Existence of positive solutions for some kinds of BVPs in Banach spaces

Lyna Benzenati; Svetlin Georgiev Georgiev; Karima Mebarki

doi:10.24193/subbmath.2021.4.10

Authors

Lyna Benzenati Abderrahmane Mira University
Svetlin Georgiev Georgiev Sofia University, Faculty of Mathematics and Informatics
Karima Mebarki Abderrahmane Mira university

DOI:

https://doi.org/10.24193/subbmath.2021.4.10

Keywords:

Boundary value problem, Green's function, unbounded interval, measure of noncompactness, fixed point index, sum operator.

Abstract

In this work, we use index fixed point theory for perturbation of expansive mappings by $\ell$-set contractions to study the existence of bounded positive solutions for a class of two-point boundary value problem (BVP) associated to second-order nonlinear differential equation posed on the positive half-line. The nonlinearity, which may exhibit a singularity at the origin, is written as a sum of two functions which behave differently. These functions, depend on the solution and their derivative, take values in a general Banach space and have at most polynomial growth conditions. An example to illustrate the main results is given.

Author Biography

Karima Mebarki, Abderrahmane Mira university

Department of Mathematics

References

Corduneanu, C., Integral equations and stability of feedback systems, Academic Press, New York, 1973.

Deimling, K., Ordinary dierential equations in Banach spaces, Springer, Berlin, Germany, 1977.

Deimling, K., Nonlinear functional analysis, Springer-Verlag, Berlin, Heidel-berg, 1985.

Djebali, S., Travelling wave solutions to a reaction-diffusion system arising in epidemiology, Nonlinear Anal. Real World Appl., 2(2001), no. 4, 417-442.

Djebali, S., Madjidi, F., Mebarki, K., Existence results for singular boundary value problems on unbounded domains in Banach spaces, Mediterr. J. Math., 11(2014), 45-74.

Djebali, S., Mebarki, K., Fixed point index theory for perturbation of expansive mappings by k-set contractions, in press . DOI: 10.12775/TMNA.yyyy.000.

Djebali, S., Mebarki, K., On the singular generalized Fisher-like equation with derivative depending nonlinearity, Appl. Math. Comput., 205(2008), no. 1, 336-351.

Djebali, S., Moussaoui, T., A class of second order bvps on infinite intervals, Electron. J. Qual. Theory Dier. Equ., 4(2006), 1-19.

Djebali, S., Saifi, O., Baoqiang, Y., Positive solutions for singular BVPs on the positive half-Line arising from epidemiology and combustion theory, Acta Math. Sci. Ser. B, 32(2012), no. 2, 672-694.

Guo, D., Cho, Y.I., Zhu, J., Partial ordering methods in nonlinear problems, Shangdon Science and Technology Publishing Press, Shangdon, 1985.

Guo, D., Lakshmikantham, V., Liu, X., Nonlinear integral equations in abstract spaces, Math. and Appl., vol. 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

Guo, D., Lakshmikantham, V., Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5, Academic Press, Boston, Mass, USA, 1988.

Kneser, A., Untersuchung und asymptotische darstellung der integrale gewisser differentialgleichungen bei grossen werthen des arguments, J. Reine Angew. Math., 116(1896), 178-212.

Lakshmikantham, V., Leela, S., Nonlinear differential equations in abstract spaces, Pergamon, Oxford, UK, 1981.

Li, P., Chen, H., Wu, Y., Existence of solutions of n-point boundary value problems on the half-line in Banach spaces, Acta Appl. Math., 110(2010), 785-795.

Liu, Y., Boundary value problems for second order differential equations on unbounded domains in a Banach space, Appl. Math. Comput., 135(2003), 569-583.

Murray, J.D., Mathematical biology, biomathematics texts, Springer-Verlag, 1989.

Potter, A.J.B., A fixed point theorem for positive k-set contractions, Proc. Edinb. Math. Soc., 19(1974), 93-102.

Zeidler, E., Nonlinear functional analysis and its applications. Vol. I: Fixed Point Theorems, Springer-Verlag, New York, 1986.