Equivalence of the Gelfand–Shilov Spaces

AbstractWe prove that there is a one to one correspondence between the Gelfand–Shilov spaceWMΩof typeWand the spaceSMpNpof generalized typeS. As an application we prove the equalityWM∩WΩ=WMΩ, which is a generalization of the equalitySr∩Ss=Srsfound by I. M. Gelfand and G. E. Shilov (“Generalized Functions, II, III,” Academic Press, New York/London, 1967)

Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

AbstractMaking use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation9f(x+y+z3)+f(x)+f(y)+f(z)=4[f(x+y2)+f(y+z2)+f(z+x2)] in the spaces of Schwartz tempered distributions and Fourier hyperfunctions

Stability of a Jensen type equation in the space of generalized functions

AbstractWe reformulate and solve the stability problem of a Jensen type functional equation3f(x+y+z3)+f(x)+f(y)+f(z)−2f(x+y2)−2f(y+z2)−2f(z+x2)=0, in the spaces of some generalized functions such as tempered distributions and Fourier hyperfunctions

Blow-Up Solutions and Global Solutions to Discrete p

We discuss the conditions under which blow-up occurs for the solutions of discrete p-Laplacian parabolic equations on networks S with boundary ∂S as follows: ut(x,t)=Δp,ωu(x,t)+λ|u(x,t)|q-1u(x,t), (x,t)∈S×(0,+∞); u(x,t)=0, (x,t)∈∂S×(0,+∞); u(x,0)=u0≥0, x∈S¯, where p>1, q>0, λ>0, and the initial data u0 is nontrivial on S. The main theorem states that the solution u to the above equation satisfies the following: (i) if 01, then the solution blows up in a finite time, provided u¯0>ω0/λ1/q-p+1, where ω0:=maxx∈S⁡∑y∈S¯‍ω(x,y) and u¯0:=maxx∈S u0(x); (ii) if 0<q≤1, then the nonnegative solution is global; (iii) if 1<q<p-1, then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results

Harmonic Functions and Inverse Conductivity Problems on Networks

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do this we introduce an elliptic operator Dw and an w-harmonic function on thegraph, with its physical interpretation been the diffusion equation on the graph, which models an electric network. After deriving the basic properties of w-harmonic functions, we prove the solvability of (direct) problems such as the Dirichlet and Neumann boundary value problems.Our main result is the global uniqueness of the inverse conductivity problem for a network under a suitable monotonicity condition