## Three positive fixed points of nonlinear operators on ordered Banach spaces.(English)Zbl 1005.47051

The authors generalize the triple fixed-point theorem of Leggett and Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. As an application of the abstract result, the authors prove the existence of three positive symmetric solutions of the discrete second-order nonlinear conjugate boundary value problem $\Delta^2 x(t-1)+f(x(t))=0, \text{for all} t\in [a+1,b+1],$
$x(a)=0=x(b+2),$ where $$f: \mathbb R\to \mathbb R$$ is continuous and $$f$$ is nonnegative for $$x\geq 0.$$

### MSC:

 47H10 Fixed-point theorems 34B15 Nonlinear boundary value problems for ordinary differential equations 39A05 General theory of difference equations 47N20 Applications of operator theory to differential and integral equations 65J15 Numerical solutions to equations with nonlinear operators 65Q05 Numerical methods for functional equations (MSC2000)
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### References:

  Zeidler, E., Nonlinear functional analysis and its applications I: fixed-point theorems, (1993), Springer-Verlag New York  Agarwal, R.P.; Wong, P.J.Y.; O’Regan, D., Positive solutions of differential, difference, & integral equations, (1999), Kluwer Academic Boston · Zbl 0923.39002  Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operations on ordered Banach spaces, Indiana university mathematics journal, 28, 673-688, (1979) · Zbl 0421.47033  Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040  Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic press San Diego · Zbl 0661.47045  Kelley, W.G.; Peterson, A.C., Difference equations: an introduction with applications, (1991), Academic Press San Diego · Zbl 0733.39001  Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian systems: difference equations, continued fractions and Riccati equations, (1996), Kluwer Academic Boston · Zbl 0860.39001  Avery, R.I., Multiple positive solutions to boundary value problems, Dissertation, (1997), University of Nebraska Lincoln  Avery, R.I., Three positive solutions of a discrete second order conjugate problem, Panamerican mathematical journal, 8, 39-55, (1998)  Avery, R.I., A generalization of the Leggett-Williams fixed point theorem, MSR hot-line, 3, 7, 9-14, (1999) · Zbl 0965.47038  Avery, R.I.; Henderson, J., Three symmetric positive solutions for a second order boundary value problems, Appl. math. lett., 13, 3, 1-7, (2000) · Zbl 0961.34014  Avery, R.I.; Peterson, A.C., Multiple positive solutions of a discrete second order conjugate problem, Panamerican mathematical journal, 8, 1-12, (1998) · Zbl 0959.39006  J. Henderson, Multiple symmetric positive solutions for discrete Lidstone boundary value problems, Dynamics of Continuous, Discrete & Impulsive Systems (to appear). · Zbl 0969.39003  J. Henderson and H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proceedings of the American Mathematical Society (to appear). · Zbl 0949.34016
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