Existence of some Positive Solutions to Fractional Difference Equations

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1 International Journal of Difference Equations ISSN , Volume 10, Number, pp 1 3 (015 Existence of some Positive Solutions to Fractional Difference Equations Deepak B Pachpatte Dr Babasaheb Ambedkar Marathwada University Department of Mathematics Aurangabad, 3100, India pachpatte@gmailcom Arif S Bagwan Pimpri Chinchwad College of Engineering Department of First Year Engineering Nigdi, Pune, 110, India arifbagwan@gmailcom Amol D Khandagale Dr Babasaheb Ambedkar Marathwada University Department of Mathematics Aurangabad, 3100, India kamoldsk@gmailcom Abstract The main objective of this paper is to study the existence of solutions to some basic fractional difference equations The tool employed is Krasnosel skii fixed point theorem which guarantees at least two positive solutions AMS Subject Classifications: 39A10, 6A33 Keywords: Fractional difference equation, existence, positive solution 1 Introduction The theory of fractional calculus and associated fractional differential equations in continuous case has received great attention However, very limited progress has been done Received March 1, 015; Accepted July 6, 015 Communicated by Martin Bohner

2 Deepak Pachpatte, Arif Bagwan and Amol Khandagale in the development of the theory of finite fractional difference equations But, recently a remarkable research work has been made in the theory of fractional difference equations Diaz and Osler [10] introduced a discrete fractional difference operator defined as an infinite series Recently, a variety of results on discrete fractional calculus have been published by Atici and Eloe [,5,7] with delta operator Atici and Sengul [6] provided some initial attempts by using the discrete fractional difference equations to model tumor growth M Holm [11] extended his contribution to discrete fractional calculus by presenting a brief theory for composition of fractional sum and difference Furthermore, Goodrich [1 3] developed some results on discrete fractional calculus in which he used Krasnosel skii fixed point theorem to prove the existence of initial and boundary value problems Following this trend, H Chen, et al [13] and S Kang, et al [1] discussed about the positive solutions of BVPs of fractional difference equations depending on parameters H Chen, et al [8], in their article provided multiple solutions to fractional difference boundary value problems using various fixed point theorems In this paper, we consider the boundary value problems of fractional difference equation of the form v y(t = h(t + v 1f(t + v 1, y(t + v 1, (11 y(v = y(v + b = 0, (1 where t [0, b] N0, f : [v 1, v+b] Nv 1 R R is continuous, h : [v 1, v+b] Nv 1 [0,, 1 < v and is a positive parameter The present paper is organized as follows In Section, together with some basic definitions, we will demonstrate some important lemmas and theorem in order to prove our main result In Section 3, we establish the results for existence of solutions to the boundary value problem (11 (1 using Krasnosel skii fixed point theorem Preliminaries In this section, let us first collect some basic definitions and lemmas that are very much important to us in the sequel Definition 1 (See [3, 7] We define t v = Γ(t + 1 Γ(t + 1 v, (1 for any t and v for which right hand side is defined We also appeal to the convention that if t + v 1 is a pole of the Gamma function and t + 1 is not a pole, then t v = 0

3 Existence of Solutions to Fractional Difference Equations 3 Definition (See [3, 7] The v th fractional sum of a function f, for v > 0 is defined as, v f(t = v f(t, a := 1 Γ(v t v (t s 1 v 1 f(s, ( for t {a + v, a + v + 1, } =: N a+v We also define the v th fractional difference for t v = 0 by v a+vf(t := N v N f(t, where t N a+v and N N is chosen so that 0 N 1 < v N Now we give some important lemmas Lemma 3 (See [3, 7] Let t and v be any numbers for which t v and t v 1 are defined Then t v = vt v 1 s=a Lemma (See [3, 7] Let 0 N 1 < v N Then v v y(t = y(t + C 1 t v 1 + C t v + + C N t v N, (3 for some C i R with 1 i N Lemma 5 (See [7] Let 1 < v and f : [v 1, v +b] Nv 1 R R be given Then the solution of fractional boundary value problem v y(t = f(t+v 1, y(t+v 1, y(v = y(v + b + 1 = 0 is given by b+1 y(t = G(t, sf(s + v 1, y(s + v 1, ( where Green s function G : [v 1, v + b] Nv 1 [0, b + 1] N0 R is defined by t G (t, s = 1 v 1 (v + b s v 1 (v + b 1 v 1 (t s 1 v 1, 0 s < t v + 1 b + 1 Γv t v 1 (v + b s v 1 (v + b 1 v 1, 0 t v + 1 < s b + 1, Lemma 6 (See [7] The Green s function G(t, s given Lemma 5 satisfies 1 G (t, s 0 for each (t, s [v, v + b] Nv [0, b + 1] N0, max t [v,v+b] Nv G (t, s = G (s + v 1, s for each s [0, b] N0 and 3 There exists a number γ (0, 1 such that min G (t, s max G (t, s = γg (s + v 1, s ] t [v,v+b] Nv Nv t [ v+b, 3(v+b for s [0, b] N0 (5

