Existence results for a fourth order multipoint boundary value problem at resonance

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1 Avalable onlne at ScenceDrect Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx Exstence results for a fourth order multpont boundary value problem at resonance Q S.A. Iyase Department of Mathematcs, Covenant Unversty, Canaanland P.M.B. 3, Ota, Ogun State, Ngera Receved November 3; receved n revsed form June 5; accepted 5 August 5 Abstrac In ths paper we present some exstence results for a fourth order multpont boundary value problem at resonance. Our man 4 tools are based on the concdence degree theory of Mawhn. 5 c 5 Producton and Hostng by Elsever B.V. on behalf of Ngeran Mathematcal Socety. Ths s an open access artcle under the CC BY-NC-ND lcense ( Keywords: Fourth order; Multpont boundary value problem; Resonance; Concdence degree 7. Introducton 9 In ths paper, we shall dscuss the solvablty of the multpont boundary value problem x (v) (t) = f (t, x(t), x (t), x (t), x (t)) (.) m x() = α x(ξ ) x () = x () =, x() = x(η) (.) = where f : [, ] R 4 R s a contnuous functon α ( m ) R, < ξ ξ < ξ m < and 3 η (, ). 4 Multpont boundary value problems of ordnary dfferental equatons arse n a varety of dfferent areas of 5 Appled Mathematcs, Physcs and Engneerng. For example Brdges of small szes are often desgned wth two supported ponts, whch leads to a standard two-pont boundary condton and brdges of Large szes are sometmes 7 contrved wth multpont supports whch corresponds to a multpont boundary condton. Boundary value problem (.) (.) s called a problem at resonance f Lx = x (v) (t) = has non-trval solutons 9 under the boundary condtons (.) that s, when dm ker L. On the nterval [, ] second order and thrd order boundary value problems at resonance have been studed by many authors (see [ 4]) and references theren. Peer revew under responsblty of Ngeran Mathematcal Socety. E-mal address: dryase@yahoo.com / c 5 Producton and Hostng by Elsever B.V. on behalf of Ngeran Mathematcal Socety. Ths s an open access artcle under the CC BY-NC-ND lcense (

2 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx Q Although the exstng lterature on solutons of multpont boundary value problems s qute large, to the best of our knowledge there are few papers that have nvestgated the exstence of solutons of fourth order multpont boundary value problems at resonance. Our motvaton for ths paper s derved from these prevous results. In what follows, we shall use the classcal spaces C k [, ], k =,, 3. For x C 3 [, ] we use the norm x = max t [,] x(t). We denote the norm n L [, ] by and on L [, ] by. We wll use the Sobolev spaces W 4, (, ) whch may be defned by W 4, (, ) = {x : [, ] R : x, x, x, x } are absolutely contnuous on [, ] wth x (v) L [, ].. Prelmnares Consder the lnear equaton Lx = x (v) (t) = (.) m x() = α x(ξ ), x () = x () =, x() = x(η). (.) = If we consder a soluton of the form x(t) = 3 a t, a R. = Then ths soluton exsts f and only f a 3 ( η 3 ) =, η (, ). In ths case (.) (.) has non-trval solutons. Hence f Lx = y then L s not nvertble. Therefore, the problem s sad to be at resonance. We shall prove exstence results for the boundary value problem (.) (.) under the condton (.4). We shall apply the contnuaton Theorem of Mawhn [5] to get our results. We present some prelmnares needed to understand ths contnuaton Theorem. Let X and Z be real Banach spaces and L : doml X Z be a lnear operator whch s Fredholm of ndex zero and P : X X, Q : Z Z be contnuous projectons such that I m P = ker L, ker Q = I ml and X = ker L ker P Z = I ml I m Q. It follows that L doml ker P I ml s nvertble and we wrte the nverse of ths map by K p. Let Ω be an open bounded subset of X such that doml Ω Φ and let N : Ω Z be an L-compact mappng, that s, the maps QN( Ω) s bounded and K p (I Q)N : Ω X s compact. In order to obtan our exstence results we shall use the followng fxed pont Theorem of Mawhn. Theorem. (See [5]). Let L be a Fredholm operator of ndex zero and let N be L-compact on Ω. Assume that the followng condtons are satsfed () Lx λn x for every (x, λ) [(doml\ker L) Ω (, )] () N x I ml for every x ker L Ω () deg(j QN ker L Ω ; Ω ker L, ) where Q : Z Z s a contnuous projecton as above and J : I m Q ker L s an somorphsm. Then the equaton L x = N x has at least one soluton n dom L Ω. We shall prove exstence results for the boundary value problem (.)-(.) when m α ξ 3 = and = m α =. = Let X = C 3 [, ], Z = L [, ]. Let L : doml X Z be defned by Lx = x (v) (.3) (.4)

