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The purpose of this paper is to study the controllability in the α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach space. To do this, we give sufficient conditions ensuring the controllability by assuming that the undelayed part admits a resolvent operator in the sense of Grimmer and that the delayed part is continuous with respect to the fractional power of the generator. The results are obtained by using the Schauder fixed-point theorem. [ABSTRACT FROM AUTHOR]

This article deals with the existence, stability and optimality of a semilinear control systems with instantaneous and non-instantaneous impulses. We study this problem by using the technique of mild solution and fixed point theory together with theory of semigroups. In the field of applied sciences impulses are of great significance for a dynamical systems, as mathematical models and dynamical systems representing any real life scenario it is not necessary to have continuous dynamics. This problem where both impulses are discussed together is opening the door for many researhers to work and generalize this concept for stochastic, fractional and delay cases by considering the qualitative properties as well. One example is also given in the last to show the applicability of the problem considered in the main system. [ABSTRACT FROM AUTHOR]

Stochastic differential equations with Poisson jumps become very popular in modeling the phenomena arising in various fields, for instance in financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. The objective of this paper is to investigate the approximate controllability for a class of control systems represented by second-order stochastic differential equations with time delay and Poisson jumps. The main technique is the theory of fundamental solution constructed through Laplace transformation. By employing the so-called resolvent condition, theory of cosine operators and stochastic analysis, we formulate and prove some sufficient conditions for the approximate controllability of the considered system. In the end an example is given and discussed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and optimal controllability of a multi-term time-fractional stochastic system with non-instantaneous impulses. Using semigroup theory, stochastic techniques, and Krasnoselskii's fixed point theorem, we first establish the existence of a mild solution. Further, we obtain that there exists an optimal state-control pair for the system under certain assumptions. Some examples are given to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we study the impulsive evolution equation with state-dependent delay by the theory of operator semigroup in Banach spaces. Under conditions that both nonlinearity and impulsive functions are Lipschitz continuous, we obtain the existence and uniqueness results of mild solution. Furthermore, we prove the differentiability of a semi-flow defined by a continuously differentiable solution operator under the appropriate condition. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability problem for a system of nonlocal integro-differential equations with impulsive effects. We start by investigating the existence and uniqueness of solutions for this system. The results are derived using the theory of resolvent operators combined with fixed point theory in a generalized Banach space. Next, we examine approximate controllability without necessarily requiring the nonlinear terms to be uniformly bounded. In particular, we do not impose here the compactness condition for either the resolvent operator or the state-dependent function in the nonlocal condition, as is commonly found in the literature. Finally, we provide an example to demonstrate the abstract results of this work. [ABSTRACT FROM AUTHOR]

In present manuscript, we bring into work the technique of lower and upper solutions incorporated with the monotone iterative technique to discuss the existence and uniqueness of extremal mild solutions for integro-differential neutral measure driven system in an ordered Banach space. A strenuous situation of Zeno behaviour arising in enormous hybrid systems can also be modeled under our general system. Measure driven system is also a more generalized concept as compared to ordinary differential equation. We established the results in space of regulated functions. Our main findings are obtained by using the monotone iterative technique, theory of Bochner–Stieltjes integral, measure of noncompactness, regulated functions and semigroup theory. In the end, we provide an application to justify our results. [ABSTRACT FROM AUTHOR]

In this article, we study weighted Stepanov-like pseudo-almost automorphic functions with infinite delay using 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 space of such functions. We also study the existence and uniqueness of (μ,ν)\left(\mu ,\nu )-Stepanov-like pseudo-almost automorphic solutions of infinite class in the α\alpha -norm for some partial functional differential equations in a Banach space with unbounded delay, using the spectral decomposition of the phase space developed by Adimy and his co-authors. An example is given to illustrate this work.

