On the fractional p-Laplacian problems

This paper deals with nonlocal fractional p-Laplacian problems with difference. We get a theorem which shows existence of a sequence of weak solutions for a family of nonlocal fractional p-Laplacian problems with difference. We first show that there exists a sequence of weak solutions for these problems on the finite-dimensional subspace. We next show that there exists a limit sequence of a sequence of weak solutions for finite-dimensional problems, and this limit sequence is a sequence of the solutions of our problems. We get this result by the estimate of the energy functional and the compactness property of continuous embedding inclusions between some special spaces.

Topological Properties of Parallel Series Language

Parallelism is a process by which a sequential string is broken down into a number of alphabets and used to speed up the acceptance of a string. To identify the parallelisable string, we have used parallel operator || and defined the language as parallel series languages. Algebraic and recognition properties of series parallel posets have been studied by Lodaya in [8]. In this paper, we have introduced finite and infinite parallel series language (parallel strings are connected sequentially). We have considered the set of all parallel series language as topological space and prefix order relation (poset relation) has been used to relate two parallel series strings. Topological concepts like limit, sequence, open set, closed set and basis for parallel series languages and their properties have been derived