Improving Results on Solvability of a Class of nth-Order Linear Boundary Value Problems

Copyright © 2016 P. Almenar and L. Jodar. This is an open access article distributed under the Creative Commons Attribution ´ License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.[EN] This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of �th-order linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numericallyThis work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.Almenar, P.; Jódar Sánchez, LA. (2016). Improving Results on Solvability of a Class of nth-Order Linear Boundary Value Problems. International Journal of Differential Equations. https://doi.org/10.1155/2016/3750530S10Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Gentry, R. D., & Travis, C. C. (1976). Comparison of eigenvalues associated with linear differential equations of arbitrary order. Transactions of the American Mathematical Society, 223, 167-167. doi:10.1090/s0002-9947-1976-0425241-xSchmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Tomastik, E. C. (1983). Comparison Theorems for Second Order Nonselfadjoint Differential Systems. SIAM Journal on Mathematical Analysis, 14(1), 60-65. doi:10.1137/0514005Hankerson, D., & Henderson, J. (1990). Positive Solutions and Extremal Points for Differential Equations. Applicable Analysis, 39(2-3), 193-207. doi:10.1080/00036819008839980Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., & Henderson, J. (1993). Focal Points and Comparison Theorems for a Class of Two Point Boundary Value Problems. Journal of Differential Equations, 103(2), 375-386. doi:10.1006/jdeq.1993.1055Eloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Eloe, P. ., Henderson, J., & Thompson, H. . (2000). Extremal points for impulsive Lidstone boundary value problems. Mathematical and Computer Modelling, 32(5-6), 687-698. doi:10.1016/s0895-7177(00)00165-5Eloe, P. W., & Ahmad, B. (2005). Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions. Applied Mathematics Letters, 18(5), 521-527. doi:10.1016/j.aml.2004.05.009Graef, J. R., & Yang, B. (2006). Positive solutions to a multi-point higher order boundary value problem. Journal of Mathematical Analysis and Applications, 316(2), 409-421. doi:10.1016/j.jmaa.2005.04.049Graef, J. R., Kong, L., & Wang, H. (2008). Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. Journal of Differential Equations, 245(5), 1185-1197. doi:10.1016/j.jde.2008.06.012Zhang, X., Feng, M., & Ge, W. (2009). Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 70(2), 584-597. doi:10.1016/j.na.2007.12.028Zhang, P. (2011). Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation. Boundary Value Problems, 2011, 1-10. doi:10.1155/2011/483057Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Eloe, P. W., & Ridenhour, J. (1994). Sign Properties of Green’s Functions for a Family of Two-Point Boundary Value Problems. Proceedings of the American Mathematical Society, 120(2), 443. doi:10.2307/2159880Hämmerlin, G., & Hoffman, K.-H. (1991). Numerical Mathematics. Undergraduate Texts in Mathematics. doi:10.1007/978-1-4612-4442-

Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

Inverse problems connected with two-point boundary value problems

For the purpose of studying those properties of a nonlinear function $f(u)$ for which the two-point boundary value problem $u''+\lambda f(u)=0 (00$, the authors construct a number of kinds of special examples. "Inverse" in the title refers to the fact that the multiplicity is specified first and then a suitable function $f$ is constructed