POSITIVE SOLUTIONS FOR NONLOCAL SEMIPOSITONE FIRST ORDER BOUNDARY VALUE PROBLEMS

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1 Dynamic Systems and Applications POSITIVE SOLUTIONS FOR NONLOCAL SEMIPOSITONE FIRST ORDER BOUNDARY VALUE PROBLEMS ERBIL ÇETIN AND FATMA SERAP TOPAL Department of Mathematics, Ege University, 35 Bornova, Izmir-Turkey ABSTRACT. We establish the existence of positive solutions for a nonlinear nonlocal first order boundary value problem by applying a variety of fixed point theorems. Emphasis is put on the fact that the nonlinear terms f i may take negative values. AMS MOS Subect Classification. 34B5, 34B8. Keywords: Positive solution, Nonlinear boundary condition, Sign-changing problem, Fixed point theorems.. INTRODUCTION Boundary value problems with nonlocal boundary conditions arise in a variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity and plasma physics can be reduced to nonlocal problems. They include two, three and multi-point boundary value problems as special cases and also they have attracted the attention of many authors [3, 6-, 5] such as Gallardo [5], Karakostas and Tsamatos [] and the references therein. While Nieto and Rodriguez-Lopez [4] considered the first order problem y t + rtyt = ft, t [, ], yt = yt, Anderson [] studied the following problem if T = R y t + rtyt = ft, yt, t [, ], y = y + yt, where the nonlinear term ft, yt is allowed to take on negative values. Zhao [6] investigated the problem if T = R y t + rtyt = ft, yt, t [, ], Received December, $5. c Dynamic Publishers, Inc.

2 44 E. ÇETIN AND F. S. TOPAL y = gy, where g denotes a nonlinear term. In 3, Anderson [] interested in the first order boundary value problem given by y t rtyt = f i t, yt, t [, ], y = y + Λ τ, yτ, τ [, ], where nonlinear continuous functions f i and Λ are all nonnegative. In [9], using Krasnosel ski`ı s fixed point theorem, Goodrich studied the existence of a positive solution to the first-order problem given by if T = R y t + ptyt = ft, yt, ya = yb + τ τ t a, b, Fs, ysds, where τ, τ [a, b] T with τ < τ, p and F are nonnegative functions and the nonlinearity f can be negative for some values of t and y. In recent paper [4], Çetin and Topal concerned with the existence and iteration of positive solutions for the nonlinear nonlocal first order multipoint problem.. y t + ptyt = f i t, yt, t [, ], y = y + g t, yt, t [, ], with =. Motivated by the above works, in this paper, we will interested in the existence of at least one, two and three positive solutions to the semipositone firstorder nonlinear boundary value problem.., where the continuous function f i : [, ] [, R is semipositone, i.e., f i t, y needn t be positive for all t [, ] and all y, p : [, ] [, is continuous and does not vanish identically and the nonlinear functions g : [, ] [, [, satisfy v t, yy g t, y v t, yy, t [, ] for some positive continuous possibly nonlinear functions v, v [,. : [, ] [,

3 NONLOCAL SEMIPOSITONE FIRST ORDER PROBLEMS 44. PRELIMINARIES In this section we collect some preliminary results that will be used in main results which give the existence of positive solutions of the main problem... We initially construct the Green function for the linear first order boundary value problem.. y t + ptyt =, t [, ], y = y. The techniques here are standard when establishing the Green function given in [4]. Lemma.. The unique solution wt of the problem.. where.3 Gt, s = wt = exp t s exp satisfies wt +expr exp R. Proof. Set wt = = t t Gt, sds Gt, sds, exp t s exp ds + exp t t exp exp + exp exp. ds + t + exp, s < t; exp t, s t t exp s exp ds exp t exp s exp ds Lemma. [4]. The Green function Gt, s which is given by.3 satisfies exp pξdξ Gs, s Gt, s exp pξdξ Gs, s. Lemma.3. The function yt is a solution of the problem.. if and only if yt = Gt, sf i s, ysds + exp t exp g t, yt where Gt, s is defined by.3