4 Deepak Pachpatte, Arif Bagwan and Amol Khandagale Now we give the solution of fractional boundary value problem (11 (1, if it exists Theorem 7 Let f : [v 1, v + b] Nv 1 R R be given A function y is a solution to the discrete fractional boundary value problem (11 (1 iff it is a fixed point of the operator F y(t = G (t, s h (s + v 1 f (s + v 1, y (s + v 1, (6 where G(t, s is given in Lemma 5 Proof From Lemma, we find that a general solution to problem (11 (1 is y (t = v h(t + v 1f(t + v 1, y(t + v 1 + C 1 t v 1 + C t v, from the boundary condition y(v = 0, y (v = v h(t + v 1f(t + v 1, y(t + v 1 t=v + C 1 (v v 1 + C (v v = 1 Γv t v (t s 1 v 1 h(s + v 1f(t + v 1, y(t + v 1 + C Γ(v 1 = C Γ(v 1 = 0, therefore, C = 0 On the other hand, using boundary condition y(v + b = 0 t=v y (v + b = v h(t + v 1f(t + v 1, y(t + v 1 t=v+b + C 1 (v + b v 1 + C (v + b v = 1 Γv t v (t s 1 v 1 h(s + v 1f(t + v 1, y(t + v 1 + C 1 (v + b v 1 = 0, C 1 (v + b v 1 = 1 Γv C 1 = 1 Γv(v + b v 1 t=v+b t v (t s 1 v 1 h(s + v 1f(s + v 1, y(s + v 1 t=v+b (v + b s 1 v 1 h(s + v 1f(s + v 1, y(s + v 1

5 Existence of Solutions to Fractional Difference Equations 5 Using C 1 and C in y(t, we get y(t = 1 Γv t v 1 + Γv(v + b v 1 t v (t s 1 v 1 h(s + v 1f(s + v 1, y(s + v 1 (v + b s 1 v 1 h(s + v 1f(s + v 1, y(s + v 1 { t v t v 1 (v + b s 1 v 1 y(t = Γv(v + b v 1 } (t s 1v 1 h(s + v 1 Γv f(s + v 1, y(s + v 1 t v 1 (v + b s 1 v 1 + Γv(v + b v 1 h(s + v 1f(s + v 1, y(s + v 1 s=t v+1 y(t = G(t, sh(s + v 1f(s + v 1, y(s + v 1 (7 Consequently, we observe that y(t implies that whenever y is a solution of (11 (1, y is a fixed point of (6, as desired Theorem 8 (See [1] Let E be a Banach space, and let K E be a cone in E Assume that Ω 1 and Ω are open sets contained in E s t 0 Ω 1 and Ω 1 Ω, and let S : K (Ω \ Ω 1 K be a completely continuous operator such that either 1 Sy y for y K Ω 1 and Sy y for y K Ω ; Or Sy y for y K Ω 1 and Sy y for y K Ω Then S has at least one fixed point in K (Ω \ Ω 1 3 Main Result To prove our main result, let us state all required theorems for the existence of positive solutions to problem (11 (1 For this, let 1 η :=, G(s + v 1, sh(s + v 1 σ := 1 γ [ 3(b+v v+1] s=[ b+v v+1] G ([ ], b v + v, s

