Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T06:18:59.545Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and monotonicity of nonlocal boundary value problems: the one-dimensional case

Part of: Qualitative theory Partial differential equations General theory in ordinary differential equations Boundary value problems Integral transforms, operational calculus Harmonic analysis in one variable Real functions

Published online by Cambridge University Press:  23 December 2020

Christopher Goodrich  and
Carlos Lizama
Christopher Goodrich
Affiliation:
School of Mathematics and Statistics, UNSW Australia, Sydney, NSW, 2052, Australia (c.goodrich@unsw.edu.au)
Carlos Lizama
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, 9160000, Santiago, Chile (carlos.lizama@usach.cl)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider nonlocal equations of the general form

\begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation}
By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself.

Keywords

Convolutionmonotonicitynonlocal differential equationfractional differential equation

MSC classification

Primary: 34B10: Nonlocal and multipoint boundary value problems 34C11: Growth, boundedness 35B09: Positive solutions 42A85: Convolution, factorization 44A35: Convolution
Secondary: 26A33: Fractional derivatives and integrals 26A48: Monotonic functions, generalizations 34A08: Fractional differential equations 34B27: Green functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 1 - 27
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abadias, L. and Miana, P. J.. Generalized Cesàro operators, fractional finite differences and Gamma functions. J. Funct. Anal. 274 (2018), 1424–1465.CrossRefGoogle Scholar
Abadias, L., Lizama, C., Miana, P. J. and Velasco, M. P.. Cesàro sums and algebra homomorphisms of bounded operators. Israel J. Math. 216 (2016), 471–505.CrossRefGoogle Scholar
Abadias, L., Lizama, C., Miana, P. J. and Velasco, M. P.. On well-posedness of vector-valued fractional differential-difference equations. Discrete Contin. Dyn. Syst. 39 (2019), 2679–2708.CrossRefGoogle Scholar
Aly, J. J.. Thermodynamics of a two-dimensional self-gravitating system. Phys. Rev. E. 49 (1994), 3771–3783.CrossRefGoogle ScholarPubMed
Anderson, D. R.. Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions. J. Math. Anal. Appl. 408 (2013), 318–323.CrossRefGoogle Scholar
Barnett, N. S., Cerone, P. and Dragomir, S. S.. Majorisation inequalities for Stieltjes integrals. Appl. Math. Lett. 22 (2009), 416–421.CrossRefGoogle Scholar
Bavaud, F.. Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation. Rev. Mod. Phys. 63 (1991), 129–148.CrossRefGoogle Scholar
Biler, P. and Nadzieja, T.. A class of nonlocal parabolic problems occurring in statistical mechanics. Colloq. Math. 66 (1993), 131–145.CrossRefGoogle Scholar
Biler, P. and Nadzieja, T.. Nonlocal parabolic problems in statistical mechanics. Nonlinear Anal. 30 (1997), 5343–5350.CrossRefGoogle Scholar
Biler, P., Krzywicki, A. and Nadzieja, T.. Self-interaction of Brownian particles coupled with thermodynamic processes. Reports Math. Phys. 42 (1998), 359–372.CrossRefGoogle Scholar
Caglioti, E., Lions, P.-L., Marchioro, C. and Pulvirenti, M.. A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 143 (1992), 501–525.CrossRefGoogle Scholar
Cao, X. and Dai, G.. Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations. Electron. J. Differential Equations 2018 (179) (2018), 1–10.CrossRefGoogle Scholar
Cianciaruso, F., Infante, G. and Petramala, P.. Non-zero radial solutions for elliptic systems with coupled functional BCs in exterior domains. Proc. Edinb. Math. Soc. (2) 62 (2019), 757–769.CrossRefGoogle Scholar
Corrêa, F. J. S. A.. On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59 (2004), 1147–1155.CrossRefGoogle Scholar
Corrêa, F. J. S. A., Menezes, S. D. B. and Ferreira, J.. On a class of problems involving a nonlocal operator. Appl. Math. Comput. 147 (2004), 475–489.Google Scholar
Dahal, R. and Goodrich, C. S.. A monotonicity result for discrete fractional difference operators. Arch. Math. (Basel) 102 (2014), 293–299.CrossRefGoogle Scholar
Diekmann, O. and Kaper, H.. On the bounded solutions of a nonlinear convolution equation. Nonlinear Anal. 2 (1978), 721–737.CrossRefGoogle Scholar
Diethelm, K.. Monotonicity of functions and sign changes of their Caputo derivatives. Fract. Calc. Appl. Anal. 19 (2016), 561–566.CrossRefGoogle Scholar
do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P.. Positive solutions for some nonlocal and nonvariational elliptic systems. Complex Var. Elliptic Equ. 61 (2016), 297–314.CrossRefGoogle Scholar
Ehrenpreis, L.. Solution of some problems of division. IV. Invertible and elliptic operators. Amer. J. Math. 82 (1960), 522–588.CrossRefGoogle Scholar
Esposito, P., Grossi, M. and Pistoia, A.. On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 227–257.Google Scholar
Gómez-Callado, M. and Jordá, E.. Regularity of solutions of convolution equations. J. Math. Anal. Appl. 338 (2008), 873–884.CrossRefGoogle Scholar
Goodrich, C. S.. On nonlocal BVPs with nonlinear boundary conditions with asymptotically sublinear or superlinear growth. Math. Nachr. 285 (2012), 1404–1421.CrossRefGoogle Scholar
