Existence results and iterative method for fully third order nonlinear integral boundary value problems

Quang A Dang; Quang Long Dang

Applications of Mathematics (2021)

  • Volume: 66, Issue: 5, page 657-672
  • ISSN: 0862-7940

Abstract

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We consider the boundary value problem u ' ' ' ( t ) = f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 g ( s ) u ( s ) d s , where f : [ 0 , 1 ] × ℝ 3 → ℝ + , g : [ 0 , 1 ] → ℝ + are continuous functions. The case when f = f ( u ( t ) ) was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.

How to cite

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Dang, Quang A, and Dang, Quang Long. "Existence results and iterative method for fully third order nonlinear integral boundary value problems." Applications of Mathematics 66.5 (2021): 657-672. <http://eudml.org/doc/297448>.

@article{Dang2021,
abstract = {We consider the boundary value problem \begin\{gather\} u^\{\prime \prime \prime \}(t)=f(t,u(t),u^\{\prime \}(t),u^\{\prime \prime \}(t)), \quad 0<t<1, \nonumber \\ u(0)=u^\{\prime \}(0)=0, \quad u(1)= \int \_0^1 g(s)u(s) \mathrm \{d\} s,\nonumber \end\{gather\} where $f\colon [0, 1] \times \mathbb \{R\}^3 \rightarrow \mathbb \{R\}^+$, $g\colon [0, 1] \rightarrow \mathbb \{R\}^+$ are continuous functions. The case when $f=f(u(t))$ was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.},
author = {Dang, Quang A, Dang, Quang Long},
journal = {Applications of Mathematics},
keywords = {fully third order nonlinear differential equation; integral boundary condition; positive solution; iterative method},
language = {eng},
number = {5},
pages = {657-672},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence results and iterative method for fully third order nonlinear integral boundary value problems},
url = {http://eudml.org/doc/297448},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Dang, Quang A
AU - Dang, Quang Long
TI - Existence results and iterative method for fully third order nonlinear integral boundary value problems
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 5
SP - 657
EP - 672
AB - We consider the boundary value problem \begin{gather} u^{\prime \prime \prime }(t)=f(t,u(t),u^{\prime }(t),u^{\prime \prime }(t)), \quad 0<t<1, \nonumber \\ u(0)=u^{\prime }(0)=0, \quad u(1)= \int _0^1 g(s)u(s) \mathrm {d} s,\nonumber \end{gather} where $f\colon [0, 1] \times \mathbb {R}^3 \rightarrow \mathbb {R}^+$, $g\colon [0, 1] \rightarrow \mathbb {R}^+$ are continuous functions. The case when $f=f(u(t))$ was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.
LA - eng
KW - fully third order nonlinear differential equation; integral boundary condition; positive solution; iterative method
UR - http://eudml.org/doc/297448
ER -

References

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