122,153 research outputs found
Existence of positive solutions for boundary value problems of fractional functional differential equations
This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order given by , , , , , . Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem
Solvability of Nth Order Linear Boundary Value Problems
Copyright Š 2015 P. Almenar and L. JĂłdar. 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.; JĂłdar SĂĄnchez, LA. (2015). Solvability of Nth Order Linear Boundary Value Problems. 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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. 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Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions
We develop the concept and the calculus of anti-self dual (ASD) Lagrangians
which seems inherent to many questions in mathematical physics, geometry, and
differential equations. They are natural extensions of gradients of convex
functions --hence of self-adjoint positive operators-- which usually drive
dissipative systems, but also rich enough to provide representations for the
superposition of such gradients with skew-symmetric operators which normally
generate unitary flows. They yield variational formulations and resolutions for
large classes of non-potential boundary value problems and initial-value
parabolic equations. Solutions are minima of functionals of the form (resp. )
where is an anti-self dual Lagrangian and where are
essentially skew-adjoint operators. However, and just like the self (and
antiself) dual equations of quantum field theory (e.g. Yang-Mills) the
equations associated to such minima are not derived from the fact they are
critical points of the functional , but because they are also zeroes of the
Lagrangian itself.Comment: 50 pages. For the most updated version of this paper, please visit
http://www.pims.math.ca/~nassif/pims_papers.htm
The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation
We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: ââCDβ(p(t)CDÎąu(t)) + f(t,u(t-Ď),u(t+θ))=0, tâ(0,1), âCDÎąu(0)=âCDÎąu(1)=(âCDÎąu(0))=0, au(t)-buâ˛(t)=Ρ(t), tâ[-Ď,0], cu(t)+duâ˛(t)=Ξ(t), tâ[1,1+θ], where ââCDÎą, âCDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-Ď,1+θ],â), 10, and ΡâC([-Ď,0],[0,â)), ΞâC([1,1+θ],[0,â)). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results
Nontrivial solutions of boundary value problems for second order functional differential equations
In this paper we present a theory for the existence of multiple nontrivial
solutions for a class of perturbed Hammerstein integral equations. Our
methodology, rather than to work directly in cones, is to utilize the theory of
fixed point index on affine cones. This approach is fairly general and covers a
class of nonlocal boundary value problems for functional differential
equations. Some examples are given in order to illustrate our theoretical
results.Comment: 19 pages, revised versio
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
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