Absolute continuities of exit measures for superdiffusions

Suppose X= X_t, X_T, P_μis a superdiffusion in ℝ^d with general branching mechanism ψ and general branching rate functionA. We discuss conditions onA to guarantee that the exit measure X_TL of the superdiffusionX from bounded smooth domains in ℝ^d have absolutely continuous states.     

Minimax results for semicoercive functionals with application to differential equations

We give minimax theorems for some class of generalized convex and semicoercive functions. We define semicoercive saddle points and give sufficient conditions for functionals to have such critical points. Then we apply this method to show the existence of solutions for partial differential systems at resonance.      

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

Galerkin projections for state-dependent delay differential equations with applications to drilling

A Galerkin projection scheme to obtain low dimensional approximations of delay differential equations (DDEs) involving state-dependent delays is developed. The current scheme is an extension of a similar, recently proposed scheme for DDEs with constant delays in the publication by P. Wahi, A. Chatterjee 2005. The resulting ordinary differential equations (ODEs) from the Galerkin scheme are easier to integrate using commercial ODE solvers, and are amenable to stability and bifurcation analysis using standard techniques. First, the application of the formulation is demonstrated through a scalar delay differential equation, and the performance of the formulation is assessed. Next, the scheme is applied to a two degrees-of-freedom model describing the coupled axial and torsional vibrations of oil well drill-strings. In both cases, the Galerkin approximations show an excellent agreements with the direct numerical simulations of the original systems.     

Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Let X, Y be real Banach spaces, T: X → YA-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. $\text{(1): Tx ? λCx = 0}$. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T₀ and C₀ at 0 with T₀A-proper and injective, then each eigenvalue of T₀x ? λC₀x = 0 of odd multiplicity is a bifurcation point for Eq. (1). Theorem 2 shows that if T and C have asymptotic derivatives T_∞ and C_∞, then each eigenvalue of T_∞x ? λC_∞x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (1). Special cases are treated when Y = X and T = I ? F with Fk-ball-contractive or when Y ≠ X and T is either of type (S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (1) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.     

Perturbations ofH-selfadjoint matrices,with applications to differential equations

In this paper we study perturbations of operators which are either selfadjoint or unitary with respect to an indefinite scalar product on a finite dimensional space (i.e.H-selfadjoint orH-unitary operators). The results allow us to describe systems of differential equations of higher order with selfadjoint coefficients which, together with all neighboring systems of the same kind, have only bounded solutions. An open problem concerning the structure of the connected components of such systems is posed.     

Comparison results for functional differential equations with impulses

Comparison principles play an important role in the qualitative and quantitative study of differential equations. In this paper, we investigate a first order functional differential equations with impulses and establish new comparison results.     

Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

LetW be the Wiener process onT=[0, 1]². Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR _ζ =(s, t) ∈ T, andx ₀ ∈ ?. Under some assumptions on the coefficients a_i, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX _ζ to have a density.     

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro-differential equations

Sufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed-point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro-differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.     

General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.     

Nonuniqueness results for ordinary differential equations

In the present paper we give general nonuniqueness results which cover most of the known nonuniqueness criteria. In particular, we obtain a generalization of the nonuniqueness theorem of CHR. NOWAK, of SAMIMI's nonuniqueness theorem and of STETTNER's nonuniqueness criterion.     

Existence results for fractional functional differential equations with impulses

In this paper, we discuss some existence results for a class of multi-point boundary value problem for impulsive fractional functional differential equations. Some sufficient conditions are obtained by using suitable fixed point theorems. Examples are also given to illustrate our results.     

General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.