6 6 Deepak Pachpatte, Arif Bagwan and Amol Khandagale where η and σ are well defined by Lemma 6 and γ is the number given by Lemma 63 In the sequel, we present some conditions on f that will imply the existence of positive solutions H1: a number r > 0 such that f(t, y ηr whenever 0 y r H: a number r > 0 such that f(t, y σr whenever γr y r f(t, y H3: lim min = + y 0 + t [v,v+b] Nv y H: lim y + f(t, y min t [v,v+b] Nv y = + For our purpose, let E be a Banach space defined by { } E = y : [v, v + b] Nv R, y(v = y(v + b = 0, (31 with norm y = max y(t, t [v, v + b] N0 Also, define the cones { K 0 = y E : 0 y(t, min y(t γ y(t t [ v+b, 3(v+b ] } (3 In order to prove our first existence result, let us prove the following important lemma Lemma 31 F (K 0 K 0 ie, F leaves the cone K 0 invariant Proof Observe that min t [ v+b, 3(v+b ] which implies F y K 0 (F y(t = min t [ v+b, 3(v+b ] G(t, sh(s + v 1 f(s + v 1, y(s + v 1 γ G(t, sh(s + v 1f(s + v 1, y(s + v 1 γ max t [v,v+b] Nv G(t, sh(s + v 1 f(s + v 1, y(s + v 1 = γ F y,

7 Existence of Solutions to Fractional Difference Equations 7 Theorem 3 Assume that distinct numbers r 1 > 0 and r > 0 with r 1 < r such that f satisfies the condition H1 at r 1 and H at r Then the fractional boundary value problem (11 (1 has at least one positive solution say y 0 satisfying r 1 y 0 r Proof As F is completely continuous operator and F : K 0 K 0, let Ω 1 = {y K 0 : y r 1 } Then for any y K 0 Ω 1, we have from H1 F y = max t [v,v+b] Nv G (t, s h (s + v 1 f (s + v 1, y (s + v 1, G (s + v 1, s h (s + v 1 f (s + v 1, y (s + v 1 ηr 1 = r 1 = y G(s + v 1, sh(s + v 1 (33 Hence, F y = y for y K 0 Ω 1 Now, let Ω = {y K 0 : y r } Then for any y K 0 Ω we have from H F y(t = G (t, s h (s + v 1 f (s + v 1, y (s + v 1, [ 3(b+v v+1] s=[ b+v v+1] G (t, s h (s + v 1 f (s + v 1, y (s + v 1 G(t, sh(s + v 1 σr = r = y (3 Hence, F y y for y K 0 Ω So, it follows from Theorem 8 that there exists y 0 K 0 such that F y 0 = y 0 i e, fractional boundary value problem (11 (1 has a positive solution, say y 0 satisfying r 1 y 0 r In the next theorem, we give the existence of at least two positive solutions Theorem 33 Assume that f satisfies condition H1 and H3 Then the fractional boundary value problem (11 (1 has at least two positive solutions, say y 1 and y such that 0 y 1 < m < y

8 8 Deepak Pachpatte, Arif Bagwan and Amol Khandagale Proof From the assumptions, ε > 0 and r > 0 with r < m s t for 0 y r, (σ + ε f (t, y y, t [v, v + b] Nv ] [ [ b v b + v Let r 1 (0, r and + v Hence, for y Ω r, we have (F y + v = G, 3(b + v ] + v, s h(s + v 1 f(s + v 1, y(s + v 1 ([ ] b v (σ + ε G + v, s h(s + v 1 y (σ + ε y s=[ 3(v+b v+1] s=[ v+b v+1] G + v, s h(s + v 1 > σ y 1 σ = y = r (35 Thus, F y > y, for y K 0 Ω r On the other hand, suppose H3 holds, then there exists τ > 0 and R 1 > 0 such that (σ + τ f(t, y y, y R 1, t [v, v + b] Nv Now let R such that, R > max (m, R 1/ γ then, we have (F y + v = G + v, s h(s + v 1 f(s + v 1, y(s + v 1 ([ ] b v G + v, s h (s + v 1 (σ + τ y [ 3(b+v v+1] (σ + τ y s=[ b+v v+1] G (t, s h (s + v 1 σ y 1 σ = R (36

