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In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a C 0 -semigroup W (·) generated by the operator - A and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding C 0 -semigroup W (·) and non-instantaneous impulsive function γ k are compact, W (·) is not compact and γ k is compact, W (·) and γ k are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is concerned with the existence of mild solutions and approximate controllability for a class of non-autonomous evolution system with nonlocal conditions in Banach spaces. First, the existence of mild solution is obtained by means of contraction mapping principle. Then sufficient conditions have been presented to establish the approximate controllability by using the sequence method. The application of the theory is explained by an example. [ABSTRACT FROM AUTHOR]

In this paper, based on the monotone iterative technique and the operator semigroup theory, a class of nonlocal problem of structural damped elastic systems with delay is studied in the case of noncompact semigroups in ordered Banach space. First, on the premise of the existence of upper and lower mild solutions, the existence and uniqueness of 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 mild solutions. Finally, as the application of abstract results, the existence and uniqueness of mild solutions and positive mild solutions for two classes of nonlocal damped beam vibration equations with delay are discussed. [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]

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 used Henstock–Kurzweil–Pettis integral instead of classical integrals. Using fixed point theorem and weak measure of noncompactness, we study the existence of weak solutions of boundary value problem for fractional integro-differential equations in Banach spaces. Our results generalize some known results. Finally, an example is given to demonstrate the feasibility of our conclusions. [ABSTRACT FROM AUTHOR]

In this article, we concern with the existence of mild solutions for damped elastic systems. The discussion is based on the operator semigroups theory and fixed point theorem. New forms of the corresponding first order evolution equation are introduced, and their well-posed property is proved by means of the operator semigroup theory. Further, we consider analyticity and exponential of the semigroup associated with the elastic system (1.1). Finally, two examples are given to illustrate availability of our 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 fractional evolution equations 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. The discussions are based on Schauder's fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical 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]

This paper is concerned with the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach space E. By employing the method of lower and upper solutions, the measure of noncompactness, and Sadovskii's fixed point theorem, we obtain the existence of extremal mild solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provided to illustrate the feasibility of our main results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space E, where the operators in linear part (possibly unbounded) depend on time t. Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [13, J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of ω -periodic solutions for second-order functional differential equation with delay in E − u ′ ′ ( t ) = f ( t , u ( t ) , u ( t − τ ) ) , t ∈ R , where E is an ordered Banach space, f : R × E × E → E is a continuous function which is ω -periodic in t and τ ≥ 0 is a constant. We first build a new maximum principle for the ω -periodic solutions of the corresponding linear equation with delay. With the aid of this maximum principle, under the assumption that the nonlinear function is quasi-monotonicity, we study the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions. [ABSTRACT FROM AUTHOR]

In this paper, we use the perturbation method and the mixed monotone iterative technique to discuss the existence of periodic solutions for impulsive evolution equations in ordered Banach spaces. Under impulsive functions satisfying broader monotone conditions and without assumption that the lower and upper solutions exist, we obtain the existence results of ω-periodic mild solutions. Moreover, an application is given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of the positive time periodic mild solutions for the evolution equation in an ordered Banach space E, $u'(t)+Au(t)=f(t,u(t))$, $t\in \mathbb{R}$, where $A:D(A)\subset E\rightarrow E$ is a closed linear operator and − A generates a positive compact semigroup $T(t)$ ( $t\geq0$) in E, the nonlinear function $f:\mathbb{R}\times E\rightarrow E$ is continuous and $f(t,x)$ is ω-periodic in t. We apply the operator semigroup theory and the Leray-Schauder fixed point theorem to obtain the existence of a positive ω-periodic mild solution under the condition that the nonlinear function satisfies a linear growth condition concerning the growth exponent of the semigroup $T(t)$ ( $t\geq0$). In the end, an example is given to illustrate the applicability of our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions to a second order evolution equation initial value problem in an ordered Banach space, which can model an elastic system with structural damping. Under monotone condition and the noncompactness measure condition on nonlinearity, we obtain the existence of extremal mild solutions and positive mild solutions. Moreover, some applications are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with nonlocal problem for fractional evolution equations with mixed monotone nonlocal term of the form where E is an infinite-dimensional Banach space, $${^CD^{q}_t}$$ is the Caputo fractional derivative of order $${q\in (0, 1)}$$ , A : D( A) ⊂ E → E is a closed linear operator and − A generates a uniformly bounded C-semigroup T( t) ( t ≥ 0) in E, $${f \in C(J\times E \times E, E)}$$ , and g is appropriate continuous function so that it constitutes a nonlocal condition. Under a new concept of coupled lower and upper mild L-quasi-solutions, we construct a new monotone iterative method for nonlocal problem of fractional evolution equations with mixed monotone nonlocal term and obtain the existence of coupled extremal mild L-quasi-solutions and the mild solution between them. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the initial value problem of fractional semilinear evolution equations with noncompact semigroup in Banach spaces. The existence of saturated mild solutions and global mild solutions is obtained under weaker conditions. Particularly, we find that the results obtained are also valid for the initial value problem of fractional ordinary differential equations in Banach spaces. The theorems proved in this paper improve and extend some related conclusions on this topic. An example is given to illustrate that our results are valuable. [Copyright &y& Elsevier]

In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E where A: D(A) ⊂ E → E is a closed linear operator and − A generates a strongly continuous semigroup T(t) ( t ⩾ 0) on E, f ∈ C( J × E × E, E), J = [0, a], 0 < t < t < ... < t < a, I ∈ C( E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that − A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a second order evolution equation in a Banach space, which can model an elastic system with structural damping. New forms of the corresponding first order evolution equation are introduced, and their well-posed property is proved by means of the operator semigroup theory. We give sufficient conditions for analyticity and exponential stability of the associated semigroups. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of mild -quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space . Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H where is a positive definite self-adjoint operator, and satisfy Caratheodory condition. We present some spectral conditions for the nonlinearity to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of the self-adjoint elliptic boundary value problems. [Copyright &y& Elsevier]

Abstract: In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro-differential equation of Volterra type in a Banach space where , , , and , . Under wide monotone conditions and the noncompactness measure condition of nonlinearity , we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. Our result improves and extends some relevant results in abstract differential equations. [Copyright &y& Elsevier]

This paper deals with the existence of ω -periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. L n u (t) = f (t , u (t) , u (t − τ)) , t ∈ R , where L n u (t) : = u (n) (t) + ∑ i = 0 n − 1 a i u (i) (t) , a i ∈ R , i = 0 , 1 , ⋯ , n − 1 , are constants, f (t , x , y) : R × E × E ⟶ E is continuous and ω -periodic with respect to t, τ > 0 . By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions for damped elastic systems in Banach spaces. The discussion is based on the operator semigroup theory and fixed point theorem. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence and uniqueness of periodic solutions for a class of fractional differential equations in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of nonlinearity, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. [ABSTRACT FROM AUTHOR]