On the Cesari fixed point method in a Banach space

<p>In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.</p> <p>The following is my formulation of the Cesari fixed point method:</p> <p>Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г. </p> <p>Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:</p> <p>(i) Py = PWy.</p> <p>(ii) y = (P + (I - P)W)y.</p> <p><u>Definition</u>. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:</p> <p>(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).</p> <p>(2) The function y just defined is continuous from PГ into B.</p> <p>(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.</p> <p><u>Definition</u>. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).</p> <p>The three theorems of this thesis can now be easily stated.</p> <p><u>Theorem 1</u> (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.</p> <p><u>Theorem 2</u>. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:</p> <p>(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖</p> <p>(2)P₂Г is convex.</p> <p>Then i(Г, W, P₁) = i(Г, W, P₂).</p> <p><u>Theorem 3.</u> If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).</p> <p>Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.</p>