Simplification method for nonlinear equations of monotone type in Banach space

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed 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].

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

Properties of solutions of nonlinear integral equations of evolution type in Banach spaces

On linear Volterra equations in Banach spaces

We study the Volterra integro-differential problems of convolution kernel type from two perspectives: complex inversion formula and the admissibility in the Salamon–Weiss sense (which leads us to consider unbounded control operators). Due to the hybrid Cauchy–Volterra context of these problems, the control operators need to have the admissibility property. First, we show the validity of the inverse formula of the Laplace transform for the resolvent operators associated with some class of Volterra integro-differential problems of convolution type in Banach spaces with the leading operator generating a $C_{0}$-semigroup. This extends and improves the results in Hille & Phillips (1957, 347–348, Theorem 11.6.1), Driouich & El-Mennaoui (1999, 72, Theorem 1), Cioranescu & Lizama (2003, 188–189, Proposition 2), Haase (2008, 84, 81–82, Theorems 4.2 and 4.1) and Fadili & Bounit (2014, 7–8, Proposition 8), including the strong version for this class on UMD spaces. Second, the sufficient or/and necessary conditions for $L^{p}$-admissibility with $p\in (1,+\infty )$ of the control operators are given in terms of the UMD geometric property of its underlying control spaces for a wide class of Volterra integro-differential problems, which provides a generalization of a result known to hold for the standard Cauchy problems shown in Bounit et al. (2010, Proposition 3.2) and the standard Volterra integro-differential problems shown in Fadili & Bounit (2014, 9–10, Theorem 12).

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.

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.

In this paper, we use a new method and combining the partial ordering method to study the existence of the solutions for the first order nonlinear impulsive integro‐differential equations of Volterra type on finite interval in Banach spaces and for the first order nonlinear impulsive integro‐differential equations of Volterra type on infinite interval with infinite number impulsive times in Banach spaces. By introducing an interim space and using progressive estimation method, some restrictive conditions on impulsive terms, used before, such as, prior estimation, noncompactness measure estimations are deleted.

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

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

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.

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).

Some theorems on the sum of nonlinear mappings of monotone type in Banach spaces