scholarly journals Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hairong Lian ◽  
Junfang Zhao
Keyword(s):  
Differential Equation ◽  
Solution Method ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Unbounded Solutions ◽  
Sturm Liouville ◽  
Infinite Intervals ◽  
Lower Solution Method

We generalize the unbounded upper and lower solution method to a third-order ordinary differential equation on the half line subject to the Sturm-Liouville boundary conditions. By using such techniques and the Schäuder fixed point theorem, some criteria are presented for the existence of solutions and positive ones to the problem discussed.

scholarly journals Existence of Solutions of Nonlinear Mixed Two-Point Boundary Value Problems for Third-Order Nonlinear Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Yongxin Gao ◽  
Fengqin Wang
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Nonlinear Differential Equation ◽  
Solution Method ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
Point Boundary ◽  
Third Order ◽  
Upper And Lower Solution ◽  
Lower Solution Method

The authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation  y′′′=f(x,y,y′,y′′),  y′(b)=h(y′(a)),  p(y(a),y(b),y′(a),y′(b))=0,  g(y(a),y(b),y′(a),y′(b),y′′(a),y′′(b))=0. Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.

scholarly journals The existence of solutions for Sturm–Liouville differential equation with random impulses and boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zihan Li ◽  
Xiao-Bao Shu ◽  
Tengyuan Miao
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Lower Solution ◽  
Iterative Sequence ◽  
Random Impulses ◽  
Sturm Liouville

AbstractIn this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

scholarly journals Existence of solutions for high order ordinary differential equations with some periodic-type boundary condition

2010 ◽  
Vol 41 (3) ◽  
pp. 293-301
Author(s):  
S. C. Jhuang ◽  
W. C. Lian ◽  
S. P. Wang ◽  
F. H. Wong
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Existence Theorem ◽  
Solution Method ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
High Order ◽  
Lower Solution Method ◽  
Type Boundary ◽  
Type Boundary Condition

We consider the following high order periodic-type boundary value problem \[ \lefteqn{(PBVP)}  \left\{\begin{array}{lll}  (E)~u^{(n)}(t)= f(t,u(t),u^{(1)}(t), \cdots, u^{(n-2)}(t), u^{(n-1)}(t))~\mbox{for}~t\in (0,T) \\  (PBC)~\left\{\begin{array}{lll}  u^{(i)}(0)=0,~0\leq i\leq n-3,\\  u^{(n-2)}(0)= u^{(n-2)}(T),\\  u^{(n-1)}(0)= u^{(n-1)}(T),  \end{array}\right.  \end{array}\right. \] where $f\in C([0,T]\times\mathbb{R}^n,\mathbb{R})$, $n\geq 2$ and satisfies the so-called Nagumo's condition. In this article, we will use a general upper and lower solution method to establish an existence theorem for solutions of $(PBVP)$.

scholarly journals Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Boundary Value Problems ◽  
Solution Method ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Lower Solution ◽  
Half Line ◽  
Upper And Lower Solution ◽  
Unbounded Solutions ◽  
Lower Solution Method ◽  
Fractional Boundary Value Problems

We establish the existence of unbounded solutions for nonlinear fractional boundary value problems on the half-line. By the upper and lower solution method technique, sufficient conditions for the existence of solutions for the fractional boundary value problems are established. An example is presented to illustrate our main result.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities

2021 ◽  
Vol 71 (6) ◽  
pp. 1459-1470
Author(s):  
Kun Li ◽  
Yanli He
Keyword(s):  
Traveling Wave ◽  
Solution Method ◽  
Traveling Wave Solutions ◽  
Lower Solution ◽  
Cooperative System ◽  
Lattice Systems ◽  
Dimensional Lattice ◽  
Wave Solutions ◽  
Higher Dimensional ◽  
Lower Solution Method

Abstract In this paper, we are concerned with the existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities. By using the upper and lower solution method and Schauder’s fixed point theorem, we establish the existence of traveling wave solutions. To illustrate our results, the existence of traveling wave solutions for a nonlocal delayed higher-dimensional lattice cooperative system with two species are considered.

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