Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T21:22:30.420Z Has data issue: false hasContentIssue false
  • English
  • Français

On Nonlinear Boundary Conditions Involving Decomposable Linear Functionals

Part of: Integral, integro-differential, and pseudodifferential operators Boundary value problems Special classes of linear operators Nonlinear operators and their properties

Published online by Cambridge University Press:  27 October 2014

Christopher S. Goodrich
Christopher S. Goodrich*
Affiliation:
Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114, USA Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA, (cgood@prep.creighton.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the existence of a positive solution to boundary-value problems with non-local nonlinear boundary conditions, the archetypical example being −y″(t) = λf(t,y(t)), t ∈ (0, 1), y(0) = H(φ(y)), y(1) = 0. Here, H is a nonlinear function, λ > 0 is a parameter and φ is a linear functional that is realized as a Lebesgue—Stieltjes integral with signed measure. By requiring φ to decompose in a certain way, we show that this problem has at least one positive solution for each λ ∈ (0, λ0), for a number λ0 > 0 that is explicitly computable. We also give a separate result that holds for all λ > 0.

Keywords

positive solutionnonlinear boundary conditionnon-local boundary conditionfixed point indexeigenvalue

MSC classification

Primary: 34B09: Boundary eigenvalue problems 34B10: Nonlocal and multipoint boundary value problems 34B15: Nonlinear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Secondary: 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 47G10: Integral operators 47H11: Degree theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 421 - 439
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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

