Noncompactness Measure Condition Research Articles

Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.

Mixed Monotone Iterative Technique for Singular Hadamard Fractional Integro-Differential Equations in Banach Spaces

This paper deals with fractional integro-differential equations involving Hadamard fractional derivatives and nonlinear boundary conditions in an ordered Banach space. The nonlinearity is allowed to be singular with respect to time variable. Under some monotonicity conditions and noncompactness measure conditions, we use the method of coupled lower and upper L-quasisolutions associated with the mixed monotone iterative technique to investigate the existence of extremal L-quasisolutions. A unique solution between coupled lower and upper L-quasisolutions is also obtained. An example is given to illustrate our theoretical results. The results got in this paper are new and enrich the existing related work.

STUDY OF FRACTIONAL ORDER DELAY CAUCHY NON-AUTONOMOUS EVOLUTION PROBLEMS VIA DEGREE THEORY

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

Extremal solutions of $ \varphi- $Caputo fractional evolution equations involving integral kernels

This paper deals with the existence and uniqueness of solution for the Cauchy problem of $ \varphi- $Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.

Monotone iterative method for retarded evolution equations involving nonlocal and impulsive conditions

In this article, we apply the perturbation technique and monotone iterative method in the presence of the lower and the upper solutions to discuss the existence of the minimal and maximal mild solutions to the retarded evolution equations involving nonlocal and impulsive conditions in an ordered Banach space X $$\displaylines{ u'(t)+Au(t)= f(t,u(t),u_t),\quad t\in [0,a],\; t\neq t_k,\cr u(t_k^+)=u(t_k^-)+I_k(u(t_k)),\quad k=1,2,\dots ,m,\cr u(s)=g(u)(s)+\varphi(s),\quad s\in [-r,0], }$$ where \(A:D(A)\subset X\to X\) is a closed linear operator and -A generates a strongly continuous semigroup T(t) \((t\geq 0)\) on X, a, r>0 are two constants, \(f:[0,a]\times X\times C_0\to X\) is Caratheodory continuous, \(0<t_1<t_2<\dots<t_m<a\) are pre-fixed numbers, \(I_k\in C(X,X)\) for \(k=1,2,\dots,m\), \(\varphi\in C_{0}\) is a priori given history, while the function \(g:C_{a}\to C_{0}\) implicitly defines a complementary history, chosen by the system itself. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the minimal and maximal mild solutions, the existence of at least one mild solutions as well as the uniqueness of mild solution between the lower and the upper solutions. An example is given to illustrate the feasibility of our theoretical results. For more information see https://ejde.math.txstate.edu/Volumes/2020/68/abstr.html

Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.

Local attractivity for integro-differential equations with noncompact semigroups

Abstract In this paper, we are devoted to study the existence and local attractivity of solutions for a class of integro-differential equations.Under the situation that the nonlinear term satisfy Carathéodory conditions and a noncompactness measure condition, we establish some existence and local attractivity of mild solutions by utilizing Mönch fixed point theorem, Kuratowski measure of noncompactness and resolvent operator theory in the sense of Grimmer.Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the right-hand real half axis 𝕉+. Moreover an example is given to illustrate our outcomes.

Cauchy problem for fractional non-autonomous evolution equations

This paper deals with the following Cauchy problem to nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness $$ \left\{\begin{array}{ll} ^CD^{\alpha}_tu(t)+A(t)u(t)= f(t,u(t),(Tu)(t), (Su)(t)),\quad t\in [0,a], \\[12pt] u(0)=A^{-1}(0)u_0 \end{array} \right. $$ in infinite-dimensional Banach space $E$, where $ ^CD^{\alpha}_t$ is the standard Caputo's fractional time derivative of order $0<\alpha\leq 1$, $A(t)$ is a family of closed linear operators defined on a dense domain $D(A)$ in Banach space $E$ into $E$ such that $D(A)$ is independent of $t$, $a>0$ is a constant, $f:[0,a]\times E\times E\times E\rightarrow E$ is a Carath\'{e}odory type function, $u_0\in E$, $T$ and $S$ are Volterra and Fredholm integral operators, respectively. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. As samples of applications, we consider the initial value problem to a class of time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition at the end of this paper.

Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

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.

Mixed monotone iterative technique for damped elastic systems in Banach spaces

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.

Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups

Abstract The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

Fractional non-autonomous evolution equation with nonlocal conditions

The aim of this paper is to discuss the existence of mild solutions for a class of time fractional non-autonomous evolution equations with nonlocal conditions and measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to k-set-contractive operator and a new estimation technique of the measure of noncompactness, we obtain the new existence results of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic by deleting the compactness condition on nonlocal function g and extending the study of fractional autonomous evolution equations in recent years to non-autonomous ones. Finally, as samples of applications, we consider a time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition and nonlocal conditions.

Monotone iterative technique for periodic problem involving Riemann\u2013Liouville fractional derivatives in Banach spaces

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.

Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups

This paper deals with the existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions and and noncompact semigroups in a real separable Hilbert space. We assume that the linear part generates an equicontinuous -semigroup, and the nonlinear functions satisfy some appropriate growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic to the case that the corresponding solution semigroup is not compact. An example is also given to demonstrate the applications of our abstract result.

Study on Sobolev type Hilfer fractional integro-differential equations with delay

In this paper, we deal with a class of nonlinear Sobolev type fractional integro-differential equations with delay using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. The definition of mild solutions for studied problem was given based on an operator family generated by the operator pair (A, B) and probability density function. Combining with the techniques of fractional calculus, measure of noncompactness and fixed point theorem, we obtain new existence result of mild solutions with two new characteristic solution operators and the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

Study a class of nonlinear fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear fractional non-autonomous evolution equations with delay by using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. Combining techniques of fractional calculus, measure of noncompactness and some fixed point theorem, we obtain new existence result of mild solutions when the associated semigroup is not compact. Furthermore, the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions. Finally, two examples will be presented to illustrate the main results.

Study on fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ(t,s), φ(t,η) and U(t), which are generated by the operator −A(t) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k-set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator −A(t) generates an analytic semigroup for every t>0. The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

Perturbation technique for a class of nonlinear implicit semilinear impulsive integro-differential equations of mixed type with noncompactness measure

By using the Arzela-Ascoli theorem, the Bellman inequality, and a monotone perturbation iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for a class of nonlinear first-order implicit semilinear impulsive integro-differential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.

Monotone iterative technique for the elastic systems with structural damping in Banach spaces

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.

Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

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 $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \end{gathered} \right.$$ 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 1 < t 2 < ... < t m < a, I k ∈ 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.