Goodrich, C. S.. On nonlinear boundary conditions involving decomposable linear functionals. Proc. Edinb. Math. Soc. (2) 58 (2015), 421–439.CrossRefGoogle Scholar
Goodrich, C. S.. The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems. J. Fixed Point Theory Appl. 19 (2017), 2629–2646.CrossRefGoogle Scholar
Goodrich, C. S.. Coercive nonlocal elements in fractional differential equations. Positivity 21 (2017), 377–394.CrossRefGoogle Scholar
Goodrich, C. S.. New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green's function. J. Diff. Equ. 264 (2018), 236–262.CrossRefGoogle Scholar
Goodrich, C. S.. Radially symmetric solutions of elliptic PDEs with uniformly negative weight. Ann. Mat. Pura Appl. (4) 197 (2018), 1585–1611.CrossRefGoogle Scholar
Goodrich, C. S.. A topological approach to nonlocal elliptic partial differential equations on an annulus. Math. Nachr., to appear. https://doi.org/10.1002/mana.201900204.Google Scholar
Goodrich, C. S. and Lizama, C.. A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity. Israel J. Math. 236 (2020), 533–589.CrossRefGoogle Scholar
Goodrich, C. S. and Lizama, C.. Positivity, monotonicity, and convexity for convolution operators. Discrete Contin. Dyn. Syst. Series A. 40 (2020), 4961–4983.CrossRefGoogle Scholar
Goodrich, C. S. and Peterson, A. C.. Discrete Fractional Calculus (Cham: Springer, 2015), doi: 10.1007/978-3-319-25562-0.CrossRefGoogle Scholar
Graef, J. and Webb, J. R. L.. Third order boundary value problems with nonlocal boundary conditions. Nonlinear Anal. 71 (2009), 1542–1551.CrossRefGoogle Scholar
Guo, D. and Lakshmikantham, V.. Nonlinear problems in abstract cones (Boston: Academic Press, 1988).Google Scholar
Infante, G.. Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence. Discrete Contin. Dyn. Syst. Ser. B 25 (2020), 691–699.Google Scholar
Infante, G. and Pietramala, P.. Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains. NoDEA Nonlinear Differential Equations Appl. 22 (2015), 979–1003.CrossRefGoogle Scholar
Infante, G., Pietramala, P. and Tenuta, M.. Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2245–2251.CrossRefGoogle Scholar
Ionescu, C. M.. The Human Respiratory System. An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics (London: Springer, 2013).Google Scholar
Jankowski, T.. Positive solutions to fractional differential equations involving Stieltjes integral conditions. Appl. Math. Comput. 241 (2014), 200–213.Google Scholar
Karakostas, G. L.. Existence of solutions for an $n$-dimensional operator equation and applications to BVPs. Electron. J. Differential Equations 2014 (71) (2014), 1–17.Google Scholar
Kilbas, A., Srivastava, H. M. and Trujillo, J. J.. Theory and Applications of Fractional Differential Equations (New York: North-Holland, 2006).Google Scholar
Lipovan, O.. Asymptotic properties of solutions to some nonlinear integral equations of convolution type. Nonlinear Anal. 69 (2008), 2179–2183.CrossRefGoogle Scholar
Liu, F., Luo, H. and Dai, G.. Global bifurcation and nodal solutions for homogeneous Kirchhoff type equations. Electron. J. Qual. Theory Diff. Equ. 29 (2020), 1–13.Google Scholar
Lizama, C.. The Poisson distribution, abstract fractional difference equations, and stability. Proc. Amer. Math. Soc. 145 (2017), 3809–3827.CrossRefGoogle Scholar
Lizama, C. and Murillo-Arcila, M.. Maximal regularity in $\ell _p$ spaces for discrete time fractional shifted equations. J. Differential Equations. 263 (2017), 3175–3196.CrossRefGoogle Scholar
Picone, M.. Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1908), 1–95.Google Scholar
Podlubny, I.. Fractional Differential Equations (New York: Academic Press, 1999).Google Scholar
Rosier, C. and Rosier, L.. On the global existence of solutions for a non-local problem occurring in statistical mechanics. Nonlinear Anal. 60 (2005), 1509–1531.CrossRefGoogle Scholar
Rostek, S.. Option Pricing in Fractional Brownian Markets (Berlin-Heidelberg: Springer, 2009).CrossRefGoogle Scholar
Stańczy, R.. Nonlocal elliptic equations. Nonlinear Anal. 47 (2001), 3579–3584.CrossRefGoogle Scholar
Vergara, V. and Zacher, R.. Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods. SIAM J. Math. Anal. 47 (2015), 210–239.CrossRefGoogle Scholar
Wang, Y., Liu, L. and Wu, Y.. Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Analysis 74 (2011), 6434–6441.CrossRefGoogle Scholar
Webb, J. R. L. and Infante, G.. Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. (2) 74 (2006), 673–693.CrossRefGoogle Scholar
Whyburn, W. M.. Differential equations with general boundary conditions. Bull. Amer. Math. Soc. 48 (1942), 692–704.CrossRefGoogle Scholar
Yan, B. and Wang, D.. The multiplicity of positive solutions for a class of nonlocal elliptic problem. J. Math. Anal. Appl. 442 (2016), 72–102.CrossRefGoogle Scholar
Yang, Z.. Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal. 65 (2006), 1489–1511.CrossRefGoogle Scholar
Yang, Z.. Positive solutions of a second-order integral boundary value problem. J. Math. Anal. Appl. 321 (2006), 751–765.CrossRefGoogle Scholar
Zeidler, E.. Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems (New York: Springer, 1986).CrossRefGoogle Scholar
Zhu, T.. Existence and uniqueness of positive solutions for fractional differential equations. Bound. Value Probl. 2019 (2019), 22.CrossRefGoogle Scholar