Search Results (14)

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

Starting from ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ${\psi }_k$-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), $\psi$-Riemann–Liouville ($\psi$-R-L), ${\psi }_k$-Riemann–Liouville (${\psi }_k$-R-L), Raina’s, k-Raina’s, and $\psi$-Raina’s fractional integrals ($\psi$-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

This paper deals with a nonlocal fractional coupled system of $(k,\psi )$-Hilfer fractional differential equations, which involve, in boundary conditions, $(k,\psi )$-Hilfer fractional derivatives and $(k,\psi )$-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. Full article

This article is a study on the $(k,s)$-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the $(k,s)$-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval $I=[0,1]$, which indicates that the upper Box dimension of the integrand $f\left(x\right)$ will not be increased by the $\sigma$-order $(k,s)$-Riemann–Liouville fractional integral ${}_k^s{D}^{-\sigma }f\left(x\right)$ where $\sigma >0$ on I. Additionally, we prove that the fractal dimension of ${}_k^s{D}^{-\sigma }f\left(x\right)$ of one-dimensional continuous functions $f\left(x\right)$ is still one. Full article

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, $(\zeta ,m)$-convex functions, s-convex functions, $(s,r)$-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, $MT$-convex functions, P-convex functions, m-convex functions, $(s,m)$-convex functions, exponentially s-convex functions, $(\beta ,m)$-convex functions, exponential-convex functions, $\overline{\zeta },\beta ,\gamma ,\delta$-convex functions, quasi-geometrically convex functions, $s-e$-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, $(\theta ,h-m)-p$-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, $(k-p)$-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and $(p,q)$-calculus are also included. Full article

In this paper, we study a coupled fully hybrid system of $(\mathbf{k},\mathsf{\Phi })$–Hilfer fractional differential equations equipped with non-symmetric $(\mathbf{k},\mathsf{\Phi })$–Riemann-Liouville ($\mathcal{RL}$) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space ($\mathcal{GBS}$). Moreover, the Ulam–Hyers ($\mathcal{UH}$) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, $(\alpha ,m)$-convex functions, $(\alpha ,m)$-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically $(\theta ,m)$-convex functions, m-harmonic harmonically convex functions, $(s,r)$-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, $(\alpha ,m)$-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable $\varphi$-convex functions, $MT$-convex functions, $(s,m)$-convex functions, p-convex functions, h-convex functions, $\sigma$-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, $\sigma ,s$-convex functions, modified $(p,h)$-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, $(\alpha ,h-m)-p$-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, $(k,s)$-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator. Full article

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green’s function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area. Full article

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators. Full article

This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled $(k,\psi )$–Hilfer fractional differential equations subject to coupled $(k,\psi )$–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results. Full article

The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of $\mathbf{q}$-digamma and $\mathbf{q}$-polygamma special functions are presented. Full article

In this article, we establish the weighted $(k,s)$-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms. Full article

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral operators involving hypergeometric function in the kernel are applied to the product of Srivastava’s polynomials and the generalized Mathieu series, containing the factor $x^{\lambda }{(x^k+c^k)}^{-\rho }$ in its argument. The results are expressed in terms of the generalized hypergeometric function and Hadamard product of the generalized Mathieu series. Corresponding special cases related to the Riemann–Liouville and Erdélyi–Kober fractional integral operators are also considered. Full article