Triple positive solutions for nonlinear boundary value problems in Banach space

AbstractIn this paper, by applying two pairs of lower and upper solutions method and the topological degree theory of strict-set-contractions, the existence of at least three positive solutions to m-point boundary value problems for second order ordinary differential equations in Banach spaces is obtained

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

Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page

A short course on positive solutions of systems of ODEs via fixed point index

We shall firstly study the existence of one positive solution of a model problem for one equation via the classical Krasnosel'ski\u\i{} fixed-point theorem. Secondly we investigate how to handle this problem via the fixed point index theory for compact maps. Thirdly we illustrate how this approach can be tailored in order to deal with non-trivial solutions for systems of ODEs subject to local BCs. The case of nonlocal and nonlinear BCs will also be investigated. Finally we present some applications to the existence of radial solutions of some systems of elliptic PDEs.Comment: 52 pages 13 figure

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