Multiple Nonnegative Solutions for a Class of Fourth-Order BVPs Via a New Topological Approach
Keywords:
ODE, fourth-order BVPs, nonnegative solution, fixed point, cone, sum of operatorsAbstract
In this paper, we study a class of fourth-order boundary value problems with integral boundary conditions. The nonlinearity may have time-singularity and change sign. Moreover, it satisfies general polynomial growth conditions. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions.
An example of application illustrates the existence result.
References
R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88.
R. P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differential and integral equations 2 (1989), 91-110.
R. P. Agarwal, S. Kelevedjiev, On the solvability of fourth-order two-point boundary value problems, Mathematics 2020, 8, 603.
K. Bachouche, A. Benmezai, S. Djebali, Positive solutions to semi-positone fourth-order ϕ-Laplacian BVPs, Positivity 21 (2017), 193-212.
S. Benslimane, S. Djebali, K. Mebarki, On the ?xed point index for sums of operators, Fixed Point Theory, 23(2022), no. 1, 143-162.
S. Djebali, T. Moussaoui, R. Precup, Fourth order p-laplacian nonlinear systems via the vector version of the Krasnosel'skii's ?xed point theorem, Mediterr. J. Math 6 (2009), no 4, 447-460.
S. Djebali, K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contraction, Top. Meth. Nonli. Anal., 54 (2019), no 2A, 613-640.
D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, Boston, Mass, USA, vol. 5, (1988).
C. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonnance, Journal of Mathe- matical Analysis and Applications, 135(1988), 208-225.
L. Lin, Y. Liu, D. Zhao, Multiple solutions for a class of nonliner fourth-order boundary value problems, Symmetry 2020, 12, 1989.
B. Liu, Positive solutions of the fourth-order two point boundary value problems, Appl. Math. Comput, 148 (2004), no. 2, 407-420.
Y. Liu, D. O'Regan, Multiplicity results for a class of fourth order semipositone m-point boundary value problems, Appl. Anal. 91(2012), 911-921.
R. Ma, H. Wang, On the existence of positive solutions of fourth-order ordinary di?erential equation, Anal. Appl. 59(1- 4)(1995), 225-231.
S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973) 460-467.
Q. Wang, Y. Guo, Y. Ji, Positive solutions for fourth?order nonlinear differential equation with integral boundary condi- tions, Discrete Dynamics in Nature and Society, Vol. 2013, Article ID 684962, 10 pages.
T. Xiang, R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. 71 (2009), no. 7-8, 3229-3239.
C. Zhai, C. Hiang, Existence of nontrivial solutions for a nonlinear fourth-order boundary value problem via iterative method, J. Nonlinear Sci. Appl. 9 (2016), 4295-4304.
Y. Zhu, P. Weng, Multiple positive solutions for a fourth-order boundary value problem, Bol. Soc. Parana. Mat, 21(2003), 9-19.
