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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 operators and maximum principles for ordinary differential equations
We show an equivalence between a classical maximum principle in differential equations and positive operators on Banach Spaces. Then we shall exhibit many types of boundary value problems for which the maximum principle is valid. Finally, we shall present extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of Green’s functions
Existence of positive solutions to th-order -Laplacian boundary value problems with integral boundary conditions
WOS: 000390602900029This work is devoted to the existence of positive solutions for an nth order p-Laplacian boundary value problem with integral boundary conditions. The proof of the main result is based on six functionals fixed point theorem. As an application, we give an example to illustrate the obtained result.TUBITAK, the Scientific and Technological Research Council of TurkeyTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK)The second author was supported by the 2219 scholarship programme of TUBITAK, the Scientific and Technological Research Council of Turkey
Existence of multiple positive solutions of higher order multi-point nonhomogeneous boundary value problem
In this paper, by using the Avery and Peterson fixed point theorem, we establish the existence of multiple positive solutions for the following higher order multi-point nonhomogeneous boundary value problem
,
,
where and are integers, for and , . We give an example to illustrate our result
General non-existence theorem for phase transitions in one-dimensional systems with short range interactions, and physical examples of such transitions
We examine critically the issue of phase transitions in one-dimensional
systems with short range interactions. We begin by reviewing in detail the most
famous non-existence result, namely van Hove's theorem, emphasizing its
hypothesis and subsequently its limited range of applicability. To further
underscore this point, we present several examples of one-dimensional short
ranged models that exhibit true, thermodynamic phase transitions, with
increasing level of complexity and closeness to reality. Thus having made clear
the necessity for a result broader than van Hove's theorem, we set out to prove
such a general non-existence theorem, widening largely the class of models
known to be free of phase transitions. The theorem is presented from a rigorous
mathematical point of view although examples of the framework corresponding to
usual physical systems are given along the way. We close the paper with a
discussion in more physical terms of the implications of this non-existence
theorem.Comment: Short comment on possible generalization to wider classes of systems
added; accepted for publication in Journal of Statistical Physic
Conjugate points for fractional differential equations
Let b \u3e 0. Let 1 \u3c α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 \u3c t \u3c b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation
Simplicial homology and Hochschild cohomology of Banach semilattice algebras
The -convolution algebra of a semilattice is known to have trivial
cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is
symmetric. We ex tend this result to all cohomology groups of degree
with symmetric coef ficients. Our techniques prove a stronger splitting result,
namely that the spli tting can be made natural with respect to the underlying
semilattice.Comment: 17pp, preprint version (revised 2006). Final version to appear in
Glasgow Math. Journal (2006
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