Controllability for a class of integro-differential evolution equations involving non-local initial conditions

This paper deals with the exact controllability for a class of first-order integro-differential evolution equations involving non-local initial conditions. By using Sadovskii fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for non-local functions. An example is given to illustrate the effectiveness of the abstract results.     

Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions and resolvent operators. Sufficient conditions for the existence are derived with the help of the fixed point theorem for multi-valued operators due to Dhage and the fractional power of operators. An example is provided to illustrate the theory.     

Numerical methods for a class of nonlinear integro-differential equations

In a previous article (Glowinski, J. Math. Anal. Appl. 41, 67–96, 1973) the first author discussed several methods for the numerical solution of nonlinear equations of the integro-differential type with periodic boundary conditions. In this article we discuss an alternative methodology largely based on the Strang’s symmetrized operator-splitting scheme. Several numerical experiments suggest that the new method is robust and accurate. It is also easier to implement than the various methods discussed by Glowinski in J. Math. Anal. Appl. 41, 67–96 (1973).     

On fractional integro-differential equations with state-dependent delay

In this paper we provide sufficient conditions for the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.     

Control problems for semi-linear retarded integro-differential equations by the Fredholm theory

ABSTRACT

In this paper, we deal with the approximate controllability for semi-linear retarded functional integro-differential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the regularity of solutions of semi-linear given equations with unbounded principal operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set of trajectories of the semi-linear retarded functional integro-differential system and that of its corresponding linear system excluded by the nonlinear term.     

Existence of mild solutions on noncompact intervals to second-order initial value problems for a class of differential inclusions with nonlocal conditions

In this paper, we investigate the existence of mild solutions on an unbounded real interval to second-order initial value problems for a class of differential inclusions with nonlocal conditions. We shall rely on a theorem of Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer's Theorem.     

Approximate solution of a class of linear integro-differential equations by Taylor expansion method

In this paper, a simple and effective Taylor expansion method is presented for solving a class of linear integro-differential equations including those of Fredholm and of Volterra types. By means of the nth-order Taylor expansion of an unknown function at an arbitrary point, a linear integro-differential equation can be converted approximately to a system of linear equations for the unknown function itself and its first n derivatives under initial conditions. The nth-order approximate solution is exact for a polynomial of degree equal to or less than n. Some examples are given to illustrate the accuracy of this method.     

Explicit stability conditions for a class of semilinear retarded systems

We consider a class of semilinear retarded systems whose non-linearities satisfy the Lipschitz type conditions. The delays may be incommensurate and “non-small”. We establish exact conditions that provide the global asymptotic stability of the zero solution. They are formulated explicitly in terms of the coefficients of the considered systems. Applications of these conditions to the robust stability of linear and non-linear systems with constant and variable delays are also discussed.     

Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations

In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.     

Existence results for dynamic inclusions on time scales with nonlocal initial conditions

This paper is mainly concerned with the existence of solutions for first order dynamic inclusions on time scales with nonlocal initial conditions. By using Bohnenblust–Karlin’s fixed point theorem and Leray–Schauder nonlinear alternative for multivalued maps, some sufficient conditions are established. An example is also included to illustrate our results.     

The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay

In this paper, we introduce the optimal control problems governed by a new class of impulsive stochastic partial neutral evolution equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, the analytic semigroup theory, fractional powers of closed operators, and suitable fixed point theorems, we prove an existence result of mild solutions for the control systems in the α-norm without the assumptions of compactness. Next, we derive the existence conditions of optimal pairs of these systems. Finally, application to a nonlinear impulsive stochastic parabolic optimal control system is considered.     

Integral boundary value problems for first order integro-differential equations with impulsive integral conditions

This paper is concerned with the integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Sufficient conditions are obtained for the existence of extremal solutions.     

Controllability of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions

In this paper, the controllability for a class of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions has been considered. The sufficient conditions for the controllability are established by using the Banach fixed point theorem and the fuzzy number whose values are normal, convex, upper semicontinuous and compactly supported interval in E_N. An example is given to illustrate the results.     

Solving nonlinear singular pseudoparabolic equations with nonlocal mixed conditions in the reproducing kernel space

We study the singular pseudoparabolic problems with nonlocal mixed conditions in the reproducing kernel space, in which all functions satisfy the nonlocal mixed conditions. Based on the reproducing kernel theorem, a very simple method for solving the singular pseudoparabolic problems with nonlocal mixed conditions is proposed. The final numerical experiment illustrates the method is efficient.     

On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments

Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.      

On a class of positive linear differential equations with infinite delay

We give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a Perron-Frobenius type theorem for positive equations. Furthermore, a novel criterion for the exponential asymptotic stability of positive equations is presented. Finally, we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations. A simple example is given to illustrate the obtained results.     

Reliable algorithms for solving integro-differential equations with applications

This paper suggests four different methods to solve nonlinear integro-differential equations, namely, He's variational iteration method, Adomian decomposition method, He's homotopy perturbation method and differential transform method. To assess the accuracy of each method, a test example with known exact solution is used. The study outlines significant features of these methods as well as sheds some light on advantages of one method over the other. The results show that these methods are very efficient, convenient and can be adapted to fit a larger class of problems. The comparison reveals that, although the numerical results of these methods are similar, He's homotopy perturbation method is the easiest, the most efficient and convenient. Moreover, we applied modified forms of He's variational iteration method and differential transform method to solve a mathematical model, which describes the accumulated effect of toxins on populations living in a closed system.     

Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

In this paper, we study the nonlocal nonlinear evolution equation

${}^C{D}_{0|t}^{\alpha }u(t,x)?(J1\left|u\right|?\left|u\right|)(t,x){+}^C{D}_{0|t}^{\beta }u(t,x)={\left|u(t,x)\right|}^p,t>0,x\in {\mathbb{R}}^d,$

where $1<\alpha <2$, $0<\beta <1$, $p>1$, $J:{\mathbb{R}}^d\to {\mathbb{R}}_{+}$, $1$ is the convolution product in ${\mathbb{R}}^d$, and ${}^C{D}_{0|t}^q$, $q\in \{\alpha ,\beta \}$, is the Caputo left-sided fractional derivative of order $q$ with respect to the time $t$. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when $1<p<1+\frac{2\beta }{d\beta +2(1?\beta )}$. Next, we deal with the system

where $1<\alpha <2$, $0<\beta <1$, $p>1$, and $q>1$. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when $1<pq<1+\frac{2\beta }{d\beta +2(1?\beta )}max\{p+1,q+1\}$. Our approach is based on the test function method.