Sublinear Case Research Articles

A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations

In this paper, we establish a new composition theorem for square-mean almost automorphic functions under conditions which are different from Lipschitz conditions in the literature. We apply this new composition theorem together with Schauder’s fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space. Finally, an interesting corollary is also given for the sub-linear growth cases.

Large solutions of coupled sublinear/superlinear elliptic equations

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u α + q(x)v β on bounded domains in R n , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.

Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay

In this paper, we prove the existence of integral solutions for a first order nondensely-defined impulsive neutral differential equation with state-dependent delay. The results are obtained by using suitable fixed point theorems. As applications of these main theorems, some consequences are obtained for the sub-linear growth cases.

Existence of Solutions for Impulsive Neutral Integro-Differential Inclusions with Nonlocal Initial Conditions via Fractional Operators

This paper is mainly concerned with existence of mild solutions for first-order impulsive neutral integro-differential inclusions with nonlocal initial conditions in α-norm. We assume that the undelayed part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed point theorem for condensing multivalued maps, a main existence theorem is established. As an application of this main theorem, a practical consequence is derived for the sublinear growth case. Finally, we present an application to a neutral partial integro-differential equation with Dirichlet and nonlocal initial conditions.

Solving sharply a singular semilinear elliptic equation

We show the existence of positive solutions to some Dirichlet problem of semilinear elliptic equations on annulus with singular nonlinearity on the boundary, and emphasize the sharpness of the results in the sublinear case.

The Dirichlet problem for second order semilinear elliptic and parabolic equations

In the present paper the Dirichlet problem for semilinear elliptic and parabolic equa- tions in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u| p�1 u, p>1, is formulated. For sublinear case (for example ln(1+|u|) or |u| p�1 u, p < 1) this condition is automatically fulfilled. Our approach gives new a priori estimate of the solution for superlinear, sublinear and linear case as well.

Existence results for a second order impulsive functional differential equation with state-dependent delay

In this paper, we study existence of mild solutions for a second order impulsive neutral functional differential equations with state-dependent delay. By using a fixed point theorem for condensing maps combined with theories of a strongly continuous cosine family of bounded linear operators, we prove the main existence theorems. As applications of these obtained results, some practical consequences are derived for the sub-linear growth cases. And an example is also given to illustrate our main results.

Fractional semi-linear parabolic equations with unbounded data

This paper is devoted to the study of semi-linear parabolic equations whose principal term is fractional, i.e. is integral and eventually singular. A typical example is the fractional Laplace operator. This work sheds light on the fact that, if the initial datum is not bounded, assumptions on the non-linearity are closely related to its behaviour at infinity. The sublinear and superlinear cases are first treated by classical techniques. We next present a third original case: if the associated first order Hamilton-Jacobi equation is such that perturbations propagate at finite speed, then the semi-linear parabolic equation somehow keeps memory of this property. By using such a result, locally bounded initial data that are merely integrable at infinity can be handled. Next, regularity of the solution is proved. Eventually, strong convergence of gradients as the fractional term disappears is proved for strictly convex non-linearity.

A note on positive evanescent solutions for a certain class of elliptic problems

This paper is devoted to the existence and properties of solutions of the following class of nonlinear elliptic differential equations Δ u ( x ) + f ( x , u ( x ) ) + g ( ‖ x ‖ ) x ⋅ ∇ u ( x ) = 0 , x ∈ R n , ‖ x ‖ > R . We prove existence of positive solutions vanishing at positive infinity. Our approach is based on the subsolution and supersolution method. The nonlinearity f covers both sublinear and superlinear cases and does not necessarily satisfy f ( x , 0 ) ≡ 0 . The asymptotic behavior of solutions is also described.

Nonlinear BVPS with functional parameters

We study the existence, nonexistence and properties of solutions for a certain class of second-order ODEs and their dependence on functional parameters, also in the case when nonlinearities are, in some sense, singular. This approach is based on variational methods and cover both sublinear and superlinear cases. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principle which enables approximation of the solution for our problem. We apply these results to obtain the existence of bounded, radial and positive classical solutions for the BVP of elliptic type. Observe that our method allows us to investigate a certain class of elliptic systems in both bounded annular domain and exterior domain.

