The sign of the Green function of an n-th order linear boundary value problem

[EN] This paper provides results on the sign of the Green function (and its partial derivatives) of ann-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Almenar, P.; Jódar Sánchez, LA. (2020). The sign of the Green function of an n-th order linear boundary value problem. Mathematics. 8(5):1-22. https://doi.org/10.3390/math8050673S12285Butler, G. ., & Erbe, L. . (1983). Integral comparison theorems and extremal points for linear differential equations. Journal of Differential Equations, 47(2), 214-226. doi:10.1016/0022-0396(83)90034-7Peterson, A. C. (1979). Green’s functions for focal type boundary value problems. Rocky Mountain Journal of Mathematics, 9(4). doi:10.1216/rmj-1979-9-4-721Peterson, A. C. (1980). Focal Green’s functions for fourth-order differential equations. Journal of Mathematical Analysis and Applications, 75(2), 602-610. doi:10.1016/0022-247x(80)90104-3Elias, U. (1980). Green’s functions for a non-disconjugate differential operator. Journal of Differential Equations, 37(3), 318-350. doi:10.1016/0022-0396(80)90103-5Nehari, Z. (1967). Disconjugate linear differential operators. Transactions of the American Mathematical Society, 129(3), 500-500. doi:10.1090/s0002-9947-1967-0219781-0Keener, 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:10.2307/1997625Schmitt, 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)90045-7Eloe, 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. (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.1994.1003Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Almenar, P., & Jódar, L. (2016). Improving Results on Solvability of a Class ofnth-Order Linear Boundary Value Problems. International Journal of Differential Equations, 2016, 1-10. doi:10.1155/2016/3750530Almenar, P., & Jodar, L. (2017). SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS. Mathematical Modelling and Analysis, 22(4), 528-547. doi:10.3846/13926292.2017.1329757Sun, 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/s13661-015-0393-6Webb, J. R. L. (2017). New fixed point index results and nonlinear boundary value problems. Bulletin of the London Mathematical Society, 49(3), 534-547. doi:10.1112/blms.12055Jiang, D., & Yuan, C. (2010). The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Analysis: Theory, Methods & Applications, 72(2), 710-719. doi:10.1016/j.na.2009.07.012Wang, Y., & Liu, L. (2017). Positive properties of the Green function for two-term fractional differential equations and its application. The Journal of Nonlinear Sciences and Applications, 10(04), 2094-2102. doi:10.22436/jnsa.010.04.63Zhang, L., & Tian, H. (2017). Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations. Advances in Difference Equations, 2017(1). doi:10.1186/s13662-017-1157-7Wang, Y. (2020). The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications. Advances in Difference Equations, 2020(1). doi:10.1186/s13662-020-02549-