3 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx 3 where doml = x W 4, m (, ), x() = α x(ξ ), x () = x () =, x() = x(η). = We defne N : X Z by settng N = f (t, x(t), x (t), x (t), x (t)). Then the boundary value problem (.) (.) 3 can be put n the form 4 Lx = N x. (.5) 5 In what follows we shall use the followng lemmas. Lemma.. If m = α = then there exsts l {,,,..., m 4} such that m = α ξ l+4. 7 Proof. Follows the same procedure as n []. Lemma.. If m = α =, m = α ξ 3 = then 9 (A) I ml = y Z : m = α ξ s τ τ y(v)dvdτ dτ ds = (B) L : dom L X Z s a Fredholm operator of ndex zero. Proof. We wll show that the problem x (v) (t) = y for y Z (.) 3 has a soluton x(t) satsfyng 4 m x() = α x(ξ ), x () = x () =, x() = x(η) (.7) 5 = f and only f m ξ α = s τ τ y(v)dvdτ dτ ds =. (.) 7 Suppose (.) has a soluton x(t) satsfyng (.7) then from (.) we have x(t) = x() + x ()t + t x () + t3 x () + t s τ τ y(v)dvdτ dτ ds. 9 Usng m = α =, m = α ξ 3 = we obtan m ξ α = s τ τ y(v)dvdτ dτ ds =, for y Z. Now suppose m ξ α = s τ τ y(v)dvdτ dτ ds =. 3 Let 4 t3 x(t) = c η 3 s τ τ η y(v)dvdτ dτ ds + t s τ τ y(v)dvdτ dτ ds 5 where c s an arbtrary constant. Then x(t) s a soluton of (.) wth m ξ α = s τ τ y(v)dvdτ dτ ds =. 7

4 4 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx For y Z, we defne the projecton Q : Z Z by where (Qy)(t) = m = A α ξ l+4 m ξ t l α = A = (l + )(l + )(l + 3)(l + 4). s τ τ y(v)dvdτ dτ ds Let y = y Qy, that s y ker Q. Then by drect calculatons we have m ξ α = s τ τ m ξ = α = s τ τ y (v)dvdτ dτ ds y(v)dvdτ dτ ds m = A α ξ l+4 So, y I ml. Hence Z = I ml + I m Q. Snce I ml I m Q = {} we obtan Z = I ml I m Q. Now ker L = {x doml : x = c, c R}. Hence, dm ker L = dm I m Q =. Hence L s a Fredholm operator of ndex zero. Let P : X X be defned by Px(t) = x(), t [, ]. ξ s τ τ v l dvdτ dτ ds =. Lemma.3. If m = α =, m = α ξ 3 =. Then the generalzed nverse K p : I ml doml ker P can be wrtten as K p y(t) = t3 s τ τ t s τ τ η 3 y(v)dvdτ dτ ds + y(v)dvdτ dτ ds. η Proof. For any y I ml, we have (L K p )y(t) = (K p y(t)) (v) = y(t) and for x doml ker P, one has (K p L)x(t) = K p (x v ) = t3 η 3 + t s τ τ = t3 η 3 s τ τ η x (v) (v)dvdτ dτ ds x (v) (v)dvdτ dτ ds x() x(η) ( η 3 )x () ( η ) + x(t) x() tx () t x () x (). Snce x doml ker P, Px(t) = x() =. Also x () = x () =. x () ( η3 ) x () (.9)