On Impulsive Hyperbolic Differential Inclusions with Nonlocal Initial Conditions

This paper is focused mainly upon existence of solutions for a second-order impulsive hyperbolic differential inclusions with nonlocal initial conditions. By using some well-known fixed-point theorems, existence theorems are established when the multivalued map has convex or nonconvex values. As applications of these main theorems, some consequences are given for the sublinear growth cases.

Asymptotics at infinity of solutions for [formula omitted]-Laplace equations in exterior domains

Let 1 < p < N , and u be a nonnegative solution of − Δ p u = f ( x , u ) on R N ∖ B 1 ¯ where f behaves like | x | − l u q near | x | = ∞ and u = 0 , for some constants q ≥ 0 and l ∈ R . We obtain asymptotic decay estimates for u . In particular, our results complete the ‘sublinear case’ q < p − 1 . A related analysis is carried out for systems like − Δ p u = f ( x , v ) , − Δ p v = g ( x , u ) , where p = 2 corresponds to a Hamiltonian system. In this way we extend and improve some known results of Mitidieri and Pohozaev, Bidaut-Véron and Pohozaev, and other authors. Our proofs use tools such as Harnack inequality, the Maximum Principle, Liouville Theorems and blow-up arguments.

Existence and nonexistence of positive solutions for second-order time scale systems

This paper is concerned with the following system on time scale T : { u i Δ Δ ( t ) + f i ( u 1 ( t ) , u 2 ( t ) , … , u n ( t ) ) = 0 , t ∈ [ 0 , T ] T , u i Δ ( 0 ) = 0 = u i ( σ 2 ( T ) ) , i = 1 , 2 , … , n , where 0 , T ∈ T and T > 0 . By using the theory of the fixed point index, we investigate the effect of σ 2 ( T ) on the existence and nonexistence of positive solution for the above system in sublinear cases. The results obtained are even new for the special cases of difference system and differential system, as well as in the general time scale setting.

Existence and nonexistence of positive solutions for quasilinear systems

The paper deals with the existence and nonexistence of positive solutions for a class of -Laplacian systems. We investigate the effect of the size of the domain on the existence of positive solution for the problem in sublinear cases. We will use fixed point theorems in a cone.

A CRITICAL PHENOMENON FOR SUBLINEAR ELLIPTIC EQUATIONS IN CONE-LIKE DOMAINS

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determined explicitly by $s$ and the geometry of the cone.

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δ u = a ( x ) f ( u ) , u ⩾ 0 in Ω, with the boundary blow-up condition u | ∂ Ω = + ∞ , where Ω is a bounded domain in R N ( N ⩾ 2 ) , a ( x ) ∈ C ( Ω ) may blow up on ∂ Ω and f is assumed to satisfy (f 1) and (f 2) below which include the sublinear case f ( u ) = u m , m ∈ ( 0 , 1 ) . For the radial case that Ω = B (the unit ball) and a ( x ) is radial, we show that a solution exists if and only if ∫ 0 1 ( 1 − r ) a ( r ) d r = + ∞ . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.

Parallel dynamics and computational complexity of network growth models

The parallel computational complexity or depth of growing network models is investigated. The networks considered are generated by preferential attachment rules where the probability of attaching a new node to an existing node is given by a power alpha of the connectivity of the existing node. Algorithms for generating growing networks very quickly in parallel are described and studied. The sublinear and superlinear cases require distinct algorithms. As a result, there is a discontinuous transition in the parallel complexity of sampling these networks corresponding to the discontinuous structural transition at alpha=1 , where the networks become scale-free. For alpha>1 , networks can be generated in constant time while for 0</=alpha<1 , logarithmic parallel time is required. The results show that these networks have little depth and embody very little history dependence despite being defined by sequential growth rules.

Little Grothendieck's theorem for sublinear operators

Let SB( X, Y) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y. Consider π 2( X, Y) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB( C( K), H) is in π 2( C( K), H) for any compact K and any Hilbert H. In the noncommutative case the problem is still open.

A New String Matching Algorithm

In this paper a new exact string-matching algorithm with sub-linear average case complexity has been presented. Unlike other sub-linear string-matching algorithms it never performs more than n text character comparisons while working on a text of length n . It requires only O ( m +σ) extra pre-processing time and space, where m is the length of the pattern and σ is the size of the alphabet.

Existence of entire explosive positive solutions of quasi-linear elliptic equations

In this paper, our main purpose is to establish the existence of entire explosive positive radial solutions for quasi-linear elliptic equations. The main results of the paper extend previously known results both for the super-linear and sub-linear cases.