Unbounded solutions of second order discrete BVPs on infinite intervals

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1 Available online at J. Nonlinear Sci. Appl ), Reearch Article Unbounded olution of econd order dicrete BVP on infinite interval Hairong Lian a,, Jingwu Li a, Ravi P Agarwal b a School of Science, China Univerity of Geocience, Beijing 00083, P. R. China. b Department of Mathematic, Texa A&M Univerity-Kingville, Kingville, Texa 78363, USA. Communicated by R. Saadati Abtract In thi paper, we tudy Sturm-Liouville boundary value problem for econd order difference equation on a half line. By uing the dicrete upper and lower olution, the Schäuder fixed point theorem, and the degree theory, the exitence of one and three olution are invetigated. An intereting feature of our exitence theory i that the olution may be unbounded. c 206 All right reerved. Keyword: Coincidence point, dicrete boundary value problem, infinite interval, upper olution, lower olution, degree theory common fixed point. 200 MSC: 47H0, 54H25, 54C60.. Introduction In thi paper, we tudy Sturm-Liouville boundary value problem for econd order difference equation on an infinite interval 2 x = f, x, x ), N, x 0 a x 0 = B, x = C,.) where x = x + x i the forward difference operator. N =, 2,, } and f : N R 2 R i continuou. a > 0, B, C R, x = lim x. Recall that the map f : N R 2 R i continuou if it map continuouly the topological pace N R 2 into R. The topology on N i the dicrete topology. By a olution x of.), we mean a equence x = x 0, x,, x n, ) which atify.). We will provide Correponding author addre: lianhr@cugb.edu.cn Hairong Lian) Received

2 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), ufficient condition on f o that the dicrete boundary value problem.) have one olution, and three olution. An important apect of our exitence theory i that the olution may be unbounded. Recently, the exitence of linear and nonlinear dicrete boundary value problem ha been tudied by many author. We refer here to ome wor uing the upper and lower olution method, e.g., ee [3, 0,, 3, 4, 5, 8, 20, 2, 24, 25, 28, 29] for finite interval problem, and [, 5, 7, 27] for infinite interval problem. Dicrete infinite interval problem have alo been tudied by everal other method in [2, 4, 6, 8, 9, 2, 6, 7, 9, 2, 22, 26]. In [5], R. P. Agarwal and D. O Regan tudied the exitence of nonnegative olution to the boundary value problem 2 xi ) + f i, xi) ) = 0, x0) = 0, lim xi) = 0. i They employed upper and lower olution on finite interval, and a diagnolization proce, to prove the exitence of at leat one nonnegative bounded) olution. Later, imilar method were ued for the exitence of olution to uch dicrete BVP and on the time cale, ee [, 7]. In [25], Y. Tian, C. C. Tidell and Weigao Ge etablihed the exitence of three bounded) olution of the dicrete boundary value problem 2 xn ) p xn ) qxn ) + f n, xn), xn) ) = 0, x0) γxl) = x 0, xn) bounded on [0, ). For thi, they aumed the exitence of two pair of upper and lower olution on finite interval, and ued the equential argument and the degree theory. A far a we now the exitence of unbounded olution for the dicrete boundary value problem ha not been tudied. The only nown wor where unbounded poitive olution of econd order nonlinear neutral delay difference equation have been etablihed i a recent contribution of Zeqing Liu et al. [7]. Since the infinite interval i noncompact, the dicuion here i more complicated compared to finite interval problem. In Section 2, we hall begin with the whole dicrete infinite interval and introduce a new Banach pace. Here dicrete Arezà-Acoli lemma i alo etablihed, which i neceary to prove that the ummation mapping i compact. In Section 3, we will how that in the preence of a pair of upper and lower olution the problem.) ha a olution. For thi we hall apply the Schäuder fixed point theorem. Here to how how eaily our reult can be applied in practice two example are alo illutrated. In Section 4, we hall employ the topological degree theory to how that the problem.), in the preence of two pair of upper and lower olution, ha three olution. 2. Definition and Green function Let N 0 be the et of all nonnegative integer and S be the pace of equence, i.e., by x S, we mean x = x } N0. For x, y S, we write x y if x y for all N 0. We conider S = x S : } lim x exit endowed with the norm x = max x, x }, x where x = x } N0, x = up +, x = up x. Becaue lim x exit, x } N0 bounded. If we denote by M = up x, then it follow that i

