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In this manuscript, we deal with the nonlocal controllability results for the fractional evolution system of 1 < r < 2 in a Banach space. The main results of this article are tested by using fractional calculations, the measure of noncompactness, cosine families, Mainardi's Wright-type function, and fixed point techniques. First, we investigate the controllability results of a mild solution for the fractional evolution system with nonlocal conditions using the Mönch fixed point theorem. Furthermore, we develop the nonlocal controllability results for fractional integrodifferential evolution system by applying the Banach fixed point theorem. Finally, an application is presented for drawing the theory of the main results. [ABSTRACT FROM AUTHOR]

The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch's fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included. [ABSTRACT FROM AUTHOR]

In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of second-order semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a second-order semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every non-initial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term u ′ (⋅) and finding a control v such that the mild solution satisfies u (b) = u b and u ′ (b) = u b ′ . The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and (α , β) -resolvent operator. Finally, an example is given to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

In this paper, we give some new results on the maximal regularity of an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces in UMD spaces. This work completes naturally the results obtained in our paper (Labbas et al. in Mediterr J Math 15(3):128, 2018) studied in H ölder spaces. Our techniques use essentially the semigroup theory and the celebrated Dore–Venni theorem (Math Z 196:124–136, 1987). [ABSTRACT FROM AUTHOR]

This paper discusses the abstract time-space fractional evolution equation with the Caputo derivative of order α ∈ (0 , 1) and fractional power operator - A β , β ∈ (0 , 1) , where - A generates a C 0 -semigroup on a Banach space. The compactness and exponential stability of the semigroup which is generated by fractional power operator - A β are investigated. With the aid of the properties of the semigroup, the existence and global asymptotic behavior of S-asymptotically periodic solutions are obtained by some fixed point theorems and related inequalities. An example to the time-space fractional diffusion equation with fractional Laplacian will be shown. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability for a class of semilinear second-order control systems with finite delay. Sufficient conditions for approximate controllability are established by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. To illustrate the applications of the obtained results, an example is provided in the end. [ABSTRACT FROM AUTHOR]

This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results. [ABSTRACT FROM AUTHOR]

The objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

An individual-based model of stochastic branching is proposed and studied, in which point particles drift in R ¯ + : = [ 0 , + ∞) toward the origin (edge) with unit speed, where each of them splits into two particles that instantly appear in R ¯ + at random positions. During their drift, the particles are subject to a random disappearance (death). The model is intended to capture the main features of the proliferation of tumor cells, in which trait x ∈ R ¯ + of a given cell is time to its division and the death is caused by therapeutic factors. The main result of the paper is proving the existence of an honest evolution of this kind and finding a condition that involves the death rate and cell cycle distribution parameters, under which the mean size of the population remains bounded in time. [ABSTRACT FROM AUTHOR]

We prove several important results concerning existence and uniqueness of pseudo almost automorphic (paa) solutions with measure for integro-differential equations with reflection. We use the properties of almost automorphic functions with measure and the Banach fixed point theorem, and we discuss two linear and nonlinear cases. We conclude with an example and some observations. [ABSTRACT FROM AUTHOR]

This paper addresses approximate controllability for a class of control systems represented by second order stochastic differential equations driven by Teugels martingales associated with a Lévy process. The main technique is the fundamental solution theory constructed through Laplace transformation. By using the so-called resolvent condition and cosine family of linear operators, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of the considered second order stochastic differential system are formulated and proved. Due to the fundamental solution theory, the nonlinear terms are only required to be partly uniformly bounded. As an illustration of the applications of the obtained results, an example is also provided in the end. [ABSTRACT FROM AUTHOR]

The aim of this work is to present new approach to study weighted pseudo almost periodic and automorphic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of μ -pseudo almost periodic and automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed in Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part are assumed to be pseudo almost periodic or pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and asymptotic stability of the positive periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space E, u ′ (t) + A u (t) = F (t , u (t) , u (t - τ)) , t ∈ R , where A : D (A) ⊂ E → E is a closed linear operator and - A generates a positive C 0 -semigroup T (t) (t ≥ 0) , F : R × E × E → E is a continuous mapping which is ω -periodic in t. Under the ordered conditions on the nonlinearity F concerning the growth exponent of the semigroup T (t) (t ≥ 0) or the first eigenvalue of the operator A, we obtain the existence and asymptotic stability of the positive ω -periodic mild solutions by applying operator semigroup theory. In the end, an example is given to illustrate the applicability of our abstract results. [ABSTRACT FROM AUTHOR]

