Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

Existence results for a class of semilinear nonlocal evolution equations

In this paper, we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of mild solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.     

Approximations for strongly singular evolution equations

The problem of specification of self-adjoint operators corresponding to singular bilinear forms is very important for applications, such as quantum field theory and theory of partial differential equations with coefficient functions being distributions. In particular, the formal expression corresponds to a non-trivial self-adjoint operator ? in the space only if d?3. For spaces of larger dimensions (this corresponds to the strongly singular case), the construction of ? is much more complicated: first one should consider the space as a subspace of a wider Pontriagin space, then one implicitly specifies ?. It is shown in this paper that Schrodinger, parabolic and hyperbolic equations containing the operator ? can be approximated by explicitly defined systems of evolution equations of a larger order. The strong convergence of evolution operators taking the initial condition of the Cauchy problem to the solution of the Cauchy problem is proved.     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

Existence results for a coupled system of nonlinear neutral fractional differential equations

Applying the monotone iterative method, we investigate the existence of solutions for a coupled system of nonlinear neutral fractional differential equations, which involves Riemann–Liouville derivatives of different fractional orders. As an application, an example is presented to illustrate the main results.     

Existence and uniqueness of periodic solution for a class of semilinear evolution equations

This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H

     

Peano type theorem for abstract parabolic equations

We consider parabolic problems with non-Lipschitz nonlinearity in the different scales of Banach spaces and prove local-in-time existence theorem.     

Existence results for partial neutral functional integrodifferential equations with unbounded delay

We prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the Leray-Schauder alternative.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations

We establish existence and uniqueness of solutions for a general class of nonlocal nonlinear evolution equations. An application of this theory to a class of nonlinear reaction-diffusion problems that arise in population dynamics is presented. Furthermore, conditions on the initial population density for this class of problems that result in finite time extinction or persistence of the population is discussed. Numerical evidence corroborating our theoretical results is given.

     

Young measure solutions of a class of forward-backward diffusion equations

This paper is devoted to Young measure solutions of a class of forward-backward diffusion equations. Inspired by the idea from a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence for such solutions, and then approximate the extreme case by the approach of regularization and establish the existence of Young measure solutions in the class of functions with bounded variation.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.     

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

In this paper, we study the existence of -periodic solutions for the problem

where is a -periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

     

Existence of three periodic positive solutions for a class of integral equations with parameters

In this paper, we establish the existence of three periodic positive solutions for a class of abstract integral equations by Leggett-Williams fixed point theorem. Using the existence results for abstract integral equations, the population models are also considered.     

Existence results for partial neutral integro-differential equation with unbounded delay

In this article, we study the existence and regularity of mild solution for a class of partial neutral integro-differential equation with unbounded delay.