A Three Solutions Theorem for Nonlinear Operator Equations in Ordered Banach Spaces

Positivity ◽  
2006 ◽  
Vol 10 (4) ◽  
pp. 647-664
Author(s):  
Xu Xian ◽  
Donal O'Regan
Keyword(s):  
Banach Spaces ◽  
Nonlinear Operator ◽  
Three Solutions ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Three Solutions Theorem ◽  
Ordered Banach Spaces

scholarly journals Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yonghui Ling ◽  
Xiubin Xu ◽  
Shaohua Yu
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Nonlinear Operator ◽  
Convergence Criterion ◽  
Second Derivative ◽  
Operator Equations ◽  
Convergence Ball ◽  
Nonlinear Operator Equations ◽  
Steffensen’S Method ◽  
Convergence Problems

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

scholarly journals Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Mingliang Song ◽  
Shuyuan Mei
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Operator ◽  
Adjoint Operator ◽  
Index Theory ◽  
Second Order ◽  
Three Solutions ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Point Reduction

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

scholarly journals Iterations converging to distinct solutions of some nonlinear operator equations in Banach space

1986 ◽  
Vol 9 (3) ◽  
pp. 583-587
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Nonlinear Operator ◽  
Operator Equations ◽  
Nonlinear Operator Equations

We examine the solvability of multilinear equations of the formMk(x,x,…,x)−k   times−=y,   k=2,3,…whereMkis ak-linear operator on a Banach spaceXandy∈Xis fixed.

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