Well posedness of an integrodifferential kinetic model of Fokker–Planck type for angiogenesis

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators.     

Cauchy–Goursat Problem for One-Dimensional Semilinear Wave Equations

We consider the initial-characteristic problem for nonlinear wave equations with positive power nonlinearity source term. Depending on the power of nonlinearity, we investigate the problem on a global existence and blow-up of solutions of initial-characteristic problem. The question on local solvability of the problem is also considered.     

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt^?1u′(t) = Au(t) on the half-line. (Here k ∈ ? is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t ^k u′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.     

The Brezis–Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains

In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue λp,s(■) with respect to the domain■. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence res...     

Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem

Abstract cyclical monotonicity is studied for a multivalued operator F : X L, where L R ^X . A criterion for F to be L-cyclically monotone is obtained and connections with the notions of L-convex function and of its L-subdifferentials are established. Applications are given to the general Monge–Kantorovich problem with fixed marginals. In particular, we show that in some cases the optimal measure is unique and generated by a unique (up to the a.e. equivalence) optimal solution (measure preserving map) for the corresponding Monge problem.     

Ulam–Hyers Stability Analysis of a Three-Point Boundary-Value Problem for Fractional Differential Equations

Ukrainian Mathematical Journal - We study the problem of existence and uniqueness of the solution of a three-point boundary-value problem for a differential equation of fractional order. Further,...     

Ulam–Hyers Stability for the Darboux Problem for Partial Fractional Differential and Integro-differential Equations via Picard Operators

In the present paper we investigate some uniqueness and Ulam’s type stability concepts of fixed point equations due to Rus, for the Darboux problem of partial differential and integro-differential equations involving the Caputo fractional derivative. Our results are obtained by using weakly Picard operators theory.     

An Efficient Algorithm for Quadratic Sum-of-Ratios Fractional Programs Problem

The quadratic sum-of-ratios fractional program problem has a broad range of applications in practical problems. This article will present an e?cient branch-and-bound algorithm for globally solving the quadratic sum-of-ratios fractional program problem. In this algorithm, lower bounds are computed by solving a series of parametric relaxation linear programming problems, which are established by utilizing new parametric linearizing technique. To enhance the computational speed of the proposed algorithm, a rectangle reducing tactic is used to reject a part of the investigated rectangle or the whole rectangle where there does not contain any global optimal solution of the quadratic sum-of-ratios fractional program problem. Compared with the known approaches, the proposed algorithm does not need to introduce new variables and constraints. Therefore, the proposed algorithm is more suitable for application in engineering.     

Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula

The ‘value’ of infinite horizon risk-sensitive control is the principal eigenvalue of a certain positive operator. For the case of compact domain, Chang has built upon a nonlinear version of the Krein–Rutman theorem to give a ‘min–max’ characterization of this eigenvalue which may be viewed as a generalization of the classical Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a nonnegative irreducible matrix. We apply this formula to the Nisio semigroup associated with risk-sensitive control and derive a variational characterization of the optimal risk-sensitive cost. For the linear, i.e., uncontrolled case, this is seen to reduce to the celebrated Donsker–Varadhan formula for principal eigenvalue of a second-order elliptic operator.     

Crank–Nicolson Scheme for Abstract Linear Systems

This paper studies the Crank–Nicolson discretization scheme for abstract differential equations on a general Banach space. We show that a time-varying discretization of a bounded analytic C₀-semigroup leads to a bounded discrete-time system. On Hilbert spaces, this result can be extended to all bounded C₀-semigroups for which the inverse generator generates a bounded C₀-semigroup. The presentation is based on C₀-semigroup theory and uses a functional analysis approach.     

Ax–Schanuel for linear differential equations

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.     

The Cauchy Problem for a Class of Semilinear Wave Equations with Piecewise Constants Data

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The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L~2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.     

Abstract Cauchy problem for the Bessel–Struve equation

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.     

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

The paper defines the Weyl–Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl–Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl–Titchmarsh function for the three dimensional Schr?dinger operator introduced by W.O. Amrein and D.B. Pearson. Relationships to the Schr?dinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Received: May 21, 2007., Revised: October 8, 2007., Accepted: October 11, 2007     

On the Uniqueness of the Positive Solution for a Semilinear Elliptic Problem with Critical Exponent

Inrecentyears,thequestionsofexistence,uniquenessandmultiplicityofsolutionstothe.problemhasbeenmuchinvestigated(see[l-ll]).WhereOisasmoothboundedregioninRN,N23and6istheLaplacianoperator.Thesolvabilityofboundaryvalueproblemdependsonthegrowthofthenonlinearityf(u)andontheregion,boundedorunbounded,unitballoranyotherregions.Wesaythatfunctionf(u)growssubcriticallyifthereexistsqe(l,'(N 2)/(N-2))suchthatliT32PIf(u)l/lulq相似文献