An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities

In this paper an existence result of homoclinic solutions to a nonlinear second-order ODE is presented. To this end, a method based on differential inequalities is used.     

Existence and fractional regularity of solutions for a doubly nonlinear differential inclusion

This article considers the issues of existence and regularity of solutions to the following doubly nonlinear differential inclusion $$\omega_t+\alpha (\omega_t)-\Delta \omega-\Delta_p{\omega} \ni f$$ where α is a maximal monotone operator in ${\mathbb{R}^2}$ and Δ _p denotes the p-Laplacian with p > 2. The investigation on fractional regularity is based on the Galerkin method combined with a suitable basis for W ^1,p , which we exhibit as a preliminary result. This approach also allows the obtaining of estimates in the so-called Nikolskii spaces, since it balances the interplay between the maximal monotone operator with the appearing higher order nonlinear terms.     

Solitary wave solutions for a certain class of nonlinear differential equations

Generalized forms of exact solitary wave solutions of the class (1.1) are investigated. The analysis rests mainly on the standard a direct algebraic method. The most general solutions are obtained, possibly having a constant term in their expansion into real exponentials. These solutions of the class (1.1) are performed under certain conditions for the relationship between the coefficients of the nonlinear, dispersive and dissipative terms. The analytical solutions of this class are of pulse-type and of kink-type solitary wave solutions and they are obtained with an arbitrary constant phase shift.     

On the growth of solutions of certain second-order nonlinear differential equations

In this paper, we determine the growth of real-valued solutions of certain second-order algebraic differential equations. Our main result, together with a result of G. Valiron, shows that if y₀ is an entire function which has only real, nonnegative coefficients in its power series around the origin, and which is a solution of a quadratic second-order algebraic differential equation, then y₀ satisfies a growth estimate of the form,y₀(x) ? exp(exp x^c), where c is a constant, for all sufficiently large x. The determination of the growth of such solutions was an open problem since the Valiron-Wiman theory fails to provide any information on growth, if the equation possesses a solution of infinite order of growth.     

Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.     

The continuation of solutions for certain nonlinear differential equations and its applications

We give some sufficient conditions for the continuation of solutions for some nonlinear differential equations. As an application, we obtain a new criterion for the oscillation of solutions of the Liénard equation.     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.