4 44 E. ÇETIN AND F. S. TOPAL In the proof of Theorem. in [], the similar result has been given. Therefore, we don t restate the proof here. Also the solution of the problem.. is given in [4] for =. The following theorems play an important role to prove our main results. Theorem.4 []. Let E = E, be a Banach space, and let P E be a cone in B. Assume Ω, Ω are bounded open subsets of E with Ω, Ω Ω, and let S : P Ω \Ω P be a continuous and completely continuous operator such that, either a Su u, u P Ω, and Su u, u P Ω, or b Su u, u P Ω, and Su u, u P Ω. Then S has a fixed point in P Ω \Ω. Theorem.5 [3]. Let P be a cone in Banach space E. Let α, β and be three increasing, nonnegative and continuous functionals on P, satisfying for some c > and M > such that x βx αx and x Mx, for all x P, c Suppose there exists a completely continuous operator T : P, c P and < a < b < c such that i Tx < c, for all x P, c; ii βtx > b, for all x Pβ, b; iii Pα, a and αtx < a for all x Pα, a. Then T has at least three positive solutions x, x, x 3 P, c satisfying < αx < a < αx, βx < b < βx 3, x 3 < c. 3. MAIN RESULTS In this section, we will give the existence results of positive solutions for the problem... Let E denote the Banach space C[, ] with the norm y = max t [,] yt. Define the cone P E by P = y E : yt y, t [, ] where = exp R +exp R exp pξdξ and we can easily see that < <. The main results of this paper are following: Theorem 3.. Assume that the following conditions are satisfied: C There exists a constant M > such that f i t, y M for all t, y [, ] [,, C lim y f i t,y y = for t [, ].

5 NONLOCAL SEMIPOSITONE FIRST ORDER PROBLEMS 443 Then there exists a positive number such that the problem.. has at least one positive solution for < <. Proof. Let xt = Mwt, where wt is the unique solution of the boundary value problem We shall show that the following boundary value problem u t + ptut = u = u + F i t, u x t, t [, ], g t, u x t, t [, ], has at least one positive solution where we ll take F i t, u x t := f i t, u x + M and u x t := maxut xt,. We define the operator T : P E with Tut := Gt, s F i s, u x sds + exp t exp g t, u x t. It is well known that the existence of positive solutions to the problem is equivalent to the existence of fixed points of the operator T in our cone P. First, it is obvious that T is completely continuous by a standard application of the Arzela-Ascoli Theorem. Also, for any u P, we get t Tut = Gt, s F i s, u x sds + + exp t exp t exp + t exp pξdξ Gs, s + exp pξdξ g t, u x t Gs, s t Gt, s F i s, u x sds F i s, u x sds F i s, u x sds + exp exp t pξdξ exp t + exp exp exp pξdξ g t, u x t Gs, s g t, u x t F i s, u x sds pξdξ Gs, s F i s, u x sds

6 444 E. ÇETIN AND F. S. TOPAL = exp So, we get TP P. exp pξdξ Gs, s g t, u x t + exp exp exp pξdξ + exp Tu = Tu. F i s, u x sds For each r >, set β = max t,y [,] [,r] v t, y. Let define M i = maxf i t, y + M : t, y [, ] [, r] and choose r exp = min r m n M i β, R. M Let Ω r = u E : u < r. Then for any u P Ω r we have u x t ut u = r and T u exp Gs, s F i s, u x sds + exp g t, u x t exp u x m M i ds + β exp exp M i + u x β exp M i + r β r = u. Therefore, we get Tu u, for u P Ω r. We set α = min t,y [,] [ R,R] v t, y for R > r. Let K i be positive real numbers such that exp K i + α. In view of C, there exists a constant N > such that for all z N and t [, ], F i t, z = f i t, z + M K i z. Now set R = r + N and Ω R = u E : u R. We shall prove that Tu u for u P Ω R.

