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In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

In this paper, we use a new technique for the treatment of systems based on the advantage of vector-valued norms and of the weak topology. We first present vector versions of the Leray-Schauder alternative and then some Krasnoselskii type fixed point theorems for a sum of two mappings. Applications are given to a system of nonlinear transport equations, and systems of mixed fractional differential equations. [ABSTRACT FROM AUTHOR]

This article is about Ulam's type stability of nth order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam's type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder's inequality and the beta integral inequality. [ABSTRACT FROM AUTHOR]

This article deals with results concerning the existence of solutions for some functional implicit of Caputo type modification of the Erdélyi-Kober fractional differential equations, with both retarded and advanced arguments. The results are based on the Mönch's and Darbo's fixed point theorem associated with the technique of measure of noncompactness, and we present an illustrative example. [ABSTRACT FROM AUTHOR]

We provide some sufficient conditions ensuring the existence of solutions of Caputo type modification of the Erdélyi-Kober fractional differential inclusions in Banach spaces with retarded and advanced arguments. We use Mönch's and Darbo's fixed point theorems and the concept of a measure of noncompactness. As illustration, an example is presented. [ABSTRACT FROM AUTHOR]

Let L = - div ⁡ ( A ⁢ ∇ ) {L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, n × n {n\times n} matrix with bounded measurable complex coefficients in ℝ n {{\mathbb{R}}^{n}}. Let L - α 2 {L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for 0 < α < n {0<\alphak-th order commutator of b and L - α 2 {L^{-\frac{\alpha}{2}}} is given by ( L - α 2 ) b , k ⁢ f ⁢ ( x ) = L - α 2 ⁢ ( ( b ⁢ ( x ) - b ) k ⁢ f ) ⁢ ( x ) . (L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x). In the paper, we mainly show that if b ∈ BMO ⁢ ( ℝ n ) {b\in\mathrm{BMO}({\mathbb{R}}^{n})} , 0 < λ < n {0<\lambda 0 {\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value problem for implicit fractional differential equations with Caputo fractional derivative. We apply the technique of measure of noncompactness and the fixed point theorems of Darbo and Mönch. As an application, two examples are included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

This work is concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a class of semilinear fractional differential equations, $D_{t}^{\alpha}x(t)=Ax(t)+D_{t}^{\alpha-1}F(t,x(t))$, $t\in \mathbb{R}$, where $1<\alpha<2$, A is a linear densely defined operator of sectorial type of $\omega<0$ on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a fractional relaxation-oscillation equation. [ABSTRACT FROM AUTHOR]

On generalized Morrey spaces with variable exponent [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions, for a class of boundary value problem for fractional differential equations involving the Riemann-Liouville fractional derivative on an unbounded interval in Banach spaces. Our main tool is the technique of measures of noncompactness combined with Mönch's fixed point theorem and the diagonalization process. [ABSTRACT FROM AUTHOR]

Two systems of fractional order functional and neutral functional differential equations with infinite delay are studied in this article, and some sufficient conditions for existence of unique solutions for these two systems are established by Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

Abstract: Of concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial condition where , , is a generator of a solution operator on a complex Banach space , the convolution integral in the equation is known as the Riemann–Liouville fractional integral, representing the delay property, is a function, and is an operator defined from into for some which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, we study the existence and uniqueness of initial value problems for nonlinear fractional differential equations and obtain a new result. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann–Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray–Schauder type. [Copyright &y& Elsevier]

Abstract: We study -asymptotically -periodic solutions of the semilinear fractional equation , considered in a Banach space , where is a linear operator of sectorial type . [Copyright &y& Elsevier]

Abstract: We study almost automorphic (mild) solutions of the semilinear fractional equation , considered in a Banach space , where is a linear operator of sectorial type . We prove the existence and uniqueness of an almost automorphic mild solution assuming that is almost automorphic in for each , satisfies some Lipschitz type conditions and takes values on . [Copyright &y& Elsevier]

In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results. [ABSTRACT FROM AUTHOR]