Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

In this article, sufficient conditions are obtained so that every solution of the neutral difference equation Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) = 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$ or every unbounded solution of Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) − u n H ( y α ( n ) ) = 0 , n ≥ n 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$ oscillates, where m=2 is any integer, Δ is the forward difference operator given by Δy n = y n+1 − y n ; Δ m y n = Δ(Δ m−1 y n ) and other parameters have their usual meaning. The non linear function L ∈ C (ℝ, ℝ) inside the operator Δ m includes the case L(x) = x. Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. Further, all the three possible cases, p n ≥ 0, p n ≤ 0 and p n changing sign, are considered. The results of this paper generalize and extend some known results.

Existence of Positive Solutions to Singular Boundary Value Problems Involving φ-Laplacian

This paper is concerned with the existence of positive solutions to singular Dirichlet boundary value problems involving φ -Laplacian. For non-negative nonlinearity f = f ( t , s ) satisfying f ( t , 0 ) ¬ ≡ 0 , the existence of an unbounded solution component is shown. By investigating the shape of the component depending on the behavior of f at ∞ , the existence, nonexistence and multiplicity of positive solutions are studied.

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Unbounded solution of characteristic singular integral equation using differential transform method

In this paper, The differential transform method is extended to solve the Cauchy type singular integral equation of the first kind. Unbounded solution of the Cauchy type singular Integral equation is discussed. Numerical results are shown to illustrate the efficiency and accuracy of the present solution.