An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings {T_n} with common fixed points, an equilibrium function G, a contraction f with coefficient 0<α<1 and a strongly positive linear bounded operator A with coefficient . Let . We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem , where h is a potential function for γf and C is the intersection of the equilibrium points and the common fixed points of the sequence {T_n}.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model

Consider the generalized growth curve model subject to R(X_m)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, and E=(ε₁,…,ε_s)^′ and are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ. We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function with C≥0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator with of to be the UMVNNQUE are obtained as well.     

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.     

A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

Mann iteration of Cesàro means for asymptotically non-expansive mappings

Let K be a nonempty closed convex subset of a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm. Suppose that T:K→K is an asymptotically non-expansive mapping and for arbitrary initial value x₀∈K, we will introduce the Mann iteration of its Cesàro means:

     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

Nontrivial solutions of some quasilinear problems via a cohomological local splitting

This paper deals with a generalization of the p-Laplacian type boundary value problem

     

A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation on a complete Riemannian manifold is considered. The existence of solutions connecting any two points is studied, i.e., for T>0 the critical points of the functional with x(0)=x₀,x(T)=x₁. When the potential V has a subquadratic growth with respect to x, J_T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of is not trivial). When V has an at most quadratic growth, i.e., , this property does not hold, but an optimal arrival time T(λ)>0 exists such that, if 0<T<T(λ), any pair of points in can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik-Schnirelman theory are used. The optimal value is fulfilled by the harmonic oscillator. These ideas work for other related problems.     

Liouville theorems for quasi-harmonic functions

Let N be a compact Riemannian manifold. A self-similar solution for the heat flow is a harmonic map from to N (n≥3), which was also called a quasi-harmonic sphere (cf. Lin and Wang (1999) [1]). (Here is the Euclidean metric in .) It arises from the blow-up analysis of the heat flow at a singular point. When and without the energy constraint, we call this a quasi-harmonic function. In this paper, we prove that there is neither a nonconstant positive quasi-harmonic function nor a nonconstant quasi-harmonic function. However, for all 1≤p≤n/(n−2), there exists a nonconstant quasi-harmonic function in .     

On an additive decomposition of the BLUE in a multiple-partitioned linear model

Necessary and sufficient conditions are derived for the BLUE in a general multiple-partitioned linear model to be the sum of the BLUEs under the k small models , …, . Some consequences and further research topics are also given.     

A nonlinear parabolic equation backward in time: Regularization with new error estimates

Consider a nonlinear backward parabolic problem in the form

     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.