Combining the approaches of functionals associated with h-concave functions
and fixed point techniques, we study the existence and uniqueness of a
solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) +
? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R)
denotes the space of all continuous functions on [0,T] equipped with
the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0,
T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued
functions in [0,T]xR.
In this paper, we establish some optimal inequalities for the squared mean
curvature in terms warping functions of a C-totally real doubly warped
product submanifold of a locally conformal almost cosymplectic manifold with
a pointwise ?-sectional curvature c. The equality case in the statement of
inequalities is also considered. Moreover, some applications of obtained
results are derived.
In the present article, we have investigated pointwise pseudo-slant
submanifolds of Kenmotsu manifolds and have sought conditions under which
these submanifolds are warped products. To this end first, it is shown that
these submanifolds can not be expressed as non-trivial doubly warped product
submanifolds. However, as there exist non-trivial (single) warped product
submanifolds of a Kenmotsu manifold, we have worked out characterizations in
terms of a canonical structure T and the shape operator under which a
pointwise pseudo slant submanifold of a Kenmotsu manifold reduces to a
warped product submanifold.
In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.
AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.
We consider the backwards topological property of pullback attractors for the nonautonomous MHD equations. Under some backwards assumptions of the nonautonomous force, it is shown that the theoretical existence result for such an attractor is derived from an increasing, bounded pullback absorbing and the backwards pullback flattening property. Meanwhile, some abstract results on the convergence of nonautonomous pullback attractors in asymptotically autonomous problems are established and applied to MHD equations.
We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.
We study an initial-boundary problem for a sixth order Cahn-Hilliard type equation, which arises in oil-water-surfactant mixtures. An existence result for the problem with a concentration dependent diffusional mobility in three space dimensions is presented.
Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3
