Nagumo viability theorem. Revisited

We consider the nonlinear ordinary differential equation u ′ ( t ) = f ( t , u ( t ) ) + h ( t , u ( t ) ) , where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and loc...

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Published inNonlinear analysis Vol. 64; no. 9; pp. 2043 - 2052
Main Author Vrabie, Ioan I.
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 01.05.2006
Elsevier
Subjects
b-compact function
Exact sciences and technology
Lipschitz function
Mathematical analysis
Mathematics
Operator theory
Ordinary differential equations
Partial differential equations
Physics
Pseudoparabolic equation
Sciences and techniques of general use
Slow fluids
Statistical physics, thermodynamics, and nonlinear dynamical systems
Tangency condition
Thermodynamics
Viable set
Tangency condition
Slow fluids
Viable set
Pseudoparabolic equation
secondary 35K70
b-compact function
Lipschitz function
80A20
Primary 34G20
47J35
b compact function
Slow fluid
Viability
80A20 Viable set
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Abstract We consider the nonlinear ordinary differential equation u ′ ( t ) = f ( t , u ( t ) ) + h ( t , u ( t ) ) , where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each ( τ , ξ ) ∈ I × K there exists T > τ such that the equation above has at least one solution u : [ τ , T ] → K is the tangency condition below lim inf s ↓ 0 1 s d ( ξ + s [ f ( τ , ξ ) + h ( τ , ξ ) ] ; K ) = 0 for each ( τ , ξ ) ∈ I × K . As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations.
AbstractList We consider the nonlinear ordinary differential equation u'(t)=f(t,u(t))+h(t,u(t)), where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f:IXK- > X a compact function, and h:IXK- > X continuous on IXK and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each (tau,&D*x)IXK there exists T > tau such that the equation above has at least one solution u:[tau,T]- > K is the tangency condition below
We consider the nonlinear ordinary differential equation u ′ ( t ) = f ( t , u ( t ) ) + h ( t , u ( t ) ) , where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each ( τ , ξ ) ∈ I × K there exists T > τ such that the equation above has at least one solution u : [ τ , T ] → K is the tangency condition below lim inf s ↓ 0 1 s d ( ξ + s [ f ( τ , ξ ) + h ( τ , ξ ) ] ; K ) = 0 for each ( τ , ξ ) ∈ I × K . As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations.
Author Vrabie, Ioan I.
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10.1016/S1874-5725(05)80005-7
10.1007/BF01695169
10.1007/978-1-4612-0091-8
10.1016/S0022-247X(02)00200-7
10.1002/cpa.3160230211
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Issue 9
Keywords Tangency condition
Slow fluids
Viable set
Pseudoparabolic equation
secondary 35K70
b-compact function
Lipschitz function
80A20
Primary 34G20
47J35
b compact function
Slow fluid
Viability
80A20 Viable set
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Snippet We consider the nonlinear ordinary differential equation u ′ ( t ) = f ( t , u ( t ) ) + h ( t , u ( t ) ) , where X is a real Banach space, I is a nonempty...
We consider the nonlinear ordinary differential equation u'(t)=f(t,u(t))+h(t,u(t)), where X is a real Banach space, I is a nonempty and open interval, K a...
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StartPage 2043
SubjectTerms b-compact function
Exact sciences and technology
Lipschitz function
Mathematical analysis
Mathematics
Operator theory
Ordinary differential equations
Partial differential equations
Physics
Pseudoparabolic equation
Sciences and techniques of general use
Slow fluids
Statistical physics, thermodynamics, and nonlinear dynamical systems
Tangency condition
Thermodynamics
Viable set
Title Nagumo viability theorem. Revisited
URI https://dx.doi.org/10.1016/j.na.2005.07.037
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