New methods for solving classical problems in the theory of nonlinear operator equations (solvability, multiple solutions, bifurcations, nonlinear resonance, potential methods, etc) are introduced and discussed. The general abstract theorems are illustrated by various applications to differential equations and boundary value problems. In particular, the problem on forced periodic oscillations is considered for equations arising in control theory.

Inhalt

Foreword.- 1: Norm estimates for solutions of integral-functional inequalities.- Distribution functions.- Estimates for solutions of the basic integral-functional inequality.- Proof of Theorem 2.2.- A second integral-functional inequality.- Proofs of Theorems 4.1-4.4.- Additional remarks.- 2: Two-sided estimates for nonlinearities.- Equations with self-adjoint and normal operators.- Solvability of equations in case the solutions do not admit a priori norm estimates.- Proofs of Theorems 8.1 and 8.2.- . Two-point boundary value problems.- . Forced oscillations in control systems.- 3: The use of arguments of leading eigenvalues.- . Use of the arguments principle.- . Joint norms of operators.- . Two-point boundary value problems (the nonquasilinear case).- . Forced oscillations in quasilinear systems.- . Forced oscillations in systems with delay.- . Remarks on forced oscillations in systems with control by derivatives.- . Extensions of the joint norm method.- 4: Weak nonlinear it ies.- . Equations with weak nonlinearities.- . Equations with normal operators.- . Auxiliary results.- . Equations with nonnormal operators.- . Integral equations with nonnegative kernels.- . Landesman-Lazer type theorems.- . Asymptotic bifurcation points.- 5: One-sided estimates for nonlinearities.- . Positive linear operators.- . Solvability of nonlinear operator equations with positive linear part.- . Equations with strictly positive operators.- . Two-point boundary value problems (the quasilinear case).- . Potential positivity of the periodic problem operator.- . Multiply-connected control systems.- . One-sided estimates in nonquasilinear problems.- . First order equations with variable coefficients.- . Variational methods.- References.- List of Symbols.