# Agarwal P.

### Certain integrals involving ℵ-functions and Laguerre polynomials

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1159-1175

UDC 517.5

Our aim is to establish certain new integral formulas involving $\aleph$ -functions associated with Laguerre-type polynomials. We
also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler
known functions, e.g., the generalized hypergeometric function $_pF_q$ in a fairly systematic way.

### Existence of global solutions for some classes of integral equations

Agarwal P., Jabeen T., Lupulescu V., O’Regan D.

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 130-148

We study the existence of $L^p$ -solutions for a class of Hammerstein integral equations and neutral functional differential equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving abstract Volterra equations is given.

### Hybrid type generalized multivalued vector complementarity problems

Agarwal P., Ahmad M. K., Salahuddin

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 7-20

We introduce a new type of generalized multivalued vector complementarity problems with moving pointed cone. We discuss the existence results for generalized multivalued vector complementarity problems under inclusive assumptions and obtain results on the equivalence between the generalized multivalued vector complementarity problems and the generalized multivalued vector variational inequality problems.

### Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 8–27

We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $. We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums $$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$

### Oscillation of certain fourth-order functional differential equations

Agarwal P., Grace S. R., O’Regan D.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 291–313

Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the form $$\frac{d^2}{dt^2} \left(a(t) \left(\frac{d^2x(t)}{dt^2}\right)^{α} \right) + q(t)f(x[g(t)])=0, \quad α>0,$$ are established.

### Interval Oscillation Criteria for Second-Order Nonlinear Differential Equations

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1161-1173

We present new interval oscillation criteria for certain classes of second-order nonlinear differential equations, which are different from the most known ones in the sense that they are based only on information on a sequence of subintervals of [*t* _{0}, ∞) rather than on the whole half-line. We also present several examples that demonstrate wide possibilities of the results obtained.