Onl p solutions of discrete Volterra equations

Summary In the paper a discrete analog to the Volterra nonlinear integral equation is discussed. Weighted norms are used to find sufficient conditions that all solutions of such equations are elements of anl ^p space.Some generalizations ofl ^p spaces are also considered and the corresponding sufficient conditions are established.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Recent results on blow-up and quenching for nonlinear Volterra equations

In this survey paper, the author examines nonlinear Volterra integral equations of the second kind with solutions that blow-up or quench. The focus is on analytical results, although a few words about numerical solutions for such equations are provided. The integral equations arise in the mathematical modeling of thermal processes within a reactive–diffusive medium. The scope of this review is on the published literature between 1997 and 2005, serving as an update to a previous review by the same author.     

Asymptotic behavior of integral equations using monotonicity

In this paper, some Volterra integral equations that arise in heat transfer are studied. In particular, sufficient conditions for asymptotically periodic solutions are given. The results are derived, in part, using the fact that the resolvent form of the equations involved yields a monotone operator.     

Periodic solutions of integro-differential equations in vector-valued function spaces

Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces Lp, the Besov spaces (and related spaces such as the Hölder-Zygmund spaces Cs) and the Triebel-Lizorkin spaces . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations.     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.     

Time dependent Desch-Schappacher type perturbations of Volterra integral equations

In this paper, we study time dependent multiplicative perturbations and unbounded additive perturbations of the Volterra integral equations. Some Desch-Schappacher type perturbation theorems, which generalize previous related results, are established by new and concise approaches.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.     

Integral equations and inequalities of Volterra type for functions defined in partially ordered spaces

The paper considers Volterra type integral operators acting in L₂(T), where T is a partially ordered topological space as well as equations and inequalities related to them. For the linear operators of this type it is shown that they are quasinilpotent. Explicit estimates for the solutions of linear integral inequalities have been obtained. Nonlinear equations and inequalities have also been considered.     

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.     

Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

We study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in R×Rd. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator u?A(t)u−ut in suitable Lp spaces.     

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.     

Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.      

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.     

Periodic solutions in n dimensions and Volterra equations

If a nonlinear autonomous n-dimensional system of ordinary differential equations has a bounded solution with a certain uniform stability property, this solution approaches an almost periodic solution with the same stability property. (More precisely, the almost periodic solution is in the set of ω-limit points of the given solution.) If the bounded solution has, in addition to the uniform stability property, an asymptotic stability property, then the solution approaches a periodic solution with the same stability properties. Practical (i.e., computable) sufficient conditions for boundedness of solutions are obtained. The results are applied to generalized Volterra equations.     

On monotonic solutions of an integral equation of Volterra type

Using a technique associated with measures of noncompactness we prove the existence of nondecreasing solutions to integral equations of Volterra type in C[0,1].     

Hölder continuous solutions for integro-differential equations and maximal regularity

We characterize existence and uniqueness of solutions for a linear integro-differential equation in Hölder spaces. Our method is based on operator-valued Fourier multipliers. The solutions we consider may be unbounded. Concrete equations of the type we study arise in the modeling of heat conduction in materials with memory.     

Existence theorems of solutions for nonlinear impulsive Volterra integral equations in Banach spaces

In this paper,the Monch fixed point theorem and an impulsive integral inequality is used to prove some existence theorems of solutions for nonlinear impulsive Volterra integral equations in Banach spaces that improve and extend the previous results.     

Asymptotic behavior of some nonlinear Volterra integral equations

Many people have looked at nonlinear Volterra integral equations of convolution type because they arise in a natural way from heat radiation problems. In this paper the author looks at such problems for systems of equations and looks at conditions for asymptotically constant solutions. Some theorems for comparing solutions are also derived. The theorems are applied to several integral equations derived from heat radiation problems.