In this paper the resolvent family { S α , β (t) } t ≥ 0 ⊂ L (X , Y) generated by an almost sectorial operator A, where α , β > 0 , X, Y are complex Banach spaces and its Laplace transform satisfies S ^ α , β (z) = z α - β (z α - A) - 1 is studied. This family of operators allows to write the solution to an abstract initial value problem of time fractional type of order 1 < α < 2 as a variation of constants formula. Estimates of the norm ‖ S α , β (t) ‖ , as well as the continuity and compactness of S α , β (t) , for t > 0 , are shown. Moreover, the Hölder regularity of its solutions is also studied. [ABSTRACT FROM AUTHOR]

We research extremal S-asymptotically ω-periodic mild solutions to elastic systems by using upper and lower solutions. In our first result, it is assumed that the semigroup is compact. In addition, in our second result, the semigroup does not need to be compact and not assuming lower and upper solutions. Our conclusion is drawn using measures of noncompactness. Finally, the existence of mild solutions for damped equations is discussed as an application. [ABSTRACT FROM AUTHOR]

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

Biologically, because of the impact of reproduction period and nonlocal dispersal of HIV-infected cells, time delay and spatial heterogeneity should be considered. In this paper, we establish an HIV infection model with nonlocal dispersal and infection age. Moreover, applying the theory of Fourier transformation and von Foerster rule, we transform the model to an integro-differential equation with nonlocal time delay and dispersal. The well-posedness, positivity, and boundedness of the solution for the model are studied. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of strict solutions to a class of differential equations with state-dependent nonlocal conditions. Using fixed point theorems and general Gronwall inequality, we give some new conclusions. Moreover, an example is given to illustrate our conclusions valuable. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number $ \Re _0^{\rho } $ ℜ 0 ρ . The results show that (i) if $ \Re _0^{\rho } \gt 1 $ ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when $ \Re _0^{\rho } \lt 1 $ ℜ 0 ρ < 1. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure. [ABSTRACT FROM AUTHOR]

In this manuscript, we employ integrated resolvent operator to derive the variation of constants formula for the solution of the impulsive integro-differential system, in nonlocal domain with finite delay function. Using the Banach fixed point theorem and integrated resolvent operator, we explore the existence of mild solution of the aforementioned system. Additionally, we establish the Hyers–Ulam stability of the system. Finally, the main result is illustrated with the help of an example. [ABSTRACT FROM AUTHOR]

In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the ψ -sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of ψ -Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work. [ABSTRACT FROM AUTHOR]

We study the local and global existence and uniqueness of a strict solution for a general class of abstract explicit neutral equations with state-dependent delay. Some examples on explicit partial neutral differential equations with state dependent delay are presented. [ABSTRACT FROM AUTHOR]

By a thorough performance comparison, we compare the recently proposed, operator-based linear clustering process on a network with classical, existing clustering algorithms. The linear clustering process produces clusters or partitions based on the eigenstructure of a linear operator on a graph that replaces nodes to 'more natural' positions by attractive and repulsive forces. Synthetic benchmarks, along with real-world networks possessing or lacking a known community structure, are considered. Our comparative analysis demonstrates that our linear clustering process generates superior partitions compared to the algorithms assessed in most instances, while of comparable computational complexity with the simplest existing clustering algorithms. [ABSTRACT FROM AUTHOR]

We characterize existence and uniqueness of periodic strong and mild solutions to an abstract second order differential equation with memory in Banach spaces. Using vector-valued Fourier multipliers we give necessary and sufficient conditions in order to ensure the well-posedness of this equation in Lebesgue, Hölder and Besov spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider the fractional delayed evolution equation of order γ ∈ (1 , 2) in ordered Banach space. In the absence of assumptions about the compactness of cosine families or related sine families, the existence results of positive solutions are studied by using some fixed point theorems and monotone iterative method under the conditions that nonlinear function satisfies the non-compactness measure conditions and some appropriate growth conditions or order conditions. [ABSTRACT FROM AUTHOR]

In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction–diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by R 0 , is larger than one, there exists a minimal wave speed c ∗ > 0 satisfying for each c > c ∗ , the system admits a nontrivial time-periodic traveling wave solution with wave speed c, and for c < c ∗ , there exist no nontrivial time-periodic traveling waves such that if R 0 ⩽ 1 , the system admits no nontrivial time-periodic traveling waves. [ABSTRACT FROM AUTHOR]

