Differential Inclusions In Banach Spaces Research Articles

Oscillation and nonoscillation for Caputo-Hadamard fractional differential inclusions in Banach spaces

Oscillation and nonoscillation for Caputo-Hadamard fractional differential inclusions in Banach spaces

Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order α∈(0,1) for a continuous (locally integrable) function u:[0,∞)→E, where E is a complex Banach space. In our definition, we do not require that the function u(·) is continuously differentiable, which enables us to consider the wellposedness of the corresponding fractional relaxation problems in a much better theoretical way. More precisely, we systematically investigate several new classes of (degenerate) fractional solution operator families connected with the use of this type of fractional derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces are also considered.

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

Boundary value problems for Caputo-Hadamard fractional differential inclusions in Banach spaces

Boundary value problems for Caputo-Hadamard fractional differential inclusions in Banach spaces

Viability result for semilinear functional differential inclusions in Banach spaces

We show the existence result of a mild solution for a semilinear functional differential inclusion, with viability, governed by a family of linear operators. We consider the case when the constraint is moving.

Finite-time attractivity for semilinear functional differential inclusions

Our aim in this paper is to analyze a sufficient condition ensuring the finite-time attractivity for the solutions to semilinear functional differential inclusions in Banach spaces. The obtained results will be applied to a class of partial differential equations.

Existence for Impulsive Semilinear Functional Differential Inclusions

In this paper we investigate the existence of solutions for first-order impulsive semilinear functional differential inclusions in Banach spaces. Sufficient condition for the existence is obtained with the well-known Covitz and Nadler’s fixed point theorem for multivalued contractions.

Finite-time attractivity of solutions for a class of fractional differential inclusions with finite delay

Our aim in this paper is to give a sufficient condition ensuring the finite-time attractivity for the zero solution to semilinear functional differential inclusions in Banach spaces, in the case where the nonlinearity function possibly has superlinear growth. Our analysis is based on the semigroup theory, the fixed point principle for condensing multi-valued maps, and local estimates of solutions. The abstract results will be applied to a class of polytope inclusions in $$C_0$$ setting.

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

Asymptotically Weyl almost periodic functions in Lebesgue spaces with variable exponents

In this paper, we introduce and analyze several different notions of Weyl almost periodic functions and Weyl ergodic components in Lebesgue spaces with variable exponent Lp(x). We investigate the invariance of (asymptotical) Weyl almost periodicity with variable exponent under the actions of convolution products, providing also some illustrative applications to abstract fractional differential inclusions in Banach spaces. The introduced classes of generalized (asymptotically) Weyl almost periodic functions are new even in the case that the function p(x) has a constant value p≥1, provided that the functions ϕ(x) and F(l,t) under our consideration satisfy ϕ(x)≠x or F(l,t)≠l(−1)/p.

Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents, Part II

In this paper, we introduce and analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with variable exponent L p ( x ) . We primarily consider the Stepanov and Weyl classes of generalized almost periodic type functions and generalized uniformly recurrent type functions. We also investigate the invariance of generalized almost periodicity and generalized uniform recurrence with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract fractional differential inclusions in Banach spaces.

Evolution Inclusions in Banach Spaces under Dissipative Conditions

We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions.

Viable Solutions for a Class of Delay Evolution Problems

In this paper, we give existence results for viable solutions in the so-called fully constrained set for functional differential inclusions in Banach spaces for a non-autonomous set-valued mapping with convex compact values. We study also the time dependent case of these invariance sets.

Nonlocal Integro-Differential Equations Without the Assumption of Equicontinuity on the Resolvent Operator in Banach Spaces

In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$ without the assumption of equicontinuity on the resolvent operator and without the assumption of separability on the Banach space X. The nonlocal initial condition is assumed to be compact. Our main result is new and its proof is based on a measure of noncompactness developed in Kamenskii et al. (Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Walter De Gruyter, Berlin, 2001) together with the well-known Monch fixed point Theorem. To illustrate our result, we provide an example in which the resolvent operator is not equicontinuous.

Coupled systems of Hilfer fractional differential inclusions in banach spaces

This paper deals with some existence results in Banach spaces for Hilfer and Hilfer-Hadamard fractional differential inclusions. The main tools used in the proofs are Monch's fixed point theorem and the concept of a measure of noncompactness.

A general class of noninstantaneous impulsive fractional differential inclusions in Banach spaces

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results.

On semilinear fractional order differential inclusions in Banach spaces

On semilinear fractional order differential inclusions in Banach spaces

Abstract degenerate fractional differential inclusions in Banach spaces

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

Abstract degenerate fractional differential inclusions in Banach spaces

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

Existence results of second-order impulsive neutral functional differential inclusions in Banach spaces

In this paper, we investigate the existence of solutions for a class of second-order impulsive neutral functional differential inclusions in Banach spaces. Sufficient conditions for the existence are derived with the help of the fixed point theorem for multivalued maps due to Dhage.