Existence of Solution for a Conformable Fractional Cauchy Problem with Nonlocal Condition

Nonlinear nonlocal Cauchy problems in Banach spaces

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ−Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra R and commutative Banach algebra of the collection of continuous functions in R.

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

The main concern of the paper is to prove the existence, uniqueness and continuous dependence of mild and classical solutions of a semilinear impulsive second-order functional-differential equation with nonlocal initial conditions. We consider the Cauchy problem in general Banach spaces, and apply the theory of strongly continuous cosine families of linear operators and the Banach fixed-point theorem.

In this paper, we discuss the existence and uniqueness of mild and classical solutions of quasilinear mixed integrodifferential equations of frac- tional orders with nonlocal condition in Banach spaces. Furthermore, we study continuous dependence of mild solutions. Our analysis is based on fractional calculus, resolvent operators and Banach's fixed point theorem.

A class of time-fractional Dirac type operators

"By using the fixed-point theorems, we give sufficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. Keywords: Fractional boundary value problem, Riesz-Caputo fractional derivative, existence and uniqueness, fixed point, nonlocal conditions. "

<p style='text-indent:20px;'>This paper is concerned with the existence of mild solutions and exact controllability for a class of fractional measure evolution systems with state-dependent delay and nonlocal conditions. We first establish an existence result of mild solutions for the concerned problem by applying an integral equation which is given in terms of probability density and semigroup theory. Then, the exact controllability are obtained by using the fractional calculus theory, Kuratowski measure of noncompactness and Mönch fixed point theorem, without imposing the Lipschitz continuity on nonlinear term. Finally, we give two applications to support the validity of the study.</p>

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integro-differential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20

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.

In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33, 34K40, 35L90, 35L70, 35L15, 35L07

In this paper, two types of fixed point theorems are employed to study the solvability of nonlocal problem for implicit fuzzy fractional differential systems under Caputo gH-fractional differentiability in the framework of generalized metric spaces. First of all, we extend Krasnoselskii?s fixed point theorem to the vector version in the generalized metric space of fuzzy numbers. Under the Lipschitz conditions, we use Perov?s fixed point theorem to prove the global existence of the unique mild fuzzy solution in both types (i) and (ii). When the nonlinearity terms are not Lipschitz, we combine Perov?s fixed point theorem with vector version of Krasnoselskii?s fixed point theorem to prove the existence of mild fuzzy solutions. Based on the advantage of vector-valued metrics and convergent matrix, we attain some properties of mild fuzzy solutions such as the boundedness, the attractivity and the Ulam - Hyers stability. Finally, a computational example is presented to demonstrate the effectivity of our main results.

In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.