3 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), x + = + x 0 + x i x x i + x M. Hence, up x + <. It i clear that S, ) i a normed linear pace. We claim that it i in fact a Banach pace. Lemma 2.. S, ) i a Banach pace. Proof. We hall prove it completene. Suppoe x n) } n= S i a Cauchy equence. Then y n) : y n) = x n) +, N 0} and z n) : z n) = x n) } n N and z n) y n), N 0} are bounded for each n N. Now ince for any N 0, } n N are Cauchy equence in R, there exit two equence y and z in S uch that y n) y 0, and z n) z 0, a n. Clearly, y and z are bounded. Let x = + )y and x = x, x 2,, x, ), then But thi mean that for each N 0, Further, x n) x 0, a n. lim n xn) = x. x = x + x = lim n xn) + lim n xn) = lim n xn) + xn) ) = lim n xn) = z, =, 2,... Hence, x n) x 0n ). The proof i completed. Lemma 2.2. If e = e } N atify e <, then the linear dicrete BVP = 2 x = e, N, x 0 a x 0 = B, x = C, ha a unique olution in S. Further, thi olution can be expreed a x = ac + B + C + G, i)e i, N 0, where a + i, i, G, i) = a +, i >. We define T : S S by T x) = ac + B + C + G, i)fi, x i, x i ), N 0. Clearly, x i a olution of problem.) if and only if x i a fixed point of the mapping T. 2.) 2.2)

4 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Definition 2.3. A function α S i called a lower olution of.) provided 2 α f, α, u ), α 0 a α 0 B, α < C 2.3) for all N and u α. If all inequalitie are trict, it will be called a trict lower olution. Definition 2.4. A function β S i called an upper olution of.) provided 2 β f, β, v ), β 0 a β 0 B, β > C 2.4) for all N and v β. If all inequalitie are trict, it will be called a trict upper olution. Definition 2.5. Let α, β be the lower and upper olution for the problem.) atifying α β. We ay that f atifie a dicrete Berntein Nagumo condition with repect to α and β if there exit poitive function ψ CN) and h C[0, + ) uch that f, x, y) ψ)h y ) for all N and α x β with h nondecreaing, and ψi) <, + d =. h) We will ue the Schäuder fixed point theorem to obtain a fixed point of the mapping T. To how the mapping i compact, the following generalized dicrete Arezà-Acoli lemma will be ued. Lemma 2.6 [4]). M B = x S : lim x exit.} i relatively compact if it i uniformly bounded and uniformly convergent at infinity. Lemma 2.7. M S i relatively compact if it i uniformly bounded and uniformly convergent at infinity, that i, for each ɛ > 0, there exit K = Kɛ) N uch that x + lim x + < ɛ, and x x < ɛ, > K for all x M. Proof. M S i relatively compact if every equence of M ha a convergent ubequence. Firt, we will x how that lim + exit for any x = x } N0 S. Since lim x exit, we can denote it limit by c. Now ɛ > 0, there exit a K = Kɛ, x) > 0 uch that which implie that x c < ɛ, > K, x K+ + K )c K )ɛ < x < x K+ + K )c + K )ɛ, and hence either x } N0 i bounded or x tend to infinity a. For the later cae, by uing the Stolz rule the dicrete L Hopital rule), we have lim x + = lim x exit.

