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1. CMB 2011 (vol 56 pp. 366)
Multiple Solutions for Nonlinear Periodic Problems We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p-1)$-superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$-Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 |
2. CMB 2011 (vol 55 pp. 214)
Positive Solutions of Impulsive Dynamic System on Time Scales In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation Categories:39A10, 34B15 |
3. CMB 2011 (vol 55 pp. 285)
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations |
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations For the $n$-th order nonlinear differential equation, $y^{(n)} = f(x, y, y',
\dots, y^{(n-1)})$, we consider uniqueness implies uniqueness and existence
results for solutions satisfying certain $(k+j)$-point
boundary conditions for $1\le j \le n-1$ and $1\leq k \leq n-j$. We
define $(k;j)$-point unique solvability in analogy to $k$-point
disconjugacy and we show that $(n-j_{0};j_{0})$-point
unique solvability implies $(k;j)$-point unique solvability for $1\le j \le
j_{0}$, and $1\leq k \leq n-j$. This result is
analogous to
$n$-point disconjugacy implies $k$-point disconjugacy for $2\le k\le
n-1$.
Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation Categories:34B15, 34B10, 65D05 |
4. CMB 2009 (vol 53 pp. 347)
Multiple Nontrivial Solutions for Doubly Resonant Periodic Problems We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.
Keywords:double resonance, generalized LL-condition, Leray-Schauder degree, Cerami condition Category:34B15 |