5 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx 5 Thus, (K p L)x(t) = x(t). Hence, 3 K p = (L doml ker P ) Man results 5 Theorem 3.. Let m = α =, m = α ξ 3 = and let f : [, ] R 4 R be a contnuous functon and suppose that f has the decomposton 7 f (t, x, y, w, z) = g(t, x, y, w, z) + h(t, x, y, w, z) (H ) Assume there exsts M > such that for all x doml\ker L f x(t) > M, t [, ] then 9 m ξ α = s τ τ [ f (v, x(v), x (v), x (v), x (v))]dvdτ dτ ds (3.) (H ) zg(t, x, y, w, z) for all (t, x, y, w, z) [, ] R 4 (3.) (a) 3 h(t, x, y, w, z) M{ x r + y + w + z θ } for < r, θ < (3.3) 4 (b) 5 z[ f (t, x, y, w, z)] ( z + )[D(t, x, y, w) + m(t)] (3.4) where D(t, x, y, w) s bounded on bounded sets and m(t) L [, ]. 7 (H 3 ) There exsts N > such that for all c R, c > N then ether m = ca α ξ l+4 t l m ξ α = s τ τ [ f (v, c,,, )]dvdτ dτ ds < (3.5) 9 or τ τ m = ca α ξ l+4 t l m ξ α = s [ f (v, c,,, )]dvdτ dτ ds > (3.) Then (.) (.) has at least one soluton n C 3 [, ] provded M < Bπ 4(B + π + 4) where B = 4 + π. 4 Proof. Set 5 Ω = {x domk \ ker L : Lx = λn x, λ [, ]} for x Ω. Snce Lx = λn x, then λ, N x I ml = ker Q hence 7 m ξ α = s τ τ { f (v, x(v), x (v), x (v), x (v))}dvdτ dτ ds =. 3

6 3 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx Thus by (H ) there exsts t [, ] such that x(t ) M. Therefore, x(t) x(t ) + t x M + x. t x (s) ds, t [, ] (3.7) We note that for x() = x(η) there exsts t (η, ) such that x (t ) = and from x (t ) = x () = there exsts t (, t ) such that x (t ) = and x () = x (t ) there exsts (, t ) such that x ( ) =. Hence for x Ω we have t t t x (s)x (v) (s)ds = λ x (s)g(s, x, x, x, x )ds + λ x (s)h(s, x, x, x, x )ds x Usng the Cauchy nequalty we have ab εa + b ε x h(t, x, x, x, x ) dt. for ε > x h(t, x, x, x, x ) dt ε From condton (H a ) we obtan the estmate Therefore, x dt + ε h(t, x, y, w, z) 4M { x r + y + w + z θ }. x ε x M ε x r + M ε x θ h(t, x, x, x, x ) dt. + M ε x + M ε x. From Holder s nequalty we get ε 3M επ 4 M επ x M x r ε + x θ. Snce θ, r < we nfer the exstence of a constant M such that provded x < M (3.) > ε + 3M επ 4 + M επ. The choce ε = 4M 4+π π mnmzes the rght hand sde of (3.9) wth a mnmum value M(B+π +4) where Bπ B = 4 + π. Hence (3.) holds provded M < Bπ 4(B + π + 4). From (3.) and x () = x () = we get x < M 3 for some M 3 > (3.) x < M 4, M 4 > (3.) (3.9)

