Asymptotic Behavior of Positive Solutions for Three Types of Fractional Difference Equations with Forcing Term
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Asymptotic Behavior of Positive Solutions for Three Types of Fractional Difference Equations with Forcing Term ¨ 2 · Jehad Alzabut3 Said R. Grace1 · Hakan Adıguzel
· Jagan Mohan Jonnalagadda4
Received: 20 March 2020 / Accepted: 30 June 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we consider three types of fractional difference equations with forcing term. The asymptotic behaviors of positive solutions for the proposed equations are extensively investigated. Our approach mainly relies on the features of discrete fractional calculus and some mathematical inequalities. To conclude our theoretical findings, we give an example demonstrating complete consistency to the main results. Keywords Positive solutions · Fractional difference equation · Asymptotic behavior Mathematics Subject Classification (2010) 26A33 · 34A08 · 34B82 · 34B15
1 Introduction Unlike its continuous counterpart and driven by their wide spread applications in numerical computations, the qualitative study of fractional difference equations has gained extensive attention in the last years. The combined efforts of a number of researchers have laid a strong foundation for the basic theory of fractional difference equations [2–5, 11]. However, we note that most of the literature on discrete fractional calculus has been devoted to the solvability of initial and boundary value problems. Results on the asymptotic behavior of solutions for fractional difference equations have been comparably scarce; the reader can refer to some results in [1, 6, 7, 12, 15–18]. Jehad Alzabut
[email protected] 1
Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12221, Egypt
2
Department of Architecture and Urban Planning, Vocational School of Arifiye, Sakarya University of Applied Sciences, Geyve, 54580, Sakarya, Turkey
3
Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
4
Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad, 500078, Telangana, India
S.R. Grace et al.
In [15–17], the authors established some comparison theorems by which they compared the solutions of equations. And the obtained theorems extended the literature on the subject. In [6], Bai and Xu studied asymptotic behavior of solutions of nonlinear fractional difference equations with damping term. They presented some oscillation results based on generalized Riccati transformation technique and some inequalities. To the best of authors’ expectation, the literature contains no result for positive solutions of forced fractional difference equations of the following type: α t ∈ N1−α , 0 < α ≤ 1, C y(t) = d(t + α) + f (t + α, x(t + α)), (1.1) y(0) = c0 , where C α is the αth-order Caputo-like delta fractional difference operator, d : N1 → (0, ∞), f : N1 × R → R is continuous with respect to both arguments, and satisfies xf (t, x) > 0 for x = 0 and t ∈ N1 = {1, 2, 3, . . .}. Motivated by the id
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