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S. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266-273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i(A, Ω, P ), is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax = μx, which extend many famous theorems such as Leray-Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works.

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We study the Ulam-Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam-Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results. KEYWORDS Caputo operator, nabla difference equation, Ulam-Hyers stability MSC CLASSIFICATION 39A70; 39A12 INTRODUCTIONUnder what conditions one can still say that the result of a theorem holds or is approximately true when altering slightly the assumption? This question was originally raised by Ulam during a talk at the University of Wisconsin in 1940. 1,2 Then, after a year, Hyers answered this question for a special case. 3 After that, considering Cauchy differences, Rassias 4 generalized the work.Since then, the concept of Ulam stability has been investigated as well as extended in several different directions. There are many papers on this issue that have been published so far. Jung et al [5][6][7]8,9 for example, have considered the Ulam-Hyers stability for differential equations. Besides, Ulam-Hyers stabilities for fractional cases have been studied, one can see in previous studies. [10][11][12][13][14][15][16] Moreover, some works have been done for difference equations. Jung and Nam, 17-20 for example, have investigated Ulam-Hyers stability for first-order difference equations, first-order matrix difference equations, and first-order inhomogeneous matrix difference equations. Besides, Ulam-Hyers stability for nonhomogeneous linear difference equations with constant stepsize has been studied by Onitsuka. 21 For the study of fractional difference equations, one can see Bas and Ozarslan, Chen et al, There are only a few papers which consider the Ulam-Hyers stability for fractional difference equations. Recently, Jagan Mohan Jonnalagadda 25 has formulated and obtained Ulam-Hyers stability for Riemann-Liouville fractional nabla difference equations. His approach was using the N-transform. To obtain the Ulam-Hyers stability result, see Jagan Mohan. 25, Theorem 8 and 10 He used the estimation of nabla Mittag-Leffler functions shown in a previous study. 25, Lemma 4 Unfortunately, it is important to mention that this argument has not been proven rigorously since it is not trivial. Because of the fact that we do not know whether the estimations F (− , t ) ≤ 1 and F , (− , t ) ≤ 1 Γ( ) are correct or not, we cannot get 25, (24), (31) easily. Thus, the proofs of Jagan Mohan 25, theorem 8 and 10 are incomplete. However, in our paper, we use a Math Meth Appl Sci. 2019;42:7461-7470.wileyonlinelibrary.com/journal/mma

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In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties.MSC: Primary 26A48; 26A51; 39A06; 39A10; 39A22; 39A30; 39B62; secondary 26A33; 34D05; 39A12; 39A30; 39A60; 39A99; 39B99

In this paper we study the nabla and delta fractional differences and obtain the following two main results:This demonstrates that there are substantial differences between the nabla fractional difference and the corresponding delta fractional difference. Moreover, we also prove two important inequalities which illustrate that, in some sense, the positivity of the ν-th order fractional difference is very closely connected to the monotonicity of f (t).Mathematics Subject Classification. 39A12, 39A70.

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