Existence results for nonlinear boundary value problems with $m$-point integral boundary condition

Year 2018, Volume: 47 Issue: 6, 1521 - 1536, 12.12.2018

Tugba Senlik Cerdik , Nuket Aykut Hamal

Abstract

In this paper, we investigate the existence of positive solutions for the nonlinear $m$-point boundary value problems with integral boundary condition. By using  fixed-point index theorem and Leggett-Williams fixed point theorem, the existence and multiplicity of positive solutions are obtained. As an application, two examples are given to demonstrate our results.

Keywords

Boundary value problems, fixed point theorems, integral boundary conditions

References

• Agarwal, R. P., O$'$Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
• Bitsadze, A. V., Samarskii, A. A., Some elementary generalizations of linear elliptic boundary value problems (in Russian), Anal. Dokl. Akad. Nauk SSSR, 185 (1969), 739-740, English translation: Soviet Math. Dokl. 10, 398-400, 1969.
• Boucherif, A., Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. 70, 364-371, 2009.
• Conti, R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. 22, 135-178, 1967.
• Corduneanu, C., Integral Equations and Applications, Cambridge University Press, Cambridge, UK, 1991.
• Feng, M., Ji, D., Ge, W., Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math. 222, 351-363, 2008.
• Gallardo, J. M., Second-order differential operators with integral boundary conditions and generation of analytic semigroups, Rocky Mountain J. Math. 30, 1265-1291, 2000.
• Guo, D., Lashmikanthan, V., Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988.
• Il'in, V. A., Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differ. Equ. 23, 979-987, 1987.
• Jankowski, T., Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions, Nonlinear Anal. 73, 1289-1299, 2010.
• Ntouyas, S. K., Nonlocal initial and boundary value problems: a survey, in: Handbook of differential equations: ordinary differential equations, Vol. II, Elsevier B. V., Amsterdam, 461-557, 2005.
• Picone, M., Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine (in Italian), Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 1-95, 1908.
• Tariboon, J., Sudsutad, W., Positive solutions of a nonlinear m-point integral boundary value problem, Appl. Math. Sci. 6, 711-729, 2012.
• Webb, J. R. L., Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15, 45-67, 2008.
• Webb, J. R. L., Infante, G., Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74, 673-693, 2006.
• Whyburn, W. M. Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 48, 692-704, 1942.
• Yoruk, F., Hamal, N. A., Existence results for nonlinear boundary value problems with integral boundary conditions on an infinite interval, Bound. Value Probl., 2012:127, 17 pp., 2012. doi:10.1186/1687-2770-2012-127.
• Zhang, X, Feng, M., Ge, W., Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces, Nonlinear Anal. 69, 3310-3321, 2008.