Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals

2009 ◽  
Vol 70 (7) ◽  
pp. 2627-2633 ◽  
Author(s):  
Hairong Lian ◽  
Peiguang Wang ◽  
Weigao Ge
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Sturm Liouville ◽  
Infinite Intervals

scholarly journals An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sh. Rezapour ◽  
B. Azzaoui ◽  
B. Tellab ◽  
S. Etemad ◽  
H. P. Masiha
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Caputo Fractional Derivatives ◽  
Control Functions ◽  
Existence Of Positive Solutions ◽  
Semilinear Differential Equation ◽  
The Given ◽  
Existence Criteria

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.

Quasilinearization and boundary value problems at resonance

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kareem Alanazi ◽  
Meshal Alshammari ◽  
Paul Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Uniqueness Of Solutions ◽  
Shift Method ◽  
At Resonance ◽  
Problem At Resonance

Abstract A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Factorization Ladders and Eigenfunctions

1949 ◽  
Vol 1 (4) ◽  
pp. 379-396 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Manufacturing Process ◽  
Boundary Value ◽  
Factorization Method ◽  
Sturm Liouville ◽  
The Subject

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

scholarly journals A monotone iterative method for boundary value problems of parametric differential equations

2001 ◽  
Vol 14 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Xinzhi Liu ◽  
Farzana A. McRae
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Extremal Solutions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Monotone Sequences

This paper studies boundary value problems for parametric differential equations. By using the method of upper and lower solutions, monotone sequences are constructed and proved to converge to the extremal solutions of the boundary value problem.

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