9 Existence of Solutions to Fractional Difference Equations 9 Hence, F y y for y K 0 Ω R Now, for any y Ω m, H1 implies that, f (t, y ηm, t [v, v + b] N v Let F y(t = G (t, s h (s + v 1 f (s + v 1, y (s + v 1 G (s + v 1, s h (s + v 1 ηm = ηm 1 η = m = y (37 Hence, F y y for y K 0 Ω m Therefore from Theorem 8, there are two fixed points y 1 and y of operator F s t 0 y 1 < m < y Theorem 3 Suppose conditions H and H hold, f > 0 for t [v, v + b] Nv Then the fractional boundary value problem (11 (1 has at least two positive solutions, say y 1 and y such that 0 y 1 < m < y Proof Suppose that H holds Then there exists ε > 0 (ε < η and 0 < r < m such (η ε that f (t, y y, 0 y r, t [v, v + b] Nv Let r 1 (0, r, then for y Ω r1, we have F y(t = G (t, s h (s + v 1 f (s + v 1, y (s + v 1 G (s + v 1, s h (s + v 1 < ηr 1 G (s + v 1, s h (s + v 1 (η ε r 1 < ηr 1 1 η = r 1 = y (38 Hence, we have F y < y for y Ω r1 On the other hand, suppose that H holds, then there exists 0 < τ < η and R 0 > 0 s t f (t, y τη, y R 0, t [v, v + b] Nv Denote M = max f(t, y Then (t,y [v,v+b] Nv [0,R 0 ] 0 f(t, y (τy + M, 0 y <

10 30 Deepak Pachpatte, Arif Bagwan and Amol Khandagale { } M Let R > max (η τ, m For y Ω R, we have F y = max t [v,v+b] Nv G (t, s h (s + v 1 f (s + v 1, y (s + v 1 G (s + v 1, s h (s + v 1 f (s + v 1, y (s + v 1 (τ y + M G(s + v 1, sh(s + v 1 = (τr + M 1 η < R = y (39 Hence, we have F y < y for y Ω R Finally, for any y Ω m, since γm y(t m for t have (F y + v = G f(s + v 1, y(s + v 1 > σγm [ 3(b+v v+1] s=[ b+v v+1] G [ b + v + v, s h(s + v 1 ] 3 (b + v,, we + v, s h (s + v 1 = m = y (310 Hence, F y > y, for y K 0 Ω m Therefore, by Theorem 8, the proof is complete Example 35 Consider the fractional boundary value problem ( y (t = 100 e(t+ t + 1 { ( y 1 t + 1 ( + y t + 1 } (311 ( y 3 ( 5 = 0, y = 0, (31 where v = 5 1 (, b = 5, f(t, y = 100 t y 1 + y, h (t = e t With a simple computation we can verify that η > 0001 f : [0, [0, [0, and h : [0, [0, and f(t, y satisfies the conditions H1 and H3, will have at least one positive solution

11 Existence of Solutions to Fractional Difference Equations 31 References [1] C S Goodrich, Continuity of solutions to discrete fractional initial value problems, Computers and Mathematics with Applications, vol 59 no 11, pp , 010 [] C S Goodrich, On a discrete fractional three-point boundary value problem, Journal of Difference Equations and Applications, vol 18, no 3, pp , 01 [3] C S Goodrich, Some new existence results for fractional difference equations, International Journal of Dynamical Systems and Differential Equations, vol 3, no1, pp 15 16, 011 [] F M Atici and P W Eloe, A transform method in discrete fractional calculus, International Journal of Difference Equations, vol, no, pp , 007 [5] F M Atici and P W Eloe, Initial value problems in discrete fractional calculus, Proceedings of American Mathematical Society, vol 137, no 3, pp , 009 [6] F M Atici and S Sengul, Modeling with fractional difference equations, Journal of Mathematical Analysis and Applications, vol 369, no 1, pp 1 9, 010 [7] F M Atici and P W Eloe, Two-point boundary value problems for finite fractional difference equations, International Journal of Difference Equations, vol 17, no, pp 5 56, 011 [8] H Chen, Y Cui, X Zhao, Multiple solutions to fractional difference boundary value problems, Abstract and Applied Analysis, vol 01, article , 01 [9] H L Gray, N F Zhang, On a new definition of the fractional difference, Mathematics of Computation, vol50, no18, pp ,1988 [10] J B Diaz, T J Osler, Differences of fractional order, Mathematics of Computation, vol 8, pp 185 0, 197 [11] M Holm, Sum and difference compositions in discrete fractional calculus, Cubo, vol 13, no 3, pp , 011 [1] R P Agarwal, M Meehan, and D O Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, UK, 001 [13] S Kang, X Zhao, H Chen, Positive solutions for boundary value problems of fractional difference equations depending on parameters, Advances in Difference Equations, vol 013, article 376, 013

12 3 Deepak Pachpatte, Arif Bagwan and Amol Khandagale [1] S Kang, Y Li, H Chen, Positive solutions for boundary value problems of fractional difference equations with nonlocal conditions, Advances in Difference Equations, vol 01, article 7, 01

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