Abstract
The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.
Highlights
It can be said that most physical or engineering phenomena can be modeled with some categories such time-dependent, fractional differential and some variables partial equations
It is important that mathematicians and researchers design complicated and more general abstract mathematical models of procedures in the format of applicable fractional SLDE [33, 39–41]
First we investigate the partial fractional Sturm–Liouville differential equation
Summary
It can be said that most physical or engineering phenomena can be modeled with some categories such time-dependent (or time-fractional), fractional differential and some variables partial equations. One of the important frameworks of problems is the Sturm–Liouville differential equation (in brief SLDE) have been in the spotlight of the mathematicians of applied mathematics, engineering and scientists of physics, quantum mechanics, classical mechanics (see, [37, 38] and the references therein) In such a manner, it is important that mathematicians and researchers design complicated and more general abstract mathematical models of procedures in the format of applicable fractional SLDE [33, 39–41]. Lemma 1 ([45]) Assume that the metric space (Z , dZ ) is complete, T is α-admissible and α-ψ -contractive mapping and there exists σ0 ∈ Z such that α(σ0, Tσ0) ≥ 1. Lemma 2 ([46]) Assume that the metric space (Z , dZ ) is complete, F is α-admissible and α-ψ -contractive multifunction and there exist σ0 ∈ Z and σ1 ∈ Fσ0 such that α(σ0, σ1) ≥ 1. H : Ja0 × Jb0 × R → Pcl(R) is an integrable bounded multifunction so that H(., ., σ ) is measurable for all σ ∈ R
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