This article considers the optimal and time optimal feedback control problems for a semi-linear neutral retarded integro-differential system. The existence of mild solutions and feasible pairs for the considered system is studied by applying theory of resolvent operators for linear neutral integro-differential evolution systems, fractional powers of operators and Schauder fixed point theorem. Then the Lagrange optimal feedback control problem for the system is investigated via limit arguments under some suitable assumptions. The time optimal feedback control problem is proposed and discussed deliberately here as well. An example is presented in the end to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we propose the approximate controllability analysis for a second semilinear system under nonlocal conditions using monotone nonlinearity as a tool. Monotonicity is a significant feature in many communications applications that use digital-to-analog converter circuits. While non-monotonicity prevents such applications from working, nonlinearity makes them work. Therefore, it is of great interest to study a problem while assuming the nonlinear function is monotonic. Nonlocal conditions have a finer effect on the solution and are additional specifications for the physical measurements beyond the conventional ones. In order to determine the controllability of the considered system, we formulate the control function. In addition to this, we have broadened the scope of the results for the impulsive system. At last, we discuss two examples. [ABSTRACT FROM AUTHOR]

This paper studies the exact controllability of semilinear q ∈ (1 , 2) -order fractional evolution systems with a weighted delay initial condition in abstract spaces. Firstly, with the aid of properties of the strongly continuous cosine family, the expression of mild solutions of the concerned problem is presented. Exact controllability results are achieved under essential conditions on f. An example as an application is also given. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems. [ABSTRACT FROM AUTHOR]

This article establishes the existence of global classical solutions to discrete coagulation equations with collisional breakage for collision kernels having linear growth. In contrast, the uniqueness is shown under additional restrictions on collision kernels. Moreover, mass conservation property and the positivity of solutions are also shown. While coagulation dominates, the occurrence of the gelation phenomenon for kernels having specific growth is also studied. [ABSTRACT FROM AUTHOR]

In this work we introduce a general formulation of resolvent family, also named resolvent operator, to describe explicit formulas for the solutions of dynamic equations on time scales of order 0 < α ≤ 1 . The treatment developed here is based on a formulation of Laplace Transform on time scales that includes continuous and discrete cases, to obtain concise expressions for such explicit formulas. Moreover, this formulation of Laplace Transform allows to obtain discrete counterparts of some important properties in the context of fractional calculus. As main results in this work, we study the relationship between an abstract resolvent family and its infinitesimal generator, along with the main properties of resolvent families, and the existence of solutions for abstract dynamic equations on time scales. In addition, we introduce formulas for resolvent families as solutions of several types of dynamic equations of order 0 < α ≤ 1 on continuous, discrete and quantum time scales. [ABSTRACT FROM AUTHOR]

We study an abstract linear viscoelastic equation in Hilbert space. It fits some convolution type of integro-differential equations which govern dynamics in viscoelastic or thermoviscoelastic materials. A new semigroup approach is proposed which is adapted from a semigroup approach designed originally to a second-order abstract linear integro-differential equation. As compared with the existing semigroup approach, the system contraction can be obtained readily. Sufficient conditions for the system exponential stability are also established. An illustrative example is given, and numerical simulations are provided. They are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of damped elastic systems with initial conditions without the assumptions that the corresponding linear system is approximately controllable. Firstly, the existence of mild solution is obtained by means of contraction mapping principle and operator semigroup theory. Secondly, using the sequence method, a new set of sufficient conditions for approximate controllability of damped elastic systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This article is dedicated to investigate the existence and uniqueness of solutions for abstract neutral differential equations with state-dependent time impulses. We employ the Banach contraction method and the Schauder’s fixed point approach to establish our results. In order to prove our findings, certain assumptions are made. Additionally, we provide an example to illustrate the application of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of the existence of mild solutions of abstract retarded functional differential equations with infinite state-dependent delay. We obtain the result concerning the existence of mild solutions for the equations with state-dependent delays as a fixed point of the solution operator of an associated abstract retarded functional differential equation with time-varying delays. We are concerned with equations that present a phenomenon of lacunary memory. We apply our results to study the existence of solutions of a state-dependent partial differential equation with infinite state-dependent delay. [ABSTRACT FROM AUTHOR]

