Kenneth Kuttler

2700 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

provided n is large enough and so ψ has a high enough power a contraction map. Hence,if one begins with w ∈ L2 (Ω,C ([0,T ] ;W )) , the sequence of iterates {ψnw}∞

n=1 must con-verge to some fixed point u in L2 (Ω,C ([0,T ] ,W )). This u is progressively measurablesince each of the iterates is progressively measurable. This fixed point is the solution to theintegral equation.

79.3 An ExampleA model for a nonlinear beam is the Gao beam in which the transverse vibrations satisfyan equation of the form

wtt +wxxxx +αwtxxxx +(1−w2x)wxx = f

To this, we can add boundary conditions and initial conditions

w(t,0) = wx (t,0) = w(t,1) = wx (t,1) = 0,w(0,x) = w0 (x) ,wt (0,x) = v0 (x)

w0 ∈ H20 ((0,1)) =V,v0 ∈ L2 ((0,1)) = H

Also let W be the closure of V in H1 ((0,1)) so W = H10 ((0,1)). Let H denote L2 ((0,1)).

These conditions correspond to a clamped beam. An equivalent norm on V is ∥u∥V =|uxx|H . Our theory allows us to include coefficient functions which are progressively mea-surable but in the interest of simplicity, this technical complication will be omitted. Also,it will follow that there is a pointwise estimate for wx (t,x) thanks to Sobolev embeddingtheorems and routine arguments involving the term wxxxx. Therefore, in the case of a de-terministic Gao beam, there is no loss of generality in replacing w2

x with q(wx) where qis a bounded and Lipschitz continuous truncation of r→ r2. To begin with, we make thisapproximation. Let Q′ (r) = q(r) and Q(0) = 0. Thus Q will have linear growth for |r|large and is an odd function. Define operators L,N mapping V to V ′.

⟨Lw,u⟩=∫ 1

0wxxuxxdx, ⟨Nw,u⟩=

∫ 1

0(Q(wx)−wx)uxdx

Then in terms of these operators, and writing in terms of the velocity v, we obtain thefollowing abstract formulation.

v′+αLv+Lw+Nw = f ∈ V ′, w(t) = w0 +∫ t

0vds, v(0) = v0.

In terms of an integral equation, this would be

v(t)− v0 +α

∫ t

0Lv(s)ds+

∫ t

0Lwds+

∫ t

0Nwds =

∫ t

0f ds

w(t) = w0 +∫ t

0vds (79.3.35)

Then the abstract version of a stochastic equation is

v(t)− v0 +α

∫ t

0Lv(s)ds+

∫ t

0Lwds+

∫ t

0Nwds =

∫ t

0f ds+

∫ t

0σ (v)dW(79.3.36)

w(t) = w0 +∫ t

0vds (79.3.37)

2700 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALSprovided n is large enough and so y has a high enough power a contraction map. Hence,if one begins with w € L? (Q,C((0,T];W)), the sequence of iterates {yw"w}"_, must con-verge to some fixed point u in L*(Q,C([0,7],W)). This u is progressively measurablesince each of the iterates is progressively measurable. This fixed point is the solution to theintegral equation. Jj79.3, An ExampleA model for a nonlinear beam is the Gao beam in which the transverse vibrations satisfyan equation of the formWet + Weexx + OWpxxexx + (1 _ We) Wry = ftTo this, we can add boundary conditions and initial conditionsw(t,0) = w,(t,0) =w(t,1) =wy(t,1) =0,w(0,x) = wo (x), wr (0,x) = vo (x)wo € Hg((0,1))=V,v0 €L7((0,1)) =HAlso let W be the closure of V in H! ((0,1)) so W = H¢ ((0,1)). Let H denote L? ((0,1)).These conditions correspond to a clamped beam. An equivalent norm on V is ||u||y =|Uxx|;,- Our theory allows us to include coefficient functions which are progressively mea-surable but in the interest of simplicity, this technical complication will be omitted. Also,it will follow that there is a pointwise estimate for w, (t,x) thanks to Sobolev embeddingtheorems and routine arguments involving the term w,,,,. Therefore, in the case of a de-terministic Gao beam, there is no loss of generality in replacing w2 with g(w,) where gis a bounded and Lipschitz continuous truncation of r > r?. To begin with, we make thisapproximation. Let Q’(r) = g(r) and Q(0) = 0. Thus Q will have linear growth for |r|large and is an odd function. Define operators L,N mapping V to V’.(Lw,u) = [ Wyxlydx, (Nw, u) = [ (Q (wy) — Wy) Uy.dxThen in terms of these operators, and writing in terms of the velocity v, we obtain thefollowing abstract formulation.v+aLlv+Lw+Nw= fev’, w(t) = wo+ [ vas, v(0) = vo.In terms of an integral equation, this would bev(t) =v +a Lv(s)as+ | Lwds+ [ nwas = [saw(t) = wot [ vds_ (79.3.35)Then the abstract version of a stochastic equation isv(t) -ro taf Lv(s)as+ |’ Lwas+ [ nwas = [ tas+ [oyae79.3.36tw(t) = wo [ vds (79.3.37)0