7 NONLOCAL SEMIPOSITONE FIRST ORDER PROBLEMS 445 Let u P Ω R, then u x t ut u = R. From Lemma. and the fact that u P, we get ut xt ut M u M and also ut xt R > N. Thus, we have R M R R = R >, F i t, u x t = F i t, u xt K i ut xt K i R >. So we obtain T u = exp pξdξ Gs, s F i s, u x sds + exp exp g t, u x t exp pξdξ + exp exp exp s exp exp g t, u x t exp s exp + exp exp exp exp exp R K i Rds v t, u x t u x t R K i + R K i + α α R = u. F i s, u x sds Therefore we get Tu u for u P Ω R. Consequently, it follows from Theorem., T has a fixed point u P such that r u R. Moreover, ut u r M. Hence for t [, ], yt = ut xt M M = M > and it can be easily seen that y is a solution of the problem... Theorem 3.. Assume that the condition C and the following condition are satisfied: C 3 lim y f i t,y y = and lim y f i t,y y = for t [, ]. Then there exists a positive number such that the problem.. has at least two positive solutions for < <.

8 446 E. ÇETIN AND F. S. TOPAL Proof. By the similar proof of Theorem, we can easily obtain that Tu u for u P Ω r for each r > and > min with β = max t,y [,] [,r ] v t, y. f If lim i t,y y y for y r, where η i > is chosen such that r exp R r P m β P n, r M i M =, there exists a positive number r < r such that f i t, y η i y exp r exp R η i + with α = min r t,y [,] [,r ] vt, y. Then F i t, u x η i u x for u x Ω r, t [, ]. So, we get Tu n η i + m α r = u for u P Ω r. Thus T has a fixed point u in Ω r \ Ω r for < < and it can be easily seen that y t := u t xt is a positive solution of the problem.. such that r y r. f If lim i t,y y =, there exists an H > such that f y i t, y ζ i y for y H, where ζ i > is chosen such that exp ζ i + with α = min t,y [,] [ r 3,r 3 ] v t, y. Then F it, u x ζ i u x for z H. Let r 3 = maxr, H. If u Ω r 3, then min t [,] ut u H and r 3 = u u x t u xt r 3. Hence F r it, u x ζ i u xt ζ 3 i, similarly in Theorem 3.. Since T u = exp pξdξ Gs, s F i s, u x sds + exp exp g t, u x t exp exp pξdξ ζ i r 3ds + exp exp v u x exp r 3 we get Tu u for u P Ω r3. ζ i + α α α r 3 = u, Then it follows from Theorem., T has a fixed point u P in Ω r3 \Ω r for < and it can be easily seen that y t := u t xt is a positive solution of the problem.. such that r y r 3.

9 NONLOCAL SEMIPOSITONE FIRST ORDER PROBLEMS 447 Consequently,.. has two positive solutions for < < such that r y r y r 3. Now, we will give the existence of at least three positive solutions of the problem... For notational convenience, we denote k and K by exp k := n exp K := n exp pξdξ where α c. α, β, = min t,y [,] [b, b ] v t, y and β = max t,y [,] [, c ] v t, y for < b < Let the increasing, nonnegative and continuous functionals φ, ϕ and ψ be defined respectively on the cone P by, φy := max yt = yt, ϕy := min yt = yt and ψy := min yt = yt. t [,] t [,] t [,] We see that, for each y P, ψy = ϕy φy. In addition, for each y P, we know that y yt ψy. That is y for all y P. Theorem 3.3. Suppose that there exist constants < a < b < b < c such that C 4 f i t, y < c K M, for t [, ], y [, c ], C 5 f i t, y > b k M, for t [, ], y [b, b ], C 6 f i t, y < a M, for t [, ], y [, a]. K Then the problem.. has at least three positive solutions y, y, y 3 and there exist < d < a such that for < < d M. d φy < a < φy + M ω, ϕy < b < ϕy 3, ψy 3 < c Proof. Firstly, we consider the modified problem.6.7. It is well known that the existence of positive solutions to the boundary value problem.6.7 is equivalent to the existence of fixed points of the operator Tut := Gt, s F i s, u x sds + exp t exp g t, u x t, where Gt, s is given by.5.