1.Cabada, A., An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl. 2011 (2011), 893753.Google Scholar
2.Goodrich, C. S., Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Computers Math. Applic. 61 (2011), 191–202.CrossRefGoogle Scholar
3.Goodrich, C. S., Existence of a positive solution to systems of differential equations of fractional order, Computers Math. Applic. 62 (2011), 1251–1268.CrossRefGoogle Scholar
4.Goodrich, C. S., Existence of a positive solution to a first-order p-Laplacian BVP on a time scale, Nonlin. Analysis 74 (2011), 1926–1936.CrossRefGoogle Scholar
5.Goodrich, C. S., The existence of a positive solution to a second-order p-Laplacian BVP on a time scale, Appl. Math. Lett. 25 (2012), 157–162.CrossRefGoogle Scholar
6.Goodrich, C. S., Positive solutions to boundary value problems with nonlinear boundary conditions, Nonlin. Analysis 75 (2012), 417–432.CrossRefGoogle Scholar
7.Goodrich, C. S., On a discrete fractional three-point boundary value problem, J. Diff. Eqns Applic. 18 (2012), 397–415.CrossRefGoogle Scholar
8.Goodrich, C. S., Nonlocal systems of BVPs with asymptotically superlinear boundary conditions, Commentat. Math. Univ. Carolinae 53 (2012), 79–97.Google Scholar
9.Goodrich, C. S., Nonlocal systems of BVPs with asymptotically sublinear boundary conditions, Appl. Analysis Disc. Math. 6 (2012), 174–193.CrossRefGoogle Scholar
10.Goodrich, C. S., On nonlocal BVPs with boundary conditions with asymptotically sub-linear or superlinear growth, Math. Nachr. 285 (2012), 1404–1421.CrossRefGoogle Scholar
11.Goodrich, C. S., On a first-order semipositone discrete fractional boundary value problem, Arch. Math. 99 (2012), 509–518.CrossRefGoogle Scholar
12.Goodrich, C. S., On nonlinear boundary conditions satisfying certain asymptotic behavior, Nonlin. Analysis 76 (2013), 58–67.CrossRefGoogle Scholar
13.Goodrich, C. S., On a nonlocal BVP with nonlinear boundary conditions, Results Math. 63 (2013), 1351–1364.CrossRefGoogle Scholar
14.Goodrich, C. S., Positive solutions to differential inclusions with nonlocal, nonlinear boundary conditions, Appl. Math. Computat. 219 (2013), 11071–11081.CrossRefGoogle Scholar
15.Graef, J. and Webb, J. R. L., Third order boundary value problems with nonlocal boundary conditions, Nonlin. Analysis 71 (2009), 1542–1551.CrossRefGoogle Scholar
16.Infante, G., Positive solutions of some nonlinear BVPs involving singularities and integral BCs, Discrete Contin. Dynam. Syst. S 1 (2008), 99–106.CrossRefGoogle Scholar
17.Infante, G., Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Analysis 12 (2008), 279–288.Google Scholar
18.Infante, G. and Pietramala, P., Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlin. Analysis 71 (2009), 1301–1310.CrossRefGoogle Scholar
19.Infante, G. and Pietramala, P., Eigenvalues and non-negative solutions of a system with nonlocal BCs, Nonlin. Stud. 16 (2009), 187–196.Google Scholar
20.Infante, G. and Pietramala, P., Perturbed Hammerstein integral inclusions with solutions that change sign, Commentat. Math. Univ. Carolinae 50 (2009), 591–605.Google Scholar
21.Infante, G. and Pietramala, P., A third order boundary value problem subject to nonlinear boundary conditions, Math. Bohem. 135 (2010), 113–121.CrossRefGoogle Scholar
22.Infante, G. and Pietramala, P., Multiple non-negative solutions of systems with coupled nonlinear BCs, Math. Meth. Appl. Sci. (2013), DOI:10.1002/mma.2957.Google Scholar
23.Infante, G. and Webb, J. R. L., Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc. 49 (2006), 637–656.CrossRefGoogle Scholar
24.Infante, G., Minhós, F. and Pietramala, P., Non-negative solutions of systems of ODEs with coupled boundary conditions, Commun. Nonlin. Sci. Numer. Simulation 17 (2012), 4952–4960.CrossRefGoogle Scholar
25.Jankowski, T., Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions, Nonlin. Analysis 75 (2012), 913–923.CrossRefGoogle Scholar
26.Kang, P. and Wei, Z., Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations, Nonlin. Analysis 70 (2009), 444–451.CrossRefGoogle Scholar
27.Kelley, W. G. and Peterson, A. C., The theory of differential equations: classical and qualitative (Prentice Hall, Englewood Cliffs, NJ, 2004).Google Scholar
28.Lan, K. Q., Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. (2) 63 (2001), 690–704.CrossRefGoogle Scholar
29.Pietramala, P., A note on a beam equation with nonlinear boundary conditions, Bound. Value Probl. 2011 (2011), 376782.Google Scholar
30.Webb, J. R. L., Nonlocal conjugate type boundary value problems of higher order, Nonlin. Analysis 71 (2009), 1933–1940.CrossRefGoogle Scholar
31.Webb, J. R. L., Solutions of nonlinear equations in cones and positive linear operators, J. Lond. Math. Soc. (2) 82 (2010), 420–436.CrossRefGoogle Scholar
32.Webb, J. R. L., Remarks on a non-local boundary value problem, Nonlin. Analysis 72 (2010), 1075–1077.CrossRefGoogle Scholar
33.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
34.Webb, J. R. L. and Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlin. Diff. Eqns Applic. 15 (2008), 45–67.CrossRefGoogle Scholar
35.Webb, J. R. L. and Infante, G., Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc. (2) 79 (2009), 238–258.CrossRefGoogle Scholar
36.Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlin. Analysis 62 (2005), 1251–1265.CrossRefGoogle Scholar
37.Yang, Z., Positive solutions of a second-order integral boundary value problem, J. Math. Analysis Applic. 321 (2006), 751–765.CrossRefGoogle Scholar
38.Yang, Z., Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlin. Analysis 65 (2006), 1489–1511.CrossRefGoogle Scholar
39.Yang, Z., Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions, Nonlin. Analysis 68 (2008), 216–225.CrossRefGoogle Scholar
40.Zeidler, E., Nonlinear functional analysis and its applications, I, Fixed-point theorems (Springer, 1986).Google Scholar