International Journal of Analysis and Applications ISSN 2291-8639 Volume 7, Number 2 (2015), 179-184 http://www.etamaths.com EXISTENCE RESULT FOR NONLINEAR INITIAL VALUE PROBLEMS INVOLVING THE DIFFERENCE OF TWO MONOTONE FUNCTIONS J.A. NANWARE Abstract. In this paper, monotone iterative technique for nonlinear initial value problems involving the difference of two functions is developed. As an application of this technique, existence of solution of nonlinear initial value problems involving the difference of two functions is obtained. 1. INTRODUCTION In the last few decades many authors pointed out that fractional deriva- tives and fractional integrals are very suitable for the description of properties of various real materials, e.g. polymers. It has been shown that new fractional - order models are more adequate than integer - order models. The advantages of fraction- al derivatives become apparent in modeling mechanical and electrical properties of real materials, and in many other fields, like theory of fractals [10]. Many dynami- cal models are described by fractional differential equations. Analytical as well as numerical methods are available for studying fractional differential equations such as power series method, compositional method, transform method and Adomain methods etc. (see details in [4, 14, 21] and references therein). The method of lower and upper solutions has been effectively used for proving the existence results for a class of variety of nonlinear problems. Monotone iter- ative technique coupled with method of lower and upper solutions is an effective mechanism that offers constructive procedure to obtain existence results in a closed set [5]. The basic theory of fractional differential equation with Riemann-Liouville fractional derivative is developed in [2, 7, 9]. In 2008, Lakshmikantham and Vatsala obtained the local and global existence of solution of Riemann-Liouville fraction- al differential equation and uniqueness of solution in [6, 8]. Recently, McRae [11] developed monotone method for Riemann-Liouville fractional differential equation with initial conditions and studied the qualitative properties of solutions of initial value problem. Recently, Nanware et.al. developed monotone method for sys- tem of Caputo fractional differential equations with periodic boundary conditions when the function is quasimonotone nondecreasing and mixed quasimonotone[3, 19], Riemann-Liouville fractional differential equations with integral boundary condi- tions when the function on the right is sum of nondecreasing and nonincreasing functions [15] and system of Riemann-Liouville fractional differential equations with 2010 Mathematics Subject Classification. 34A12,34C60. Key words and phrases. Fractional differential equations; initial value problems; lower and upper solutions, existence result. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 179 180 NANWARE integral boundary conditions when the function is quasimonotone nondecreasing [12, 16, 20]. Monotone method is successfully applied to obtain existence and u- niqueness of solutions of these problems [12, 13, 17, 18]. Monotone iterative technique for the following initial value problem u′ = f(t,u) −g(t,u), u(0) = u0, where f and g are in C([J × R,R]) and nondecreasing in u, uniformly in t, is de- veloped by Bhaskar and McRae [1]. In this paper, monotone iterative technique is developed for nonlinear initial value problems involving the difference of two monotone functions with Riemann-Liouville fractional derivative and successfully applied this technique to obtain existence of solution of the problem. The paper is organized in the following manner: Basic definitions and results are considered in the second section. Monotone iterative technique is developed in the third section and the technique is successively employed to prove existence result. In the last section some remarks are given. 