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Fractional Calculus Approach
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## Abstract

In this article we study the existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales. The main tool employed here is Schauder’s fixed point theorem. The results obtained here extend the work of Culakova, Hanustiakova and Olach [12]. Two examples are also given to illustrate this work.

## Abstract

The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form

${A(u)=finΩu=0on∂Ω$

where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and fW−1,p (.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

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## Abstract

An accelerated of the steepest descent method for solving unconstrained optimization problems is presented. which propose a fundamentally different conjugate gradient method, in which the well-known parameter βk is computed by an new formula. Under common assumptions, by using a modified Wolfe line search, descent property and global convergence results were established for the new method. Experimental results provide evidence that our proposed method is in general superior to the classical steepest descent method and has a potential to significantly enhance the computational efficiency and robustness of the training process.

## Abstract

Let 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.