![]() ![]() In addition, by making consequent use of the fixed point index for compact maps, short and simple proofs are obtained for most of the “classical” results contained in M. This paper presents in a unified manner most of the recent work in this field. But, of course, the abstract techniques and results of this paper apply also to a variety of other problems which are not considered here. In order to demonstrate the importance of the abstract results, there are given some nontrivial applications to nonlinear elliptic boundary value problems. Moreover, solvability and bifurcation problems for fixed point equations depending nonlinearly on a real parameter are investigated. By means of iterative techniques and by using topological tools, fixed point theorems for completely continuous maps in ordered Banach spaces are deduced, and particular attention is paid to the derivation of multiplicity results. This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. Our main results are applied to establish the existence and multiplicity of positive symmetric solutions for an elliptic system in an annulus.ġ Preliminaries.- Sobolev spaces and embedding theorems.- Critical point.- Cone and partial order.- Brouwer Degree.- Compact map and Leray-Schauder Degree.- Fredholm operators.- Fixed point index.- Banach's Contract Theorem, Implicit Functions Theorem.- Krein-Rutman theorem.- Bifurcation theory.- Rearrangements of sets and functions.- Genus and Category.- Maximum principles and symmetry of solution.- Comparison theorems.- 2 Cone and Partial Order Methods.- Increasing operators.- Decreasing operators.- Mixed monotone operators.- Applications of mixed monotone operators.- Further results on cones and partial order methods.- 3 Minimax Methods.- Mountain Pass Theorem and Minimax Principle.- Linking Methods.- Local linking Methods.- 4 Bifurcation and Critical Point.- Introduction.- Main results with parameter.- Equations without the parameter.- 5 Solutions of a Class of Monge-Ampere Equations.- Introduction.- Moving plane argument.- Existence and non-existence results.- Bifurcation and the equation with a parameter.- Appendix.- 6 Topological Methods and Applications.- Superlinear system of integral equations and applications.- Existence of positive solutions for a semilinear elliptic system.- 7 Dancer-Fucik Spectrum.- The spectrum of a self-adjoint operator.- Dancer-Fucik Spectrum on bounded domains.- Dancer-Fucik point spectrum on RN.- Dancer-Fucik spectrum and asymptotically linear elliptic problems.- 8 Sign-changing Solutions.- Sign-changing solutions for superlinear Dirichlet problems.- Sign-changing solutions for jumping nonlinear problems.- 9 Extension of Brezis-Nirenberg's Results and Quasilinear Problems.- Introduction.- W01,p(OMEGA) versus C01(OMEGA ) local minimizers.- Multiplicity results for the quasilinear problems.- Uniqueness results.- 10 Nonlocal Kirchhoff Elliptic Problems.- Introduction.- Yang index and critical groups to nonlocal problems.- Variational methods and invariant sets of descent flow.- Uniqueness of solution for a class of Kirchhoff-type equations.- 11 Free Boundary Problems, System of equations for Bose-Einstein Condensate and Competing Species.- Competing system with many species.- Optimal partition problems.- Schrodinger systems from Bose-Einstein condensate.- Bibliography. Our nonlinearities f and g cover the following three cases: the first with both superlinear, the second with both sublinear, and the last with one sublinear and the other superlinear. ![]() We overcome the difficulty arising from the difference between the two kernels k1(t,s)g1(s)k1(t,s)g1(s) and k2(t,s)g2(s)k2(t,s)g2(s) by defining certain integral constants, and use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing Jensen’s inequality for concave functions and nonnegative matrices. We are concerned with the existence and multiplicity of positive solutions for the system of nonlinear Hammerstein integral equationsu(t)=∫abk1(t,s)g1(s)f1(s,u(s),v(s))ds,v(t)=∫abk2(t,s)g2(s)f2(s,u(s),v(s))ds,where ki∈C×,R+,fi∈C×R+2,R+, and gi∈Cgi∈C is almost everywhere positive on (i=1,2)(i=1,2). ![]()
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