Existence of three solutions for perturbed nonlinear fractional p -Laplacian boundary value systems with two control par
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Existence of three solutions for perturbed nonlinear fractional p-Laplacian boundary value systems with two control parameters Fares Kamache1,2 · Rafik Guefaifia1,2 · Salah Boulaaras3,4 Received: 17 March 2020 / Revised: 21 June 2020 / Accepted: 23 June 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we use two control parameters to study a class of perturbed nonlinear fractional p-Laplacian differential systems, where we prove the existence of three weak solutions by using the variational method and Ricceri’s critical points theorems respecting some necessary conditions on the primitive function of nonlinear terms Fu and Fv . Keywords Nonlinear fractional Dirichlet boundary value problems · p-Laplacian type · Variational method · Critical point theory Mathematics Subject Classification 35J60 · 35B30 · 35B40
On the occasion of the 80th birthday of the third author’s mother, Mrs. Fatma Bint Al-Tayeb Zeghdoud.
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Salah Boulaaras [email protected]; [email protected] Fares Kamache [email protected] Rafik Guefaifia [email protected]
1
Department of Mathematics and Computer Science, Larbi Tebessi University, Tebessa, Algeria
2
Laboratory of Mathematics, Informatics and systems (LAMIS), Larbi Tebessi University, Tebessa, Algeria
3
Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Buraydah, Kingdom of Saudi Arabia
4
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran , Algeria
F. Kamache et al.
1 Introduction The fractional differential equations can generally be seen as the study of differential equations with the fractional calculus application. With its use, the natural phenomena and mathematical models in several areas of science and engineering can be precisely described. The Fractional differential equations have also many uses in different domains like engineering, physics, chemistry, biology, mechanics, biophysics, and other fields (see [1–7]). As a result, many improvements have been made in the theory of fractional calculus and fractional ordinary and partial differential equations ([2,5,8–15]). Several studies have explored the existence and different solutions for nonlinear fractional initial and boundary value problems through the use of several tools and techniques of nonlinear analysis (see for example [16–20]. Some of these ways are the fixed point theorems, critical point theory,the monotone iterative methods, the coincidence degree theory, and variational methods. Motivated by different papers interested in this area, we are interested in this article in the existence of results for the following perturbed fractional differential system: ⎧ 1 α α ⎪ + μ |u (t)| p−2 u (t) t DT w (t) p−2 p w1 (t)0 Dt u (t) ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ , v (t)) a.e. t ∈ [0, T ] , ⎪ ⎨ = λFu (t, u (t) , v (t)) + δG u (t, u (t) β β 1 D w2 (t)0 Dt v (t) + μ |v (t)| p−2 v (t) ⎪ ⎪ t T w2 (t) p−2 p ⎪ ⎪ ⎪ ⎪ = λFv (t, u (t) , v (t)) + δG v (t, u (t) , v (t)) a.e. t ∈ [0, T ] , ⎪
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