanek S., “Sturm–Liouville and focal higher order BVPs with singularities in phase variables”, Georgian Math. J., 10 (2003), 165–191 | MR | Zbl

[23] Petryshyn W. V., “Solvability of various boundary value problems for the equation $x''=f(t,x,x',x'')-y$”, Pacific J. Math., 122 (1986), 169–195 | MR | Zbl

[24] Petryshyn W. V., Fitzpatrick P. M., “Galerkin method in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations”, Trans. Amer. Math. Soc., 238 (1978), 321–340 | DOI | MR | Zbl

[25] Petryshyn W. V., Yu Z. S., “Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest-order derivative”, J. Math. Anal. Appl., 89 (1982), 462–488 | DOI | MR | Zbl

[26] Petryshyn W. V., Yu Z. S., “Solvability of Neumann BV problems for nonlinear second order ODE's which need not be solvable for the highest order derivative”, J. Math. Anal. Appl., 91 (1983), 244–253 | DOI | MR | Zbl

[27] Tineo A., “Existence of solutions for a class of boundary value problems for the equation $x''=F(t,x,x',x'')$”, Comment. Math. Univ. Carolin., 29 (1988), 285–291 | MR | Zbl

[28] Zhang Z., Wang J., “On existence and multiplicity of positive solutions to singular multi-point boundary value problems”, J. Math. Anal. Appl., 295 (2004), 502–512 | DOI | MR | Zbl