2013, 2013(special): 217-226. doi: 10.3934/proc.2013.2013.217

The role of lower and upper solutions in the generalization of Lidstone problems

1. 

Centro de Investigação em Matemática e Aplicações da U.E. (CIMA-CE), Rua Romão Ramalho 59, 7000-671 Évora

2. 

School of Sciences and Technology. Department of Mathematics, University of Évora, Research Center in Mathematics and Applications of the University of Évora, (CIMA-UE), Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received  September 2012 Revised  February 2013 Published  November 2013

In this the authors consider the nonlinear fully equation
          \begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}
    They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a one-sided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.
Citation: João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
References:
[1]

P. Drábek, G. Holubová, A. Matas, P. Nečessal, Nonlinear models of suspension bridges: discussion of results,, Applications of Mathematics, 48 (2003), 497.   Google Scholar

[2]

J. Fialho, F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009), 1519.   Google Scholar

[3]

M.R. Grossinho, F.M. Minhós, A.I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, 62 (2005), 1235.   Google Scholar

[4]

M. R. Grossinho, F. Minhós, A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition,, Nonlinear Anal., 70 (2009), 4027.   Google Scholar

[5]

M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems,, Nonlinear Studies, 12 (2005), 165.   Google Scholar

[6]

C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation,, Appl. Anal., 26 (1988), 289.   Google Scholar

[7]

C. P. Gupta, Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type,, Differential and Integral Equations, 4 (1991), 397.   Google Scholar

[8]

A.C. Lazer, P.J. Mckenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,, SIAM Review 32 (1990) 537-578., 32 (1990), 537.   Google Scholar

[9]

T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order,, Appl. Math. Comp., 159 (2004), 11.   Google Scholar

[10]

F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem,, Discrete Contin. Dyn. Syst., (2005), 662.   Google Scholar

[11]

F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations,, Nonlinear Anal., (2009), 281.   Google Scholar

[12]

M. Šenkyřík, Fourth order boundary value problems and nonlinear beams,, Appl. Analysis, 59 (1995), 15.   Google Scholar

show all references

References:
[1]

P. Drábek, G. Holubová, A. Matas, P. Nečessal, Nonlinear models of suspension bridges: discussion of results,, Applications of Mathematics, 48 (2003), 497.   Google Scholar

[2]

J. Fialho, F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009), 1519.   Google Scholar

[3]

M.R. Grossinho, F.M. Minhós, A.I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, 62 (2005), 1235.   Google Scholar

[4]

M. R. Grossinho, F. Minhós, A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition,, Nonlinear Anal., 70 (2009), 4027.   Google Scholar

[5]

M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems,, Nonlinear Studies, 12 (2005), 165.   Google Scholar

[6]

C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation,, Appl. Anal., 26 (1988), 289.   Google Scholar

[7]

C. P. Gupta, Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type,, Differential and Integral Equations, 4 (1991), 397.   Google Scholar

[8]

A.C. Lazer, P.J. Mckenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,, SIAM Review 32 (1990) 537-578., 32 (1990), 537.   Google Scholar

[9]

T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order,, Appl. Math. Comp., 159 (2004), 11.   Google Scholar

[10]

F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem,, Discrete Contin. Dyn. Syst., (2005), 662.   Google Scholar

[11]

F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations,, Nonlinear Anal., (2009), 281.   Google Scholar

[12]

M. Šenkyřík, Fourth order boundary value problems and nonlinear beams,, Appl. Analysis, 59 (1995), 15.   Google Scholar

[1]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[2]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[3]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[4]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[5]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[6]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[7]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[8]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[9]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134

[10]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[11]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172

[12]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[13]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002

[14]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[15]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521

[16]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067

[17]

Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166

[18]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[19]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297

[20]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

 Impact Factor: 

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]