122,153 research outputs found

    Existence of positive solutions for boundary value problems of fractional functional differential equations

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    This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order α\alpha given by Dαu(t)+f(t,ut)=0 D^{\alpha} u(t) + f(t, u_t) = 0, t∈(0,1)t \in (0, 1), 2<α≤32 < \alpha \le 3, u′(0)=0 u^{\prime}(0) = 0, u′(1)=bu′(η)u^{\prime}(1) = b u^{\prime}(\eta), u0=ϕu_0 = \phi. 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

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    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 JJ-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. 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    Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

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    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 I(u)=L(u,Λu)I(u)=L(u, \Lambda u) (resp. I(u)=∫0TL(t,u(t),u˙(t)+Λtu(t))dtI(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+\Lambda_{t}u(t))dt) where LL is an anti-self dual Lagrangian and where Λt\Lambda_{t} 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 II, but because they are also zeroes of the Lagrangian LL 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

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    We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative:   CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1),  CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈[1,1+θ], where   CDα,  CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-τ,1+θ],ℝ), 10, and η∈C([-τ,0],[0,∞)), ξ∈C([1,1+θ],[0,∞)). 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

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    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

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    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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