5 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Now, conider the equence x n) } n N S. Since y n) : y n) = xn) +, N 0 } n N B and } z n) : z n) = x n), N 0 B, n N 0 the condition of Lemma 2.7 guarantee that they both have convergent ubequence. generality, we write thee convergent ubequence a y n) and z n) atifying Without lo of lim n yn) = y, and lim n zn) = z. Now following the dicuion a in Lemma 2., we can how that where x = x }, x = + )y and x = z. x n) x 0, n, 3. The exitence of a olution Theorem 3.. Aume that H ) The dicrete boundary value problem.) ha one pair of upper and lower olution α and β in S atifying α β. H 2 ) f CN 0 R 2, R) atifie the Berntein-Nagumo condition with repect to α and β. H 3 ) There exit γ > uch that up + ) γ ψ) <. N Then the dicrete boundary value problem.) ha at leat one olution x atifying α x β, x R, where R > 0 i a contant independent of the olution x). Proof. We chooe η, R > C uch that β α 0 η max up N R h) d > M η up N, up N β + ) γ inf N } β 0 α, α + ) γ + N ) 2 + i) γ + i) γ 2 + i) γ + i) γ, where C i the nonhomgeneou boundary value, and M = up + ) γ ψ), N = max α, β }. N Define the auxiliary function F 0, F : N R 2 R a follow and F 0, x, y) = f, β, y ) x β 2 + x β ), x > β, f, x, y), α x β, f, α, y ) x α x α ), x < α, ) F 0, x, R, y > R, F, x, y) = F 0, x, y), R y R, ) F 0, x, R, y < R.

6 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Clearly, F i a continuou function on N R 2 and atifie F, x, y) ψ)hr) + 2, for all, x, y) N R2. Conider the modified boundary value problem 2 x = F, x, x ), N, x 0 a x 0 = B, x = C. To complete the proof, it uffice to how that 3.) ha at leat one olution x = x } N0 α x β, and x R, N 0. uch that We divided the proof into the following three tep. Step. Problem 3.) ha a olution. To how that the problem 3.) ha a olution, we define the operator T : S S a ) T x) = ac + B + C + G, i)f i, xi, x i, N0. 3.2) From Lemma 2.2, we can ee that the fixed point of T are the olution of 3.). We will prove that T : S S i completely continuou and ha at leat one fixed point from the Schäuder fixed point theorem. For any x S, becaue G, i)f i, x i, x i ) a + ) ψi)hr) + i ) 2 <, for any N 0, from the definition of T, we have lim T ) x) = lim T x) + T x) ) ) = C + G +, i) G, i) F i, xi, x i = C. lim Thu, T S S. Next, for any convergent equence x m) x a m in S, we have T x m) T x m) ) T x = up T x) + + G, i) m) up F i, x i, x m) ) + i ) F i, xi, x i m) maxa, } F i, x i, x m) ) i ) F i, xi, x i and 0, a m T x m) ) T x) = up G +, i) G, i)) m) F i, x 0, a m. i F i, x m) i, x m) i ) F i, xi, x i ), x m) i ) F i, xi, x i ) ) 3.)

7 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Therefore, T i continuou. Finally, we will how that T i compact, that i, T map bounded ubet of S into relatively compact et. For thi, let B be any bounded ubet of S, then there exit a contant r > 0 uch that x r, x B. For any x B, we have and T x = up = up T x) + ac + B + C + + max ac + B, C } + maxa, } < + G, i) + F i, xi, x i ) hr)ψi) + i ) 2 T x) = up T x) = up C + ) G +, i) G, i))f i, xi, x i C + C + ) F i, xi, x i hr)ψi) + i 2 ). Thu, T B i uniformly bounded. From Lemma 2.7, if T B i uniformly convergent at infinity, then T B i relatively compact. In fact, T x) + lim T x) + ac + B + C ) G, i) = C ) F i, xi, x i ac + B + C C + + G, i) + hr)ψi) + i ) 2 0, a + and T x) lim T x) = ) G +, i) G, i))f i, xi, x i G +, i) G, i) hr)ψi) + i ) 2 0, a +. Hence, we find that T B i relatively compact. Therefore, T : S S i completely continuou. Now chooe N > max L, α, β }, where L = max ac + B, C } + maxa, } ψi)hr) + i ) 2 3.3)

