The Existence of Positive Solutions for Singular Semi-Positive Non-Local Boundary Value Problems with Nonlinear Term Depending on Derivatives

2018 ◽  
Vol 07 (08) ◽  
pp. 1028-1039
Author(s):  
宇 赵
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Local Boundary ◽  
Existence Of Positive Solutions ◽  
Non Local

scholarly journals NONLINEAR NON-LOCAL BOUNDARY-VALUE PROBLEMS AND PERTURBED HAMMERSTEIN INTEGRAL EQUATIONS

2006 ◽  
Vol 49 (3) ◽  
pp. 637-656 ◽  
Author(s):  
Gennaro Infante ◽  
J. R. L. Webb
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Positive Solutions ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Compact Set ◽  
Local Boundary ◽  
Flow Problems ◽  
Hammerstein Integral Equations ◽  
Non Local

AbstractMotivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

scholarly journals Positive solutions of some nonlocal boundary value problems

2003 ◽  
Vol 2003 (18) ◽  
pp. 1047-1060 ◽  
Author(s):  
Gennaro Infante ◽  
J. R. L. Webb
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Point Boundary ◽  
General Behaviour ◽  
Nonlocal Boundary Value Problems ◽  
Existence Of Positive Solutions

We establish the existence of positive solutions of somem-point boundary value problems under weaker assumptions than previously employed. In particular, we do not require all the parameters occurring in the boundary conditions to be positive. Our results allow more general behaviour for the nonlinear term than being either sub- or superlinear.

scholarly journals Positive Solutions of a Singular Positone and Semipositone Boundary Value Problems for Fourth-Order Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Chengjun Yuan
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Difference Equations ◽  
Fixed Point Theorems ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Order Difference ◽  
Existence Of Positive Solutions

This paper studies the boundary value problems for the fourth-order nonlinear singular difference equationsΔ4u(i−2)=λα(i)f(i,u(i)),i∈[2,T+2],u(0)=u(1)=0,u(T+3)=u(T+4)=0. We show the existence of positive solutions for positone and semipositone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone.

scholarly journals Existence of positive solutions for nonlinear third-order m-point boundary value problems on time scales

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 925-935
Author(s):  
Ilkay Karacaa ◽  
Fatma Tokmaka
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Time Scales ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Growth Conditions ◽  
Point Boundary ◽  
Double Positive ◽  
Third Order ◽  
Existence Of Positive Solutions

In this paper, we investigate the existence of double positive solutions for nonlinear third-order m-point boundary value problems with p-Laplacian on time scales. By using double fixed point theorem, we establish results on the existence of two positive solutions with suitable growth conditions imposed on the nonlinear term. As an application, we give an example to demonstrate our main result.

scholarly journals MULTIPLE POSITIVE SOLUTIONS OF RESONANT AND NON-RESONANT NON-LOCAL FOURTH-ORDER BOUNDARY VALUE PROBLEMS

2011 ◽  
Vol 54 (1) ◽  
pp. 225-240 ◽  
Author(s):  
J. R. L. WEBB ◽  
M. ZIMA
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Local Boundary ◽  
Existence And Multiplicity ◽  
Resonant Problem ◽  
Non Local ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractWe study the existence of positive solutions for equations of the form where 0 < ω < π, subject to various non-local boundary conditions defined in terms of the Riemann–Stieltjes integrals. We prove the existence and multiplicity of positive solutions for these boundary value problems in both resonant and non-resonant cases. We discuss the resonant case by making a shift and considering an equivalent non-resonant problem.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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