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Volterra Integral Equations, Differential Equations and Flow-invariance
| Content Provider | Scilit |
|---|---|
| Author | Corduneanu, C. Sandberg, I. |
| Copyright Year | 2000 |
| Description | Let X be a Banach space of norm || • || and le t / : [a,b] x X —• Xbe a continuous function. A solution of class C1 ([«,£]) to the Cauchy problem (CP) in X (1.1) satisfies the Volterra integral equation (1.2) and vice versa (i.e., (1.1) and (1.2) are equivalent in X). In other words, the (ODE) x' =f(t,x) and the Volterra equation (1.2) have the same set of solutions in X. This is not true in the case of a general Volterra integral equation differential equation in C([a,b],X) (Larrieu [3]) (1.4) Io) If x is a solution of the (VIE) (1.3) on [to,c] then the function y : [to,c] —> C([a,b];X) given by (2.1) is a solution of (I A). 2°) If y is a solution 0/(1.4), i.e., if Io) The key fact here is the property (2.3) Therefore y given by (2.1) satisfies (2.4) for all s e [to,c] so (y(t))(t) = x(t) as x is a solution of (1.3). Thus ^(¿)(i) = x(t) so (2.4) can be rewritten as (2.5) This means that y satisfies (2.2), i.e., (1.4). 2o) Viceversa, if y satisfies (2.2), i.e., if (2.5) holds, then (2.6) 3 Flow-invariance Problems In spite of the fact that a (VIE) in X can be reduced to an (ODE) in C([a, b],X), the flow-invariance of a set D with respect to a (VIE) is not a natural problem. Let us prove why. Consider the particular case of (1.3) with/(¿) = xo e D. In view of Definition 1.1, it would be very legitimate to say that D is flow-invariant with respect to (VIE) (3.1) is for every to e [a, b) and XQ e D, the corresponding solution x of (3.1) remains in D for all t > to in the domain of x. This leads to the following necessary condition for D to be a (FIS) of (3.1). Book Name: Volterra Equations and Applications |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2004-0-03597-2&isbn=9780429177927&doi=10.1201/9781482287424-36&format=pdf |
| Ending Page | 316 |
| Page Count | 8 |
| Starting Page | 309 |
| DOI | 10.1201/9781482287424-36 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2000-01-10 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Volterra Equations and Applications Operations Research and Management Science Function Volterra Flow Invariance Satisfies Equation Differential X Is a Solution |
| Content Type | Text |
| Resource Type | Chapter |