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic analysis. Finally, an example is given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the (α , β) -resolvent operator, we concern with the term u ′ (⋅) and finding a control v such that the mild solution satisfies u (b) = u b and u ′ (b) = u b ′ . Finally, we present an application to support the validity study. [ABSTRACT FROM AUTHOR]

In this paper, we consider the monotone travelling wave solutions of a reaction–diffusion epidemic system with nonlocal delays. We obtain the existence of monotone travelling wave solutions by applying abstract existence results. By transforming the nonlocal delayed system to a non-delayed system and choosing suitable small positive constants to define a pair of new upper and lower solutions, we use the contraction technique to prove the asymptotic stability (up to translation) of monotone travelling waves. Furthermore, the uniqueness and Lyapunov stability of monotone travelling wave solutions will be established with the help of the upper and lower solution method and the exponential asymptotic stability. [ABSTRACT FROM AUTHOR]

In this paper we consider a neutral differential equation with variable delays and give some new conditions for the boundedness and stability results by using of Krasnoselskii's fixed point theorem. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved, which improves and generalizes some results due to Burton and Furumochi (Dyn Syst Appl 11:499–519, 2002), and Jin and Luo (Nonlinear Anal 68:3307–3315, 2008). Finally, an example is exhibited to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we study controllability results for non-densely defined impulsive fractional differential equation by applying the concepts of semigroup theory, fractional calculus, and Banach Fixed Point Theorem. An example is also discussed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the approximate controllability for a class of non-instantaneous impulsive stochastic fractional differential equation driven by fractional Brownian motion in Hilbert space. The results are obtained by using the ρ -resolvent family, and fixed point techniques. Finally, we have given an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

In this paper, controllability results for a class of multi-term time-fractional differential systems with state-dependent delay have been studied. The concept of fractional calculus, measure of noncompactness and Mönch fixed-point theorem has been implemented to obtain a new set of controllability results. Finally, an application is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In our manuscript, we organize a group of sufficient conditions for the approximate controllability of second-order evolution hemivariational inequalities. By applying a suitable fixed-point theorem for multivalued maps, we prove our results. Lastly, we present an example to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

Recently, systems of coupled renewal and retarded functional differential equations have begun to play a central role in complex and realistic models of population dynamics. In view of studying the local asymptotic stability of equilibria and (mainly) periodic solutions, we propose a pseudospectral collocation method to approximate the eigenvalues of the evolution operators of linear coupled equations, providing rigorous error and convergence analyses and numerical tests. The method combines the ideas of the analogous techniques developed separately for renewal equations and for retarded functional differential equations. Coupling them is not trivial, due to the different state spaces of the two classes of equations, as well as to their different regularization properties. [ABSTRACT FROM AUTHOR]

By applying monotone iterative technique coupled with the method of lower and upper solutions, sufficient conditions are obtained for the existence of mild solution of Sobolev type fractional integro-differential equations with fractional nonlocal conditions. The results are obtained under monotonicity and measure of noncompactness conditions of the nonlinear functions. At last, an example is given to show the applicability of main results. [ABSTRACT FROM AUTHOR]

In this article, we shall study the approximate controllability of certain non-autonomous second-order nonlinear differential problems with finite delay in the infinite dimensional space. Sufficient conditions are proposed and proved for the controllability of such systems. Further, we briefly discuss the approximate controllability of impulsive as well as integro-differential problem. We establish these results by utilizing Schauder's fixed-point analysis approach. Finally, the application of the proposed results is presented by giving an example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept called α -order cosine-resolvent family, by using the theory of fractional calculus, the concepts of measure of noncompactness and Hybrid fixed point theorem, we consider the existence of PC-mild solutions for a class of impulsive fractional hybrid evolution equation in a Banach space. Furthermore, we obtain some sufficient conditions for approximate controllability of our concern problem. At the end, an example is given to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability for a class of semilinear second-order stochastic evolution systems with infinite delay. The main technique is the fundamental solution theory constructed through Laplace transformation. Some sufficient conditions for the approximate controllability result is obtained via the so-called resolvent condition and cosine family of linear operators. Duo to the fundamental solution theory applied, the nonlinear terms are only required to be partly uniformly bounded. Finally, an example is provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The paper is concerned with the study of some partial neutral functional integrodifferential equations with nondense domain. Using the integrated resolvent operator theory, we derive some results concerning the existence and regularity of solutions. Finally, an example is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this article, we deal with the existence of mild L-quasi-solutions for damped elastic systems in Banach spaces. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Under monotone condition and the noncompactness measure condition on nonlinearity, we obtain the existence of extremal mild solutions. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

