Initial value problems for first order impulsive integro‐differential equations of Volterra type in Banach spaces

We study an initial value problem for a class of integro-differential equations of Volterra type in a real Banach space. Using method of upper and lower solutions and M¨ onch and Von Harten theorem, we obtain an existence theorem of coupled quasi-solutions, which is an extension of those established by Y. Chen and W. Zhuang in [1].

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

In this paper, we study Volterra integral inclusions defined in a Banach space.The necessity of studying such mathematical objects, comes from control theory and engineering problems.Recall that every control system (finite or infinite dimensional), under minimal hypotheses on its data, has an equivalent formulation in which the dynamics are described by an inclusion (differential, integral or functional inclusion).In this inclusion description, the control variable does not appear explicitly ("deparametrization" of the system).This equivalent inclusion description of the system, plays an important role when studying the relaxed (i.e.''convexified") system (see [24]).On the other hand, recently Glashoff-Sprekels [8], [9], studied the problem of thermostatic regulation, in which the heating devices controlling the temperature of the system, are governed by a relay switch, and established that the system dynamics can be modeled via an integral inclusion.Finally, recently Leitmann and his coworkers [15], [16], advocated a nonstochastic approach to the robustness of uncertain control systems, which is based on differential and integral inclusions.The results in the paper, extend the single-valued works of Szufla [26] and Vaughn [29] and the multivalued ones by Ragimkhanov [25], Lyapin [17] (who studied integral inclusions in R n ) and by Papageorgiou [20], [21] (who considered integral inclusions in Banach spaces).

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll‐Nardzewski and of Saint‐Beuve.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

On linear Volterra equations in Banach spaces

Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems.

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.

We study convergences of Mann and Ishikawa iteration processes for mappings of asymptotically quasi‐nonexpansive type in Banach spaces.

Chapter 3 - Fractional Evolution Inclusions with Hille-Yosida Operators

This work deals with the existence of solutions for a class of nonlinear nonlocal fractional functional differential equations of neutral type in Banach spaces. In particular, we prove the existence of solutions with the assumptions that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear operator \(-A(t)\) generates analytic semigroup for each \(t \ge 0\). We also investigate global existence of solution and study the continuous dependence of solution on initial data. We conclude the article with an application to the developed results.

In this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

PERIODIC BOUNDARY VALUE PROBLEMS FOR NONLINEAR IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES