We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We em...
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We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.-S. Cheung (2006). Furthermore, we apply our result to a boundaryvalue problem of a partial difference equation for boundedness, uniqueness, and continuous dependence. Copyright (c) 2009 W.-S. Wang and X. Zhou.
Let T be a time scale such that 0,T is an element of T. By using a monotone iterative method, we present some existence criteria for positive solution of a multiple point general Dirichlet-Robin BVP on time scales wit...
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Let T be a time scale such that 0,T is an element of T. By using a monotone iterative method, we present some existence criteria for positive solution of a multiple point general Dirichlet-Robin BVP on time scales with the singular sign-changing nonlinearity. These results are even new for the corresponding differential (T = R) and difference equation (T = Z) as well as in general time scales setting. As an application, an example is given to illustrate the results. The interesting point here is that the sign-changing nonlinear term is involved with the first-order derivative explicitly, and the singularity may occur at u = 0, t = 0, and t = T. Copyright (C) 2009 You-Hui Su et al.
The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the correspondi...
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The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the corresponding region. A forced Cahn-Hilliard-type equation in a unit disc Omega is considered as an example. The current work focuses on revealing the mechanism of nonlinear smoothing, i.e., on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem. To this end convolutions of Rayleigh functions with respect to the Bessel index are employed. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424;V. Varlamov, Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains, J. Math. Anal. Appl. 327 (2007) 1461-1478]. In order to reveal the effect of nonlinear smoothing, anisotropic Sobolev spaces H-s,H-alpha (Omega) are introduced. They are based on the Sobolev spaces H-s(Omega) "weighted" by tangential derivatives, so that the index alpha is responsible for the smoothness in theta and s is the usual Sobolev index. Global-in-time solutions of the mixed problem in question are constructed, and additional smoothness with respect to the angular coordinate is established in the anisotropic Sobolev spaces. The above mentioned special functions are essentially used for this purpose. (C) 2008 Published by Elsevier Ltd.
We prove the unique solvability of a boundaryvalue problem for a system of fractional partial differential equations in a, rectangular domain and construct the solution in closed form.
We prove the unique solvability of a boundaryvalue problem for a system of fractional partial differential equations in a, rectangular domain and construct the solution in closed form.
In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equ...
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In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equation. We show that these eigenfunctions form a Riesz basis in the elliptic part of the domain. In addition, we prove the Riesz basis property on [0, pi/2] of the system of cosines occurring in the expressions for the eigenfunctions. Earlier, the Riesz basis property was proved for the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with continuous solution gradient.
In the space L(2) [0,pi], we consider the operators L = L(0) + V, L(0) = -y '' + (v(2) - 1/4)r(-2)y (v >= 1/2) with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a ...
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In the space L(2) [0,pi], we consider the operators L = L(0) + V, L(0) = -y '' + (v(2) - 1/4)r(-2)y (v >= 1/2) with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L(2)[0,pi] satisfying the condition integral(pi)(0) r(epsilon)(pi-r)(epsilon)vertical bar V(r)vertical bar dr < infinity, epsilon is an element of [0,1]. We prove the trace formula Sigma(infinity)(n=1)[mu n-lambda(n) - Sigma(m)(k=1) alpha((n))(k)] = 0.
On the interval (0, pi), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) is an element of L(1) (0, ...
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On the interval (0, pi), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) is an element of L(1) (0, pi) [q(x) is an element of L(2)(0, pi)], we establish exact asymptotic formulas for the eigenvalues of this problem.
We study nonlinear boundaryvalueproblems of the form [Psi u']' + F(x;u', u) = g, u(0) = u(1) = 0, where Psi is a coercive continuous operator from L-p to L-q, and F(x;u", u', u) = g, u(0) = u(1)...
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We study nonlinear boundaryvalueproblems of the form [Psi u']' + F(x;u', u) = g, u(0) = u(1) = 0, where Psi is a coercive continuous operator from L-p to L-q, and F(x;u", u', u) = g, u(0) = u(1) = 0;first- and second-order partial differential equations Phi(x(1);x(2);u(1)', u(2)', u) = 0, Sigma(infinity)(i=1)[Psi(i)(u(xi)')](xi)' + F(x;... , u(xi)', ... , u) = g(i);and general equations F(x;... , u(xi)", ... , ... , u(i)', ...;u) = g(x) of elliptic type. We consider the corresponding boundaryvalueproblems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.
We present a Friedmann-Robertson-Walker quantum cosmological model in the presence of Chaplygin gas and perfect fluid for early and late time epochs. In this work, we consider perfect fluid as an effective potential a...
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We present a Friedmann-Robertson-Walker quantum cosmological model in the presence of Chaplygin gas and perfect fluid for early and late time epochs. In this work, we consider perfect fluid as an effective potential and apply Schutz's variational formalism to the Chaplygin gas which recovers the notion of time. These give rise to Schrodinger-Wheeler-DeWitt equation for the scale factor. We use the eigenfunctions in order to construct wave packets and study the time dependent behavior of the expectation value of the scale factor using the many-worlds interpretation of quantum mechanics. We show that contrary to the classical case, the expectation value of the scale factor avoids singularity at quantum level. Moreover, this model predicts that the expansion of Universe is accelerating for the late times. (c) 2007 Elsevier B.V. All rights reserved.
We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4 pi(2) E with growing multiplicity N -> infinity, and co...
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We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4 pi(2) E with growing multiplicity N -> infinity, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is const root E. Our main result is that the variance of the volume normalized by root E is bounded by O(1/root N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.
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