Existence of weak solutions for a new class of fractional boundary value impulsive systems with Riemann–Liouville derivatives

Abstract

The existence of three solutions for perturbed systems of impulsive non-linear fractional di¤erential equations together with Lipschitz continuous non-linear terms is discussed in this paper. The idea relies on variational methods. Moreover, an example is presented in order to summarize the feasibility and e¤ectiveness of the main results. 1. Introduction In this work, we aim to study the perturbed impulsive fractional di¤erential system: (1.1) 8><>>>>: tD i T (ai (t) 0D i t ui (t)) = Fui (t; u) + Gui (t; u) + hi (ui (t)) ; t 2 [0; T ] ; t 6= tj 4 tD i 1 T ( c 0D i t ui) (tj) = Iij (ui (tj)) ; j = 1; 2; ;m ui (0) = ui (T ) = 0 ; for 1 i n; where u = (u1; u2; ; un) ; n 2; 0 < i 1 for 1 i n; > 0; > 0; T > 0; ai 2 L1 ([0; T ]) ; ai = ess inf t2[0;T ] ai (t) > 0 i.e kaik1 = inf C 2 R; jaij C ; 0D i t and tD i T designate the left and right Riemann–Liouville fractional derivatives of order i respectively, F;G : [0; T ] R ! R are quanti…able with respect to t, for all u 2 R; continuously di¤erentiable in u, for approximately every t 2 [0; T ] such that F1) For every > 0 and every 1 i n, n supj j & jF ( ; )j ; supj j & jG ( ; )j o 2 L ([0; T ]) ; for any > 0 with = ( 1; 2; ; n) and j j = qPn i=1 2 i : F2) F (t; 0; ; 0) = 0; for every t 2 [0; T ] : hi : R ! R is a Lipschitz continuous function with the Lipschitz constant Li > 0; i:e: Date : October 10th, 2020. 1991 Mathematics Subject Classi…cation. 35J60, 35B30, 35B40. Key words and phrases. AMS subject classi…cations: (2000) 35J60 35B30 35B40. *Corresponding author. 1 10 Oct 2020 10:16:52 PDT 200120-Boulaaras Version 4 Submitted to J. Integr. Eq. Appl. 2 FARES KAMACHE, SALAH BOULAARAS ,3 ; RAFIK GUEFAIFIA jhi ( 1) hi ( 2)j Li j 1 2j for every 1; 2 2 R; satisfying hi (0) = 0 for 1 i n; Iij 2 C (R;R) for i = 1; ; n; j = 1; ;m; 0 = t0 < t1 < t2 < < tm < tm+1 = T: The operator is de…ned as 4 tD i 1 T ( c 0D i t ui) (tj) =t D i 1 T ( c 0D i t ui) t+j t D i 1 T ( c 0D i t ui) t j where tD i 1 T ( c 0D i t ui) t+j = lim t!t+j tD i 1 T ( c 0D i t ui) (t) and tD i 1 T ( c 0D i t ui) t j = lim t!t j tD i 1 T ( c 0D i t ui) (t) and 0D i T represents the left Caputo fractional derivatives with an i order. While, Fui and Gui respectively represent the partial derivatives of F and G following ui for 1 i n: Recently, Fractional Di¤erential Equations (FDEs) have emerged as important approaches that can be used to describe numerous phenomena in several …elds of science such as physics and engineering. In fact, many applications related to viscoelasticity and porous media, as well as electrochemistry and electromagnetic can be modeled through the FDEs (see [15], [27], [33], [32] and [41] ). In the last few years, many papers focused on the search of solutions to boundary value problems for FDEs, we may refer to ([42], [4], [5], [13], [16], [18], [20], [21], [28], [34], [37], [38], [39] and [40]) and the references therein. It is worthy of note that Zhang et al. [46] have checked the existence of di¤erent solutions for a class of fractional advection–dispersion equations outcoming from a symmetric transition of the mass ‡ux by considering a variational structure and applying the the critical point theory. Moreover, Chen and Tang [13] have worked on the existence and multiplicity of solutions when considering such a fractional boundary value problem: ( d dt 1 2 0D t (u 0 (t)) + 12 tD T (u 0 (t)) + rF (t; u (t)) = 0; t 2 [0; T ] ; u (0) = u (T ) = 0 ; where T > 0; > 0; 0 < 1; 0D t ; and tD T are the left and right Riemann– Liouville fractional integrals of order ; respectively, F : [0; T ] R ! R is a given function and rF (t; x) is the gradient of F at x and F (t; :) are respectively super quadratic, asymptotically quadratic and sub-quadratic. In [25], employing a light version of [[9], Theorem 2.1], using a suitable oscillating response of the non-linear function F , it will be possible to determine the exact assembly of the coe¢ cient in which the system (1.1) will in…nitely accept several weak solutions (Theorem 3.1) for any non-negative arbitrary function G : [0; T ] R ! R measurable in [0; T ] and of class C R growing at in…nity, when choosing small enough. Changing the oscillating shape condition at in…nity, with an analogous one at zero, it will be possible to get a serie of pairwise distinct weak solutions converging to zero (Theorem 3.4). In [50], when = 0, problem (1.1) admits at least two nontrivial and nonnegative solutions (Theorem 3.2) : With this regard, we aim through this work to study the abundance of nontrivial and nonnegative solutions for system (1.1) with Lipschitz continuous impulsive e¤ects. Using certain logical 10 Oct 2020 10:16:52 PDT 200120-Boulaaras Version 4 Submitted to J. Integr. Eq. Appl.