10 448 E. ÇETIN AND F. S. TOPAL To make use of property i of Theorem., we choose u Pψ, c. Then ψu := min t [,] ut = ut = c. If we are recalling that u ψu c, we have u x t ut u c for all t [, ]. From C 4, we get ψtu = Tut = = = Gt, s F i s, u x sds + exp t exp g t, u x t exp pξdξ Gs, s exp exp exp c exp F i s, u x sds + f i + Mds + c K M + M + c c K n + c β n K + β = c, u x m β exp u x m β exp where K = n exp pξdξ m. β Then condition i of Theorem. holds. Secondly, we show that ii of Theorem. is fulfilled. For this, we select u Pϕ, b. Then ϕu = min t [,] ut = yt = b. This means ut b for all t [, ]. Since u P and < d < b, we have M M β u ut u x t u xt b Mωt b M ω b M b M b M = b. Note that u ϕu b. So we have b u xt b for all t [, ]. Therefore ϕtu = Tut = Gt, s + exp t exp exp pξdξ F i s, u x sds g t, u x t Gs, s f i + Mds

11 = NONLOCAL SEMIPOSITONE FIRST ORDER PROBLEMS exp exp exp pξdξ + exp exp exp exp exp exp exp exp b R exp where k = n exp R m holds. α u x exp exp exp exp n α α b k + m f i + M pξdξ b pξdξ α = b, m b k + b k n + b m α α. Then condition ii of Theorem. Finally, we verify that iii of Theorem. is also satisfied. We note that ut = a for t [, ] is a member of Pφ, a since φu = a < a. Therefore Pφ, a. Now let u Pφ, a. Then φu = max t [,] ut = ut = a. This means u x t ut a, for all t [, ]. we have Then by condition C 6 of this theorem and since φtu = Tut = = max v t, y max v t, y = β t,y [,] [,a] t,y [,] [, c ], Gt, s F i s, u x sds + exp t exp g t, u x t exp pξdξ Gs, s exp exp exp F i s, u x sds + f i + Mds + a K M + M + a a K n + a β u x m β exp u x m β exp β

12 45 E. ÇETIN AND F. S. TOPAL = where K = n a exp exp Theorem. is satisfied. n K + pξdξ m β m β = a,. Hence condition iii of Consequently it follows from Theorem. that T has at least three fixed points which are positive solutions u, u and u 3 belonging to Pψ, c of.6.7 such that < φu < a < φu, ϕu < b < ϕu 3, ψu 3 < c, and so there exists d > such that d max t [,] u t = φu. Similarly in Theorem 3. and Theorem 3., we can easily show that y i t := u i t xt, i =,, 3 are the solutions of the problem... Firstly, we shall show that y i t are positive. Since the solutions of the problem.6.7 satisfy the inequalities d φu < a < φu, ϕu < b < ϕu 3, ψu 3 < c and the definitions of the functionals φ, ϕ and ψ, we obtain u i d for i =,, 3. Thus y i t = u i t xt = u i t Mωt u i t M u i t M u it u i M u i t >, d since < < d M. Also, y it i =,, 3 satisfy the problem.. similarly in Theorem 3. and Theorem 3.. So, it follows that y, y and y 3 are positive solutions of the problem... In addition, we get d φy φu < a < φu < φu + M ω, ϕy ϕu < b ϕy 3 ϕu 3, and ψu 3 < c. REFERENCES [] D. R. Anderson, Existence of solutions for first-order multi-point problems with changing-sign nonlinearity, J Difference Equ. Appl., 4: , 8. [] D. R. Anderson, Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions, J. Math. Anal. Appl., 48: 38 33, 3. [3] A. Cabada and D. R. Vivero, Existence of solutions of first-order dynamic equations with nonlinear functional boundary value conditions, Nonlinear Anal., 63: , 5. [4] E. Çetin and F. S. Topal, Existence of solutions for a first order nonlocal boundary value problem with changing-sign nonlinearity, Turk. J. Math., 39: , 5. [5] J. M. Gallardo, Second order differential operators with integral boundary conditions and generation of semigroups, Rocky Mt. J. Math., 3: 65 9,.

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