8 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), and et Ω = x S, x < N }. Then for any x Ω, it i eay to ee that T x < N, and thu T Ω Ω. The Schäuder fixed point theorem now guarantee that the operator T ha at leat one fixed point in S, which i a olution of BVP 3.). Step 2: Every olution x of the problem 3.) atifie α x β. We aume that the right hand inequality doe not hold. Then x β ha a poitive maximum in N 0. The poitive maximum doe not occur at infinity becaue lim x β ) < 0. If the poitive maximum occur at 0, then x 0 β 0 ) 0. However, we have x 0 a x 0 β 0 a β 0 ) = x 0 β 0 ) a x 0 β 0 ) > 0, which i a contradiction to the left boundary condition. If the poitive maximum occur at 0 N, then and x 0 β 0 > 0, x 0 β 0 ) 0, x 0 β 0 ) 0 2 x 0 β 0 ) 0. However, it follow from 2.4) and 3.) that 2 ) x 0 = F 0, x 0, x 0 = f ) x 0 β 0 0, β 0, x x 0 β 0 ) 2 x 0 β 0 β x 0 β 0 ) < 2 β 0, which i a contradiction. Thu, x β hold for all N 0. The proof for x α i imilar. Step 3: If the olution x of the problem 3.) atifie α x β, then x R, N 0. We claim that x > η doe not hold for all N 0. Otherwie, without lo of generality, we can uppoe that x > η for all N 0, but then it follow that β α 0 x x 0 = x i > η β α 0, N, which i a contraction. Thu there mut exit a N 0 uch that x η. If x R doe not hold for all N, then there exit 2 N 0 uch that x 2 > R. Proceeding with thi argument, we may uppoe 2 > and 0 x η < R x 2, η x R, < < 2. Let I = i : < i 2, x i > x i } and Ī =, 2 ] N 0 \ I. Then, we have R η 2 h) d i= + i I xi x i h x i ) h) d = i I xi xi x i d j Ī x i h) d j Ī h x j ) xj x j xj x j d h) d = 2 i= + 2 i= + x i + x i ) 2 2 x i h x i ) x i + x i ) ψi) M i= + 2 x i + i) γ + i= x i + i) γ

9 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), M 2 M x ) γ + x ) γ x ) γ x + ) γ +M up up N x β + ) γ inf N 2 + i) γ + i) γ 2 + i) γ + i) γ ) α + ) γ + N ) ) 2 + i) γ + i) γ 2 + i) γ + i) γ, which i a contradiction. Hence, x R, N 0. Here we note that the erie 2 + i) γ + i) γ 2 + i) γ + i) γ i convergent. In a imilarly way, we can alo how that x R for all N 0. Hence there exit a R > 0, independent of every olution x of.), uch that x R. Example. Conider the Sturm-Liouville boundary value problem involving the econd order difference equation 2 x + 3/2 x )2 + x ) + ) 4 = 0, N, 3.4) x 0 x 0 =, x =. Clearly, BVP 3.4) i a particular cae of.) with 3/2 y)2 + x) f, x, y) = + ) 4, and a =, B =, C =. Conider the upper and lower olution of 3.4) defined by α =, β = 2 + 3, N 0. Here the function f i continuou and we will how that it atifie the Berntein Nagumo condition with repect to α and β. In fact, when N 0, x 2 + 3, y R, it follow that ft, x, y) = 3/2 y)2 + x ) + ) 4 ) ) 2 up n N 0 + ) 2 y + 3/2) 3 y + 3/2). + ) 2 Set ψ) = +) 2, h) = 3 + 3/2) and < γ 2, then = + ) 2 <, up + ) γ + ) 2 = up N h) d = 3 N <, + ) 2 γ d =. + 3/2 Hence, all condition of Theorem 3. are atified, and thu the problem 3.4) ha at leat one nontrival olution x atifying x for all N.

