Topological Methods in the Theory of Nonlinear Integral Equations

a₁ assume B₁ ball Banach space bifurcation points boundary bounded bounded set Chapter characteristic value completely continuous linear completely continuous operator completely continuous vector completes the proof conditions of Theorem cone consider continuous branch continuous functional continuous linear operator continuous vector field contraction mapping converge corresponding defined denote E₁ eigen eigenfunctions eigenvalue eigenvectors element equal equation existence theorems F₁ finite-dimensional fixed point follows Fréchet derivative functional F G₁ gradient H₁ H₂ Hammerstein Hilbert space homotopic implies integral operator K₁ kernel K(s L₁ Lemma method n-dimensional simplexes non-negative nonlinear operator norm null vectors operator f P₁ positive definite kernel positive number proposition prove radius rotation S₁ satisfies the condition sequence solution space LP spectrum sphere sub-space Suppose symmetric T₁ Theorem 2.1 topological weakly continuous functional whence zero Фо