27,091 research outputs found
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-
Positive solutions for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals
We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existenceand multiplicity results of positive solutions are obtained. The results obtained in this paperimprove and generalize some well-known results
A Caputo Boundary Value Problem in Nabla Fractional Calculus
Boundary value problems have long been of interest in the continuous differential equations context. However, with the advent of new areas like Nabla Fractional Calculus, we may consider such problems in new contexts. In this work, we will consider several right focal boundary value problems, involving a Caputo fractional difference operator, in the Nabla Fractional Calculus context. Properties of the Green\u27s functions for each of these boundary value problems will be investigated and, in the case of a particular boundary value problem, used to establish the existence of positive solutions to a nonlinear version of the boundary value problem.
Adviser: Allan C. Peterso
An algorithm for positive solution of boundary value problems of nonlinear fractional differential equations by Adomian decomposition method
In this paper, an algorithm based on a new modification, developed by Duan and Rach, for the Adomian decomposition
method (ADM) is generalized to find positive solutions for boundary value problems involving nonlinear fractional
ordinary differential equations. In the proposed algorithm the boundary conditions are used to convert the nonlinear
fractional differential equations to an equivalent integral equation and then a recursion scheme is used to obtain the
analytical solution components without the use of undetermined coefficients. Hence, there is no requirement to solve
a nonlinear equation or a system of nonlinear equations of undetermined coefficients at each stage of approximation
solution as per in the standard ADM. The fractional derivative is described in the Caputo sense. Numerical examples
are provided to demonstrate the feasibility of the proposed algorithm
Positive solutions of nonlocal boundary value problem for higher order fractional differential system
In this paper, we study existence and multiplicity results for a coupled system of nonlinear nonlocal boundary value problems for higher order fractional differential equations of the type (see PDF) where (see PDF) is Caputo fractional derivative. We employ the Guo-Krasnosel’skii fixed point theorem to establish existence and multiplicity results for positive solutions. We derive explicit intervals for the parameters _ and μ for which the system possess the positive solutions or multiple positive solutions. Examples are included to show the applicability of the main results
Eigenvalue problems for fractional differential equations with mixed derivatives and generalized p-Laplacian
This paper reports the investigation of eigenvalue problems for two classes of nonlinear fractional differential equations with generalized p-Laplacian operator involving both Riemann–Liouville fractional derivatives and Caputo fractional derivatives. By means of fixed point theorem on cones, some sufficient conditions are derived for the existence, multiplicity and nonexistence of positive solutions to the boundary value problems. Finally, an example is presented to further verify the correctness of the main theoretical results and illustrate the wide range of their potential applications
Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order
We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo's fractional order derivative. The existence results are obtained by means of the Avery-Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative
Topological methods on solvability, multiplicity and bifurcation of a nonlinear fractional boundary value problem
In this paper, we first prove new properties of the -stably solvable maps for a class of decomposable operators in the form of , where is a bounded linear operator and is nonlinear. This class of maps is important in applications as many differential equations can be written as . Secondly, three different approaches, the -stably solvable maps, fixed point index and iterative method are applied to study a nonlinear fractional boundary value problem involving a parameter . We obtain intervals of that correspond to at least two, one and no positive solutions, respectively. Thirdly, convergence of the eigenvalues and the corresponding eigenvectors for the associated Hammerstein-type integral operator are proved. This paper seems to be the first to apply the theory of -stably solvable operators in studying boundary value problems
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: 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
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