### Coercive functionals and their relationship to multiplicity of solution to nonlocal boundary value problems

#### Abstract

#### Keywords

#### References

D.R. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions, J. Math. Anal. Appl. 408 (2013), 318–323.

D.R. Anderson and J. Hoffacker, Existence of solutions for a cantilever beam problem, J. Math. Anal. Appl. 323 (2006), 958–973.

A. Cabada, G. Infante and F.A.F. Tojo, Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications, Topol. Methods Nonlinear Anal. 47 (2016), 265–287.

A. Cabada, G. Infante and F.A.F. Tojo, Nonlinear perturbed integral equations related to nonlocal boundary value problems, Fixed Point Theory 19 (2018), 65–92.

A. Cabada and S. Tersian, Multiplicity of solutions to a two-point boundary value problem for a fourth-order equation, Appl. Math. Comput. 219 (2013), 5261–5267.

F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl. 33 (2017), 317–347.

F. Cianciaruso and P. Pietramala, Multiple positive solutions of a (p1 , p2 )-Laplacian system with nonlinear BCs, Bound. Value Probl. 163 (2015), 18 pp.

D.R. Dunninger and H. Wang, Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions, Ann. Polon. Math. 69 (1998), 155–165.

C.S. Goodrich, Positive solutions to boundary value problems with nonlinear boundary conditions, Nonlinear Anal. 75 (2012), 417–432.

C.S. Goodrich, On nonlinear boundary conditions involving decomposable linear functionals, Proc. Edinb. Math. Soc. (2) 58 (2015), 421–439.

C.S. Goodrich, Coercive nonlocal elements in fractional differential equations, Positivity 21 (2017), 377–394.

C.S. Goodrich, A new coercivity condition applied to semipositone integral equations with nonpositive, unbounded nonlinearities and applications to nonlocal BVPs, J. Fixed Point Theory Appl. 19 (2017), 1905–1938.

C.S. Goodrich, The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems, J. Fixed Point Theory Appl. 19 (2017), 2629–2646.

C.S. Goodrich, New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green’s function, J. Differential Equations 264 (2018), 236–262.

C.S. Goodrich, Radially symmetric solutions of elliptic PDEs with uniformly negative weight, Ann. Mat. Pura Appl. (4) 197 (2018), 1585–1611.

J.R. Graef, L. Kong and H. Wang, A periodic boundary value problem with vanishing Green’s function, Appl. Math. Leet. 21 (2008), 176–180.

J. Graef and J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009), 1542–1551.

G. Infante, Nonlocal boundary value problems with two nonlinear boundary conditions, Commun. Appl. Anal. 12 (2008), 279–288.

G. Infante, F. Minhós and P. Pietramala, Non-negative solutions of systems of ODEs with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4952–4960.

G. Infante and P. Pietramala, Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal. 71 (2009), 1301–1310.

G. Infante and P. Pietramala, Eigenvalues and non-negative solutions of a system with nonlocal BCs, Nonlinear Stud. 16 (2009), 187–196.

G. Infante and P. Pietramala, A cantilever equation with nonlinear boundary conditions, Electron. J. Qual. Theory Differ. Equ. 2009, Special Edition I, No. 15, 14 pp.

G. Infante and P. Pietramala, Perturbed Hammerstein integral inclusions with solutions that change sign, Comment. Math. Univ. Carolin. 50 (2009), 591–605.

G. Infante and P. Pietramala, A third order boundary value problem subject to nonlinear boundary conditions, Math. Bohem. 135 (2010), 113–121.

G. Infante and P. Pietramala, Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions, Math. Methods Appl. Sci. 37 (2014), 2080–2090.

G. Infante and P. Pietramala, Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains, NoDEA Nonlinear Differential Equations Appl. 22 (2015), 979–1003.

G. Infante, P. Pietramala and M. Tenuta, Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2245–2251.

T. Jankowski, Positive solutions to fractional differential equations involving Stieltjes integral conditions, Appl. Math. Comput. 241 (2014), 200–213.

G.L. Karakostas, Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differential Equations (2014), no. 71, 17 pp.

G.L. Karakostas and P.Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal. 19 (2002), 109–121.

G.L. Karakostas and P.Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations (2002), no. 30, 17 pp.

K.Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. (2) 63 (2001), 690–704.

K.Q. Lan and W. Lin, Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations, J. Lond. Math. Soc. (2) 83 (2011), 449–469.

R.W. Leggett and L.R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688.

Y. Li, Existence of positive solutions for the cantilever beam equations with fully nonlinear terms, Nonlinear Anal. Real World Appl. 27 (2016), 221–237.

X. Liu and J. Wu, Positive solutions for a Hammerstein integral equation with a parameter, Appl. Math. Lett. 22 (2009), 490–494.

R. Ma and C. Zhong, Existence of positive solutions for integral equations with vanishing kernels, Commun. Appl. Anal. 15 (2011), 529–538.

M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1908), 1–95.

Y. Song, A nonlinear boundary value problem for fourth-order elastic beam equations, Bound. Value Probl. 191 (2014), 11 pp.

J.R.L. Webb, Boundary value problems with vanishing Green’s function, Commun. Appl. Anal. 13 (2009), 587–595.

J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc. (2) 74 (2006), 673–693.

J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 45–67.

J.R.L. Webb and G. Infante, Nonlocal boundary value problems of arbitrary order, J. Lond. Math. Soc. (2) 79 (2009), 238–258.

W.M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 48 (1942), 692–704.

J. Xu and Z. Yang, Positive solutions for a system of nonlinear Hammerstein integral equations and applications, J. Integral Equations Appl. 24 (2012), 131–147.

Z. Yang, Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal. 62 (2005), 1251–1265.

Z. Yang, Positive solutions of a second-order integral boundary value problem, J. Math. Anal. Appl. 321 (2006), 751–765.

Z. Yang, Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear Anal. 65 (2006), 1489–1511.

Z. Yang, Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions, Nonlinear Anal. 68 (2008), 216–225.

Z. Yang, Positive solutions for a system of nonlinear Hammerstein integral equations and applications, Appl. Math. Comput. 218 (2012), 11138–11150.

Q. Yao, Monotonically iterative method of nonlinear cantilever beam equations, Appl. Math. Comput. 205 (2008), 432–437.

E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer, New York, 1986.

### Refbacks

- There are currently no refbacks.