## Combien de chaussettes font la paire ? Les mathématiques by Rob Eastaway

By Rob Eastaway

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5 gives a fixed point theorem which can be called the fixed point theorem of domain expansion and compression. In this chapter, we also give some applications of the general results to nonlinear ordinary and partial differential equations. 7 ΤΙχο,ά Points ofi Monotone, Op&iatohA. (a) Fixed Points of Increasing Operators. Let P P. 1. 1 x2 (x19x2 (xl,xz (χ\9χ2 6 D) e ^) Αχλ to be decreasing if χλ decreasing if < Ax2» Αχλ implies « (xl,x2 and <_ be the partial or E. is said to be increasing if <^ Ax2, strictly increasing if and strongly increasing if Ax2 χ χ <_ x2 < x2 A : D -+ E Αχγ implies implies e D) Let D be a subset of dering defined by x E be a cone in real Banach space o in case P ^ φ.

O Since e e P, P. For any r > 0 there is a x e E9 we have and therefore ^ ^ a; - a—: e < x < — e . v — — v Thus x S E Μ β and e <ίΝ· It follows # equivalent to = #, Vxeff. 4) and conclusion (i), [ ·|| | ·|| . 1 shows that from a normal cone P , which is solid. e applications as in Chapter 2. 1. e e P5, Then Consider e > Θ. 3. It is easy to show that Let e(t) E Ξ 1, V t e Ω. 5) such that \\x\\ £ M||a;|| , V x e E = £°°(Ω). |*! 2. Let E = C[0,l] 0 £ £ £ 1}. , g(£) is the solution of the two-point boundary value problem x" = -1, *(0) = *(1) = 0.

5 λ. \\οΛλλο.. e-Norm be a cone in real Banach space = {x 6 E | there exists λ > 0 E e > Θ. 1. 1) becomes a normed linear space under the norm ll e d the Let cone V x e E . P x e E . e-norm of the element be normal. Then the following conclusions hold: (i) E is a Banach space with e-norm, and there exists a constant e M > 0 such t h a t \\x\\ £ M\X\\ f o r any x e E ; (ii) P (iii) e if = # P Il ·|| Proof. ΠP e o = E and the is equivalent to the original norm || · | . is solid and We first prove (i).