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-

Solvability of Nth Order Linear Boundary Value Problems

Copyright © 2015 P. Almenar and L. Jódar. 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.This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.Almenar, P.; Jódar Sánchez, LA. (2015). Solvability of Nth Order Linear Boundary Value Problems. International Journal of Differential Equations. 2015:1-19. https://doi.org/10.1155/2015/230405S1192015Almenar, P., & Jódar, L. (2014). The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions. Abstract and Applied Analysis, 2014, 1-17. doi:10.1155/2014/126713Greguš, M. (1987). Third Order Linear Differential Equations. doi:10.1007/978-94-009-3715-4Polya, G. (1922). On the Mean-Value Theorem Corresponding to a Given Linear Homogeneous Differential Equations. Transactions of the American Mathematical Society, 24(4), 312. doi:10.2307/1988819Sherman, T. (1965). Properties of solutions ofn-th order linear differential equations. Pacific Journal of Mathematics, 15(3), 1045-1060. doi:10.2140/pjm.1965.15.1045Muldowney, J. S. (1979). A Necessary and Sufficient Condition for Disfocality. Proceedings of the American Mathematical Society, 74(1), 49. doi:10.2307/2042104Nehari, Z. (1967). Disconjugate Linear Differential Operators. Transactions of the American Mathematical Society, 129(3), 500. doi:10.2307/1994604Ahmad, S., & Lazer, A. C. (1978). AnN-Dimensional Extension of the Sturm Separation and Comparison Theory to a Class of Nonselfadjoint Systems. SIAM Journal on Mathematical Analysis, 9(6), 1137-1150. doi:10.1137/0509092Ahmad, S., & Lazer, A. C. (1980). On nth-order Sturmian theory. Journal of Differential Equations, 35(1), 87-112. doi:10.1016/0022-0396(80)90051-0Elias, U. (1975). The extremal solutions of the equation Ly + p(x)y = 0. Journal of Mathematical Analysis and Applications, 50(3), 447-457. doi:10.1016/0022-247x(75)90001-3Elias, U. (1977). Nonoscillation and Eventual Disconjugacy. Proceedings of the American Mathematical Society, 66(2), 269. doi:10.2307/2040944Elias, U. (1978). Eigenvalue problems for the equation Ly + λp(x) y = 0. Journal of Differential Equations, 29(1), 28-57. doi:10.1016/0022-0396(78)90039-6Deimling, K. (1985). Nonlinear Functional Analysis. doi:10.1007/978-3-662-00547-7Gentry, 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. doi:10.2307/1997522Schmitt, 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)90045-7Keener, 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:10.2307/1997625Tomastik, E. C. (1983). Comparison Theorems for Second Order Nonselfadjoint Differential Systems. SIAM Journal on Mathematical Analysis, 14(1), 60-65. doi:10.1137/0514005Kreith, K. (1984). A class of hyperbolic focal point problems. Hiroshima Mathematical Journal, 14(1), 203-210. doi:10.32917/hmj/1206133155Hankerson, D., & Peterson, A. (1988). Comparison Theorems for Eigenvalue Problems for nth Order Differential Equations. Proceedings of the American Mathematical Society, 104(4), 1204. doi:10.2307/2047613Hankerson, 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., Hankerson, D., & Henderson, J. (1992). Positive Solutions and $J$-Focal Points for Two Point Boundary Value Problems. Rocky Mountain Journal of Mathematics, 22(4), 1283-1293. doi:10.1216/rmjm/1181072655Eloe, 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/s13661-015-0393-6Lemmens, B., & Nussbaum, R. (2013). Continuity of the cone spectral radius. Proceedings of the American Mathematical Society, 141(8), 2741-2754. doi:10.1090/s0002-9939-2013-11520-0Eloe, 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:10.2307/215988

Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems

We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations. We can compute branches of solutions with limit points, bifurcation points, etc. Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations

Nonlinear Diffusion and Image Contour Enhancement

The theory of degenerate parabolic equations of the forms $$u_t=(\Phi(u_x))_{x} \quad {\rm and} \quad v_{t}=(\Phi(v))_{xx}$$ is used to analyze the process of contour enhancement in image processing, based on the evolution model of Sethian and Malladi. The problem is studied in the framework of nonlinear diffusion equations. It turns out that the standard initial-value problem solved in this theory does not fit the present application since it it does not produce image concentration. Due to the degenerate character of the diffusivity at high gradient values, a new free boundary problem with singular boundary data can be introduced, and it can be solved by means of a non-trivial problem transformation. The asymptotic convergence to a sharp contour is established and rates calculated.Comment: 29 pages, includes 6 figure

Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci

In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to {prove the existence of obstructions arising from} conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values $\sigma \le 1$ at which a second ODE (the Jacobi equation) has a solution vanishing at $0$ and $\sigma$. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter $\lambda$, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of $\lambda$ of interest. Here we present an alternative approach that takes advantage of the presence of a parameter $\lambda$. Rather than search for conjugate points $\sigma \le 1$ at a fixed value of $\lambda$, we search for a set of special parameter values $\lambda_m$ (with corresponding Jacobi solution \bfzeta^m) for which $\sigma=1$ is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any $\lambda$ equals the number of these \bfzeta^m for which \langle \bfzeta^m, \Op \bfzeta^m \rangle < 0, where \Op is the Jacobi differential operator at $\lambda$. This computation is particularly simple when $\lambda$ appears linearly in \Op. We apply this approach to the elastic strut, in which the force appears linearly in \Op, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE