Search

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetry properties of positive solutions for nonlocal parabolic equations in the whole space. We obtain various asymptotic maximum principles for carrying out the asymptotic method of moving planes. With the help of these results, we show that if the equation converges to a symmetric one, then the solutions will converge to radially symmetric functions. The methods and techniques used here can be easily applied to study a variety of nonlocal parabolic equations with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P) (− Δ) s u = λ u γ + | u | 2 α ∗ − 2 u | x | α in Ω , u > --> 0 in Ω , u = 0 in R N ∖ Ω , $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < 2 s ∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$ , where 2 s ∗ = 2 N N − 2 s and 2 α ∗ = 2 (N − α) N − 2 s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (− Δ) s u (x) = − 1 2 ∫ R N u (x + y) + u (x − y) − 2 u (x) | y | N + 2 s d y , for all x ∈ R N. $$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P). [ABSTRACT FROM AUTHOR]

In this paper we consider the following fractional system F ( x , u ( x ) , v ( x ) , F α ( u ( x ) ) ) = 0 , G ( x , v ( x ) , u ( x ) , G β ( v ( x ) ) ) = 0 , $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: F α ( u ( x ) ) = C n , α P V ∫ R n f ( u ( x ) − u ( y ) ) x − y n + α d y , G β ( v ( x ) ) = C n , β P V ∫ R n g ( v ( x ) − v ( y ) ) x − y n + β d y. $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy}. \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system. [ABSTRACT FROM AUTHOR]

We introduce three representation formulas for the fractional p-Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional p-Laplace operator in order to have continuous dependence as p → 2 and s → 0+, 1−. A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional p-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional p-Laplacian on manifolds, as well as alternative characterizations of the Ws, p(ℝn) seminorms. [ABSTRACT FROM AUTHOR]

For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C[0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere. [ABSTRACT FROM AUTHOR]

In this article, we establish a technique for transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. The technique is applied to non-integer order delta difference equation with some boundary conditions. Furthermore, the summation formulation for impulsive fractional difference equation is utilized to construct fixed point operator which in turn are used to discuss existence of solutions. [ABSTRACT FROM AUTHOR]