In this paper, using the Riemann‐Liouville fractional integral with respect to another function and the ψ−Hilfer fractional derivative, we propose a fractional Volterra integral equation and the ...fractional Volterra integro‐differential equation. In this sense, for this new fractional Volterra integro‐differential equation, we study the Ulam‐Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed‐point theorem. As an application, we present the Ulam‐Hyers stability using the α‐resolvent operator in the Sobolev space
W1,1(R+,C).
In this paper, we consider the new class of the fractional differential equation involving the Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of ...Ulam‐Hyers on the compact interval Δ = a,b and on the infinite interval I = a,∞). Our analysis is based on the application of the Banach fixed‐point theorem and the Gronwall inequality involving generalized Ψ‐fractional integral. At last, we performed out an application to elucidate the outcomes got.
The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational ...methods in ψ-fractional space Hpα,β;ψ(Ω). The results obtained in this paper are the first to make use of the theory of ψ-Hilfer fractional operators with p-Laplacian.
We introduce a new version of
ψ$$ \psi $$‐Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an ...integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a
ψ$$ \psi $$‐Riemann–Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function
ψ$$ \psi $$, opening new directions of investigation. Finally, we end the article with comments and future work.
In the present paper, we investigate the Hardy–Littlewood type and the integration by parts result for
ψ$$ \psi $$–Riemann–Liouville fractional integrals. Also, we attack the integration by parts for ...the
ψ$$ \psi $$–Riemann–Liouville and
ψ$$ \psi $$–Hilfer fractional derivatives. To finish, we investigated Sobolev‐type inequalities involving the
ψ$$ \psi $$–Riemann–Liouville and the
ψ$$ \psi $$–Hilfer fractional derivatives in weighted space.
This paper is divided into two stages. In the first stage, we investigated a new approach for the
ψ$$ \psi $$‐Riemann–Liouville fractional integral and the Faa di Bruno formula for the
ψ$$ \psi ...$$‐Hilfer fractional derivative. In addition, we discussed other properties involving the
ψ$$ \psi $$‐Hilfer fractional derivative and the
ψ$$ \psi $$‐Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the
ψ(·)$$ \psi \left(\cdotp \right) $$ function are investigated and the
ψ$$ \psi $$‐Riemann–Liouville fractional integral and
ψ$$ \psi $$‐Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the
ψ$$ \psi $$‐Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro‐differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.
The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational ...methods in ψ-fractional space Hsub.p sup.α,β;ψ(Ω). The results obtained in this paper are the first to make use of the theory of ψ-Hilfer fractional operators with p-Laplacian.
In this article, we investigate the existence and multiplicity of solutions for a fractional differential equations with p-Laplacian equation at resonance in the \(\psi\)-fractional space ...\(H^{\alpha, \beta; \psi}_p\). In addition, we show that the energy functional satisfies the Palais-Smale condition. For more information see https://ejde.math.txstate.edu/Volumes/2024/34/abstr.html
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