Existence of solution to fractional differential equation with fractional integral type boundary conditions

Abstract

This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problem Dγfrakturufalse(ℓfalse)=ψfalse(ℓ,frakturufalse(λℓfalse)false),0.1emℓ∈frakturZ=false[0,1false], with fractional integral boundary conditions p1u(0)+q1u(1)=1Γ(γ)∫01(1−ρ)γ−1g1(ρ,u(ρ))dρ, and p2u′(0)+q2u′(1)=1Γ(γ)∫01(1−ρ)γ−1g2(ρ,u(ρ))dρ, where γ ∈ (1, 2], 0 < λ < 1, D denotes the Caputo fractional derivative (in short CFD), ψ,g1,g2:frakturZ×frakturR→frakturR are continuous functions and frakturpi,frakturqifalse(i=1,2false) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end.

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

“…where g ϑ (s) = g(s, ϑ(s), D β 0+ ϑ(s), D γ 0+ ϑ(s)), G is defined in (4). In order to prove that problem (1), (2) has a solution, we just have to show that operator G has a fixed point.…”

“…When λ 2 = 0, 0 β 1, there has been a great deal of literature on the fractional differential equation of such boundary conditions; see [2,4,9,15,16,30]. As for β = 0, for example, in [2], D α u(t) + f t, u(t), D µ u(t) = 0, a.e.…”

Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

“…In [11], Shoaib et al studied other existence results via f-contractions of Nadler type in 2020. After that, recently, Ali et al [12] considered a nonlinear fractional differential equation equipped with the integral type boundary conditions and proved the existence results with the help of topological degree theory.…”

Existence and uniqueness of solution for fractional differential equations with integral boundary conditions and the Adomian decomposition method

“…$$ where $$$ 2&amp;amp;lt;\alpha &amp;amp;lt;3 $$$, $$$ 0&amp;amp;lt;\lambda &amp;amp;lt;2 $$$, $$$ {}&amp;amp;#x0005E;C{D}&amp;amp;#x0005E;{\alpha } $$$ is the Caputo fractional derivative and $$$ f:\left[0,1\right]\times \left[0,\infty \right)\to \left[0,\infty \right) $$$ is a continuous function, establishing the existence of positive solutions with the help of the Guo–Krasnoselskii fixed point theorem 9 . Given the fact that problems with integral boundary conditions arise naturally in many applied fields of science, like thermal conduction problems, semiconductor problems, chemical engineering, blood flow problems, underground water problems, hydrodynamic problems and population dynamics, and include multipoint and nonlocal integral boundary value conditions as special cases, the work of Cabada and Wang 9 originated a strong research on integral boundary value problems of nonlinear multiterm fractional differential equations; see, for example, previous studies 10–13 . Among recent methods that are useful for such kind of fractional differential equations, we can mention the monotone iterative technique, 14 the topological degree theory, 15 and fixed point approaches 16,17 …”

“…9 Given the fact that problems with integral boundary conditions arise naturally in many applied fields of science, like thermal conduction problems, semiconductor problems, chemical engineering, blood flow problems, underground water problems, hydrodynamic problems and population dynamics, and include multipoint and nonlocal integral boundary value conditions as special cases, the work of Cabada and Wang 9 originated a strong research on integral boundary value problems of nonlinear multiterm fractional differential equations; see, for example, previous studies. [10][11][12][13] Among recent methods that are useful for such kind of fractional differential equations, we can mention the monotone iterative technique, 14 the topological degree theory, 15 and fixed point approaches. 16,17 Generally, most fractional differential equations do not have exact/analytical solutions.…”