scholarly journals SIGN-CHANGING SOLUTIONS OF (e1, B)-LIMIT INCREASING OPERATOR EQUATION

2011 ◽  
Vol 53 (3) ◽  
pp. 535-554
Author(s):  
XU XIAN
Keyword(s):  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Index Method ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Increasing Operator ◽  
Sturm Liouville ◽  
Sign Changing Solutions ◽  
Clear Description

AbstractIn this paper, by using the fixed point index method first we obtain some existence and multiplicity results for sign-changing solutions of an (e1, B)-limit increasing operator equation. The main results can be applied to many non-linear boundary value problems to obtain the existence and multiplicity results for sign-changing solutions. We also give a clear description of locations of these sign-changing solutions through strict lower and upper solutions. As an example, in the last section we obtain some existence and multiplicity results for sign-changing solutions of some Sturm–Liouville differential boundary value problems.

scholarly journals Pairs of Nodal Solutions for a Class of Nonlinear Problems with One-sided Growth Conditions

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
Alberto Boscaggin ◽  
Fabio Zanolin
Keyword(s):  
Weight Function ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Growth Conditions ◽  
Second Order ◽  
Nodal Solutions ◽  
Multiplicity Results ◽  
Sturm Liouville ◽  
Growth Restrictions

AbstractBoundary value problems of Sturm-Liouville and periodic type for the second order nonlinear ODE uʺ + λf(t, u) = 0 are considered. Multiplicity results are obtained, for λ positive and large, under suitable growth restrictions on f(t, u) of superlinear type at u = 0 and of sublinear type at u = ∞. Only one-sided growth conditions are required. Applications are given to the equation uʺ + λq(t)f(u) = 0, allowing also a weight function q(t) of nonconstant sign.

scholarly journals Multiplicity results for asymmetric boundary value problems with indefinite weights

2004 ◽  
Vol 2004 (11) ◽  
pp. 957-979
Author(s):  
Francesca Dalbono
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Asymptotically Linear ◽  
Topological Methods ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Rotation Numbers ◽  
Indefinite Weights ◽  
Nodal Properties

We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the formu″+f(t,u)=0,u(0)=u(T)=0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.

scholarly journals Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 680
Author(s):  
Chan-Gyun Kim
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Multiplicity Results ◽  
Asymptotic Behaviors ◽  
Nonlocal Boundary Value Problems ◽  
Strong Singularity ◽  
Existence And Multiplicity ◽  
Caratheodory Condition

In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.

scholarly journals Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yulian An
Keyword(s):  
Boundary Value Problem ◽  
Weight Function ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Constant Sign ◽  
Dirichlet Boundary Value Problem ◽  
Multiplicity Results ◽  
Sign Changing Solutions ◽  
Exact Multiplicity

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problemsu″+a(t)f(u)=0,t∈(0,1),u(0)=0, andu(1)=0, wheref∈C(ℝ,ℝ)satisfiesf(0)=0and the limitsf∞=lim|s|→∞(f(s)/s),f0=lim|s|→0(f(s)/s)∈{0,∞}. Weight functiona(t)∈C1[0,1]satisfiesa(t)>0on[0,1].

scholarly journals Multiple positive solutions of Strum-Liouville equations with singularities

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Zenggui Wang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity

The existence of multiple positive solutions for Strum-Liouville boundary value problems with singularities is investigated. By applying a fixed point theorem of cone map, some existence and multiplicity results of positive solutions are derived. Our results improve and generalize those in some well-known results.

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