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, a class of nonlocal problem of structural damped elastic systems with delay are studied in the case of noncompact semigroups in ordered Banach space. Firstly, we introduce the concept of lower S-asymptotically ω -periodic solution and upper S-asymptotically ω -periodic solution, on the premise of the existence of upper and lower S-asymptotically ω -periodic solutions, the existence of maximal and minimal S-asymptotically ω -periodic mild solutions for the elastic system are obtained. Then, an existence theorem of positive mild solutions for elastic system is obtained without assuming lower and upper S-asymptotically ω -periodic solutions. Finally, as the application of abstract results, the existence and uniqueness of S-asymptotically ω -periodic mild solutions and positive mild solutions for a classes of nonlocal damped elastic systems with delay are discussed. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of stochastic elastic systems with structural damping and infinite delay in Hilbert spaces. The estimation of the control function is discussed, where the expression of the control function is constructed by the defined resolvent operator. Under this estimate, the existence of mild solutions for this system is obtained by the Schauder fixed point theorem and the stochastic analysis theory, and sufficient conditions for the approximate controllability are formulated and proved by using the so-called resolvent operator type condition. Finally, an example is given to illustrate the applicability of our conclusion.

The paper sheds light on Hilfer’s controllability facts of neutral fractional system. Originally, the mild solution is derived using semigroup theory and the Laplace transform approach. Controllability of the Hilfer fractional system in non-dense domain using integral contractor which employs the sequence technique with the advantage that the nonlinear function does not meet the Lipschitz condition. To support the computed results, an appropriate examples are discussed. [ABSTRACT FROM AUTHOR]

In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

The initial-boundary value problem for higher-order delay partial differential equations is considered. We study the existence of its solution and also propose a method for finding approximate solutions. We are established a sufficient conditions for the existence and uniqueness of the solution to the problem under consideration. Introducing new unknown functions, we reduce the considered problem to an equivalent problem consisting of a family of problems for delay differential equation with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for delay partial differential equations are obtained in terms of the initial data. Unique solvability to the initial-boundary value problem for higher-order delay partial differential equations is interconnected with unique solvability to the family of problems with parameters for delay differential equations. [ABSTRACT FROM AUTHOR]

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order r ∈ (1 , 2) via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type (P , κ , r , γ) and Martelli's fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained. [ABSTRACT FROM AUTHOR]

In this article, we presented some sufficient conditions for the approximate controllability of a class of fractional differential systems with fixed delay in Banach space. The existence of a mild solution is discussed with the help of the fixed point theorem. Controllability results are obtained using generalized Gronwall's inequality, Cauchy sequence, and the basics of functional analysis. As a last note, we have included an example for the illustration of the obtained theory. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the existence and approximate controllability for fractional stochastic Volterra–Fredholm integro-differential systems having order 1 < r < 2. The primary outcomes are obtained by applying concepts and ideas from fractional calculus, multivalued maps, the theory of cosine family, Martelli and Dhage, and Leray–Schauder fixed point techniques. We begin by emphasizing the existence, and then demonstrate the approximate controllability of the considered system. Additionally, we determine the approximate controllability outcomes for the system with infinite delay. At last, an application is established for drawing the theoretical conclusions of primary outcomes. [ABSTRACT FROM AUTHOR]

In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation ( D α ) involving time-fractional derivative of order α ∈ (0 , 1) . For the semi discrete formulation of the equation ( D α ), we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz–Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations have a weak singularity near time t = 0 . Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the equation ( D α ) on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted H 1 (Ω) norm. We prove that the developed numerical scheme has an accuracy rate of O (P - 1 + N - (2 - α)) in L ∞ (0 , T ; L 2 (Ω)) as well as in L ∞ (0 , T ; H 0 1 (Ω)) , where P and N are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples. [ABSTRACT FROM AUTHOR]