By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results.
Citation: |
[1] | H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709. doi: 10.1137/1018114. |
[2] | P. Benevieri, A. Calamai, M. Furi and M. P. Pera, On general properties of $n$-th order retarded functional differential equations, Rend. Istit. Mat. Univ. Trieste, 49 (2017), 73-93. doi: 10.13137/2464-8728/16206. |
[3] | A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753. |
[4] | A. Cabada and R. Jebari, Multiplicity results for fourth order problems related to the theory of deformations beams, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 489-505. doi: 10.3934/dcdsb.2019250. |
[5] | A. Calamai and G. Infante, Nontrivial solutions of boundary value problems for second order functional differential equations, Ann. Mat. Pura Appl., 195 (2016), 741-756. doi: 10.1007/s10231-015-0487-x. |
[6] | A. Calamai and G. Infante, An affine Birkhoff–Kellogg type result in cones with applications to functional differential equations, Math. Methods Appl. Sci., to appear. doi: 10.1002/mma.8665. |
[7] | F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl., 33 (2017), 317-347. doi: 10.1016/j.nonrwa.2016.07.004. |
[8] | R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital., 22 (1967), 135-178. |
[9] | S. Djebali and K. Mebarki, Fixed point index on translates of cones and applications, Nonlinear Stud., 21 (2014), 579-589. |
[10] | C. S. Goodrich, Pointwise conditions for perturbed Hammerstein integral equations with monotone nonlinear, nonlocal elements, Banach J. Math. Anal., 14 (2020), 290-312. doi: 10.1007/s43037-019-00017-1. |
[11] | D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988. |
[12] | J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. |
[13] | G. Infante, Positive solutions of differential equations with nonlinear boundary conditions, Discrete Contin. Dyn. Syst., Suppl. Vol. 2003, (2003), 432-438. |
[14] | G. Infante, On the solvability of a parameter-dependent cantilever-type BVP, Appl. Math. Lett., 132 (2022), 108090, 7 pp. doi: 10.1016/j.aml.2022.108090. |
[15] | G. Infante and J. R. L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc., 49 (2006), 637-656. doi: 10.1017/S0013091505000532. |
[16] | T. Jankowski and R. Jankowski, Multiple solutions of boundary-value problems for fourth-order differential equations with deviating arguments, J. Optim. Theory Appl., 146 (2010), 105-115. doi: 10.1007/s10957-010-9658-5. |
[17] | G. L. Karakostas and P. Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal., 19 (2002), 109-121. doi: 10.12775/TMNA.2002.007. |
[18] | A. Khanfer and L. Bougoffa, A cantilever beam problem with small deflections and perturbed boundary data, J. Funct. Spaces, 2021 (2021), Article ID 9081623, 9 pp. doi: 10.1155/2021/9081623. |
[19] | Y. Li, Existence of positive solutions for the cantilever beam equations with fully nonlinear terms, Nonlinear Anal. Real World Appl., 27 (2016), 221-237. doi: 10.1016/j.nonrwa.2015.07.016. |
[20] | R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7 (2007), 257-279. |
[21] | Y. Ma, C. Yin and G. Zhang, Positive solutions of fourth-order problems with dependence on all derivatives in nonlinearity under Stieltjes integral boundary conditions, Bound Value Probl., 2019 (2019), Paper No. 41, 22 pp. doi: 10.1186/s13661-019-1155-7. |
[22] | S. K. Ntouyas, Nonlocal initial and boundary value problems: A survey, Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, Elsevier B. V., Amsterdam, (2005), 461-557. |
[23] | M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1908), 95. |
[24] | A. Štikonas, A survey on stationary problems, Green's functions and spectrum of Sturm-Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control, 19 (2014), 301-334. doi: 10.15388/NA.2014.3.1. |
[25] | J. R. L. Webb, Compactness of nonlinear integral operators with discontinuous and with singular kernels, J. Math. Anal. Appl., 509 (2022), Paper No. 126000, 17 pp. doi: 10.1016/j.jmaa.2022.126000. |
[26] | J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693. doi: 10.1112/S0024610706023179. |
[27] | M. Wei, Y. Li and G. Li, Lower and upper solutions method to the fully elastic cantilever beam equation with support, Adv. Difference Equ., 2021 (2021), Paper No. 301, 13 pp. doi: 10.1186/s13662-021-03402-z. |
[28] | W. M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc., 48 (1942), 692-704. doi: 10.1090/S0002-9904-1942-07760-3. |
[29] | G. Zhang, Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities, Electron. J. Qual. Theory Differ. Equ., 2022, Paper No. 11, 27 pp. doi: 10.14232/ejqtde.2022.1.1. |