10 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Example 2. Conider the Sturm-Liouville boundary value problem involving the econd order difference equation 2 x x + + ) 4 in x + x 2 + ) x = 0, N, x 0 3 x 0 = 0, x = 3.5) 2. Clearly, BVP 3.5) i a particular cae of.) with f, x, y) = y + + ) 4 in x + x 2 + ) x, and a = 3, B = 0, C = 2. Conider the upper and lower olution of 3.5) defined by α = 4, β = + 4, N 0. Here the function f i continuou and we will how that it atifie the Berntein Nagumo condition with repect to α and β. In fact, when N 0, 4 x + 4, y R, it follow that ft, x, y) = y + + ) 4 in x + x 2 + x ) + 4) 2 ) ) 2 + up n N 0 + ) 2 + up x n N 0 + ) 2 y + ) 9 y + ). + ) 2 Set ψ) = +) 2, h) = 9 + ) and < γ 2, then = + ) 2 = π2 6 <, up + ) γ + ) 2 = up N h) d = 9 N <, + ) 2 γ d =. + Hence, Theorem 3. guarantee that problem 3.5) ha at leat one nontrival olution x atifying 4 x + 4 for all N. 4. The multiplicity reult Here we hall how that in the preence of two pair of upper and lower olution the problem.) ha at leat three olution. Theorem 4.. Suppoe that the following condition hold. H 4 ) The dicrete boundary value problem.) ha two pair of upper and lower olution β j), α j), j =, 2 in S with α 2), β ) trict and α ) α 2) β 2), α) β ) β 2), α2) β ), N 0. Suppoe further that condition H 2 ) and H 3 ) hold with α, β replaced by α ), β 2) repectively. Then the problem.) ha at leat three olution x ), x 2) and x 3) atifying α j) x j) β j) j =, 2), x3) β ) and x 3) α 2), N 0.

11 H. Lian, J. Li, R. P. Agarwal, J. Nonlinear Sci. Appl ), Proof. Define the truncated function F 2, the ame a F in Theorem 3. with α, β replaced by α ) and β 2) repectively. Conider the modified difference equation 2 ) x = F 2, x, x, N, 4.) x 0 a x 0 = B, x = C. To how that the problem 4.) ha at leat three olution, we define a mapping T 2 : S S T 2 x) = ac + B + C + ) G, i)f 2 i, xi, x i, N0. 4.2) A in Theorem 3., T 2 i completely continuou. By uing the degree theory, we will how that T 2 ha at leat three fixed point which coincide with the olution of 4.). Chooe N 2 > maxl 2, α ), β 2) }, where L 2 ha the ame expreion a L in 3.3) except that R given by α, β i now defined by α ), β 2). Set Ω 2 = x S, x < N 2 }. Then for any x Ω 2, it follow that T 2 x < N 2. Thu, T 2 Ω 2 Ω 2, and o we have degi T 2, Ω 2, 0) =. Set } Ω α 2) = x Ω 2, x > α 2), N 0, } Ω β) = x Ω 2, x < β ), N 0. Becaue α 2) β ), α ) α 2) β 2) and α ) β ) β 2), we have Ω α 2), Ω β), Ω 2 \ Ω α 2) Ω β), Ω α 2) Ω β) =. Noticing that α 2), β ) are trict lower and upper olution, there i no olution on Ω α 2) Ω β). Therefore To how that degi T 2, Ω 2, 0) =degi T 2, Ω 2 \ Ω α 2) Ω β), 0) we define another mapping T 3 : Ω 2 Ω 2 by T 3 x) = ac + B + C + + degi T 2, Ω α 2), 0) + degi T 2, Ω β), 0). degi T 2, Ω α 2), 0) =, ) G, i)f 3 i, xi, x i, N0. where the function F 3 i imilar to F 2 except α ) i replaced by α 2). Similar to the proof of Theorem 3., we find that x i a fixed point of T 3 only if α 2) x β 2), N 0. So degi T 3, Ω \ Ω α 2), 0) = 0. Thu from the Schäuder fixed point theorem and T 3 Ω 2 Ω 2, we have degi T 2, Ω 2, 0) =. Furthermore, degi T 2, Ω α 2), 0) =degi T 3, Ω α 2), 0) Similarly, we have degi T 2, Ω β), 0) =. And then =degi T 3, Ω 2, 0) + degi T 3, Ω 2 \ Ω α 2), 0) =. degi T 2, Ω 2 \ Ω α 2) Ω β), 0) =. Uing the propertie of the degree, we conclude that T 2 ha at leat three fixed point x ) Ω α 2), x 2) Ω β) and x 3) Ω 2 \ Ω α 2) Ω β), which are the claimed three different olution of the BVP 4.). Similarly, we can how that α ) x i) β 2) and x i) R. Thu they are the olution of BVP.).

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