83,643 research outputs found
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
Non-negative solutions of systems of ODEs with coupled boundary conditions
We provide a new existence theory of multiple positive solutions valid for a wide class of
systems of boundary value problems that possess a coupling in the boundary conditions.
Our conditions are fairly general and cover a large number of situations. The theory is illustrated
in details in an example. The approach relies on classical fixed point index
Multiple Positive solutions of a -Laplacian system with nonlinear BCs
Using the theory of fixed point index, we discuss existence, non-existence,
localization and multiplicity of positive solutions for a -Laplacian
system with nonlinear Robin and/or Dirichlet type boundary conditions. We give
an example to illustrate our theory.Comment: arXiv admin note: text overlap with arXiv:1408.017
Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions
We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space , involving a linear functional but, although we seek positive solutions, the functional is not assumed to be positive for all positive . The results are new even for the classic boundary conditions of clamped or hinged ends when , because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs
Existence of three solutions for a first-order problem with nonlinear non-local boundary conditions
Conditions for the existence of at least three positive solutions to the
nonlinear first-order problem with a nonlinear nonlocal boundary condition
given by
&& y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1],
&& \lambda y(0) = y(1) + \sum_{j=1}^n \Phi_j(\tau_j,y(\tau_j)), \quad
\tau_j\in[0,1], are discussed, for sufficiently large . The
Leggett-Williams fixed point theorem is utilized.Comment: outline, 6 page
A short course on positive solutions of systems of ODEs via fixed point index
We shall firstly study the existence of one positive solution of a model
problem for one equation via the classical Krasnosel'ski\u\i{} fixed-point
theorem. Secondly we investigate how to handle this problem via the fixed point
index theory for compact maps. Thirdly we illustrate how this approach can be
tailored in order to deal with non-trivial solutions for systems of ODEs
subject to local BCs. The case of nonlocal and nonlinear BCs will also be
investigated. Finally we present some applications to the existence of radial
solutions of some systems of elliptic PDEs.Comment: 52 pages 13 figure
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