7 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx 7 and from (3.7) we derve x M + x < M + M 3 = M 5. (3.) Now usng condton (H b ) of Theorem 3. we ge x x v x + D(t, x, x, x ) + m(t) 4 and hence 5 t log e x x (s)x (v) (s) t x + ds = log e ( x (s) + ) D + m (3.3) where the constant D depends on M 3, M 4 and M 5. Snce x ( ) = we nfer from (3.3) that 7 x < e N = M (3.4) where N = D + m. 9 Hence x = max{ x, x, x, x max{m 4, M 3, M 5, M }}. Therefore, Ω s bounded. Le Ω = {x ker L : N x I ml}. 4 For x Ω, we have x = c R, thus 5 m ξ α = s τ τ [ f (v, x(v),,, )]dvdτ dτ ds =. (3.5) Then we have by H 3 and (3.5) that 7 x = c N whch shows that Ω s bounded. 9 We defne the somorphsm J : I m Q ker L by J(c) = c, c R. If (3.5) holds we set Ω 3 = {x ker L : λx + ( λ)j QN x = }, λ [, ] 3 For c Ω 3, we obtan 4 ( λ)a m ξ s τ τ λc = t l α [ f (v, c,,, )]dvdτ dτ ds 5 m α ξ l+4 = = f λ = then c = and f c > N then from (3.5) we have λc = ( λ)c A m ξ s τ τ t l α [ f (v, c,,, )]dvdτ dτ ds < 7 m α ξ l+4 = = whch contradcts λc. Thus Ω 3 s bounded. If (3.) holds, then let 9 Ω 3 = {x ker L : λx + ( λ)j QN x =, λ [, ]}. 3 Followng the above argument we can show that Ω 3 s bounded. 3

8 S.A. Iyase / Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx Q Let Ω be a bounded open subset of X such that = 3 Ω Ω. By the Arzela Ascol Theorem we can show that K p (I Q)N : Ω X s compact [5]. So N s L-compact. Thus we have shown that () Lx λn x for every (x, λ) [doml \ ker L Ω (, )] () N x I ml for every x ker L Ω. Fnally we shall prove that () of Theorem. s satsfed. Defne H(x, λ) = ±λx + ( λ)qn x, we have H(x, ) = ±x, H(x, ) = QN x. Thus H(x, λ) s a homotopy from the dentty ±I to QN and s such that H(x, λ) for every x Ω ker L. Therefore deg(j QN ker L Ω, Ω ker L, ) = deg(h(, ), Ω ker L, ) = deg(h(, ), Ω ker L, ) = deg(±i, Ω ker L, ). Then by Theorem. Lx = N x has at least one soluton n doml Ω. In other words (.) (.) has at least one soluton n C 3 [, ]. Uncted references [], [7], [], [9], [], [], [] and [3]. References [] Du Z, Ln Xaoje, Ge Wegao. On a thrd order multpont boundary value problem at resonance. J. Math. Anal. Appl. 5;3:7 9. [] Feng W, Webb JRL. Solvablty of three-pont boundary value problems at resonance. Nonlnear Anal. TMA ;47:47. [3] Gupta CP. A second order m-pont boundary value problem at resonance. Nonlnear Anal. TMA 995;4:43 9. [4] Nagle RK, Pothoven KL. On a thrd-order nonlnear bvp at resonance. J. Math. Anal. Appl. 995;95:4 59. [5] Mawhn J. Topologcal degree methods n non-lnear bvp. In: NSF CBMS regonal conference seres n math. 4. Provdence: Amercan Math. Soc.; 979. [] Agarwal R. On fourth order bvp arsng on beam analyss. Dfferental Integral Equatons 99;:9. [7] Aftabzadeh AR. Exstence and unqueness theorems for fourth order boundary value problems. J. Math. Anal. Appl. 9;:45. [] Aftabzadeh A, Gupta CP, Xu J. Exstence and unqueness theorems for three ponts bvp. SIAM J. Math. Anal. Appl. 99;(3):7. [9] Feng W, Webb JR. Solvablty of m-pont bvp wth nonlnear growth. J. Math. Anal. Appl. 997;:47. [] Gupta CP. Exstence and unqueness theorems for the bendng of an elastc beam equaton. Appl. Anal. 9;9:9 34. [] Iyase SA. Exstence and Unqueness theorems for a class of three pont fourth-order boundary value problems. J. Ngeran Math. Soc. ; 9:39 5. [] Iyase SA. Exstence and unqueness of solutons of a fourth order four pont bvp. J. Ngeran Math. Soc. 999;:3 4. [3] Lu B. Postve solutons of fourth order two pont bvp. Appl. Math. Comput. 4;4:7.