This article studies approximate controllability for a new class of semilinear control systems involving state-dependent delay in Hilbert space setting. We formulate some new sufficient conditions which ensure the existence of mild solution for the considered system via the Schauder fixed point theorem. We use the theory of fundamental solution and fractional powers of operator, to establish our major results. At last, two examples are constructed to substantiate the application of obtained results. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of mild and classical solutions for neutral partial functional integro-differential equations with non-local condition. We discuss the problems by making full use of resolvent operators theory newly established in the literature, theory of fractional power, α -norm and Schauder's fixed point theorem. Some results on existence and regularity of solutions of the considered equation are obtained. Finally, an example is presented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximation of the solutions to fractional stochastic integro-differential equations with finite delay in a separable Hilbert space. With the help of projection operator, we construct the finite dimensional approximate solutions of the given system. The theory of α-order cosine family of linear operators and stochastic version of the well-known Banach fixed point theorem are applied to show the existence and uniqueness of approximate solution. At last, we show that these approximate solutions form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is a solution of the original problem. Moreover, we show the convergence of Faedo-Galerkin approximate solution. Finally, we demonstrate an example to show the applications of these abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we approximate the solution of Fredholm integro-differential equations of the second kind by using exponential spline function. The proposed method reduces to the system of algebraic equations. The convergence analysis of the method has been discussed. The numerical examples are presented to illustrate the applications of the method and to compare the computed results with the method in Chen et al. [A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation, J. Comput. Appl. Math. 290 (2015), pp. 633–640]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of the extremal solutions for a class of impulsive fractional integro-differential equation with order 1 < α < 2 , combine upper and lower solution method with the theory of (α , β) -resolvent family, a series of results about the solutions to such differential equations is obtained. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Finally, an example is given to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The purpose of the work is to investigate the topological structure of the solution set of an initial value problems for nonlinear fractional differential equations in Banach space. We prove that the solution set of the problem is nonempty, compact and, an R δ -set by introducing a new regular measure of noncompactness in the weighted space of continuous functions. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic fractional integro-differential equation with deviated argument in a separable Hilbert space. We used the semigroup theory of linear operators to study the existence, uniqueness and convergence of approximate solutions to the given equation with the help of stochastic version of the well-known Banach fixed point theorem. Moreover, the convergence of Faedo–Galerkin approximation of the solution is shown. In the last, we give an example to illustrate the applications of the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015). [ABSTRACT FROM AUTHOR]

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term g(x)f(t) on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of f(t) that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of f(t) such that the voltage output is equal to some given function v(t) (with g(x) being given as well). The third control problem deals with an explicit construction of both the force profile, g(x), and the control, f(t), which generate the desired voltage output v(t). Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems. [ABSTRACT FROM AUTHOR]

In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps in a Hilbert space. Nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii's fixed point theorem. Finally, an example is provided to illustrate the effectiveness of the obtained result. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. Some examples related to partial neutral differential equations are presented. [ABSTRACT FROM AUTHOR]

In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, α -norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the stability for some integro-differential equations with impulses driven by fractional Brownian motion with noncompact semigroup in Hilbert spaces. We assume that the linear part has a resolvent operator not necessary compact but is operator norm continuous. Sufficient conditions for the existence of mild solutions are obtained using the Hausdorff measure of noncompactness and the Mönch fixed point theorem. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution system of parabolic type with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green's function and constructing a control function involving Gramian controllability operator. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. The discussions are based on Schauder's fixed-point theorem as well as the theory of evolution family. An example is also given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the controllability and exponential stability results for a class of nonlinear neutral stochastic functional differential control systems in the presence of infinite delay driven by Rosenblatt process in real separable Hilbert spaces. Firstly, by using stochastic analysis approach, fractional calculus theory and a fixed point technique, we prove the controllability result for the mild solutions of the nonlinear control systems with the condition that the associated linear system is controllable. Secondly, we discuss the exponential stability of mild solutions of the nonlinear neutral stochastic differential systems driven by Rosenblatt process. Finally, an example is presented to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]