2. DEFINITIONS AND BASIC RESULTS The Riemann-Liouville fractional derivative of order q, (0 < q < 1) [21] is defined as [0D q t ]u(t) = 1 Γ(1 −q) d dt ∫ t 0 u(s) (t−s)q ds.(2.1) Consider the following Riemann-Liouville fractional differential equation (2.2) [0D q t ]u(t) = f(t,u(t)) −g(t,u(t)), t ∈ J = [0,T] with initial condition (2.3) u(0) = u0 where f,g ∈ C(J × R,R) are both nondecreasing in u(t), uniformly in t. This is called a nonlinear initial value problem (IVP). Definition 2.1. A pair of functions v(t) and w(t) in Cp(J,R) are called ordered lower and upper solutions of the nonlinear IVP (2.2) − (2.3) if [0D q t ]v(t) ≤ f(t,v(t)) −g(t,v(t)), v(0) ≤ u0 [0D q t ]w(t) ≥ f(t,w(t)) −g(t,w(t)), w(0) ≥ u0 Definition 2.2. The functions v(t) and w(t) in Cp(J,R) are called coupled lower and upper solutions of the nonlinear IVP (2.2) − (2.3) if [0D q t ]v(t) ≤ f(t,v(t)) −g(t,w(t)), v(0) ≤ u0 [0D q t ]w(t) ≥ f(t,w(t)) −g(t,v(t)), w(0) ≥ u0 Lemma 2.1. [2] Let m ∈ Cp(J,R) and for any t1 ∈ (0,T] we have m(t1) = 0 and m(t) < 0 for 0 ≤ t < t1. Then it follows that Dqm(t1) ≥ 0. Lemma 2.2. [6] Let {u�(t)} be a family of continuous functions on J, for each � > 0 where Dqu�(t) = f(t,u�(t)), u�(t0) = u�(t)(t− t0)1−q}t=t0 and |f(t,u�(t))| ≤ M for t0 ≤ t ≤ T . Then the family {u�(t)} is equicontinuous on [t0,T]. Theorem 2.1. [11] Let v,w ∈ Cp(J,R),f ∈ C([t0,T] ×R,R) and NONLINEAR INITIAL VALUE PROBLEMS 181 : (i) Dqv(t) ≤ f(t,v(t)) and : (ii)Dqw(t) ≥ f(t,w(t)), t0 < t ≤ T. Assume f(t,u) satisfies the Lipschitz condition f(t,x) −f(t,y) ≤ L(x−y), x ≥ y,L > 0. Then v0 < w0, where v0 = v(t)(t−t0)1−q|t=t0 and w0 = w(t)(t−t0)1−q|t=t0, implies v(t) ≤ w(t), t ∈ [t0,T]. 3. MAIN RESULTS In this section we develop monotone iterative technique for nonlinear IVP (2.2)− (2.3). As an application of the technique we prove the existence of solution of nonlinear initial value problem (2.2) − (2.3). Theorem 3.1. Assume that: (i): f(t,u(t)) and g(t,u(t)) in C[J ×R,R] are nondecreasing in u(t), (ii): v0(t) and w0(t) in C(J,R) are coupled lower and upper solutions of IVP (2.2) − (2.3) such that v0(t) ≤ w0(t), t ∈ J = [0,T]. Then there exist monotone sequences {vn(t)} and {wn(t)} in C(J,R) such that {vn(t)}→ v(t) and {wn(t)}→ w(t) as n →∞, uniformly and monotonically on J and the functions v(t) and w(t) are the coupled minimal and maximal solutions of nonlinear IVP (2.2) − (2.3) respectively. Proof : Consider the following coupled linear system of fractional differential e- quations with initial conditions (LIVP) (3.1) vn+1(t) = f(t,vn) −g(t,wn), vn+1(0) = u0 [0D q t ]wn+1(t) = f(t,wn) −g(t,vn), wn+1(0) = u0 Since the functions f(t,u) and g(t,u) are continuous on J × R, the solutions of LIVP (3.1) exist on J. We claim that v0(t) ≤ v1(t) ≤ w1(t) ≤ w0(t) on J. For this, set p(t) = v1(t) −v0(t) then we have [0D q t ]p(t) = [0D q t ]v1(t) − [0D q t ]v0(t) ≥ f(t,v0) −g(t,w0) −f(t,v0) + g(t,w0) [0D q t ]p(t) ≥ 0 p(0) = 0 By applying Theorem 2.1, we get v0(t) ≤ v1(t). Similarly, we can prove w1(t) ≤ w0(t) on J. Also, we prove that v1(t) ≤ w1(t) on J. Set p(t) = w1(t) −v1(t). Then we have [0D q t ]p(t) = [0D q t ]w1(t) − [0D q t ]v1(t) ≥ f(t,w0) −g(t,v0) −f(t,v0) + g(t,w0) [0D q t ]p(t) ≥ 0 p(0) ≥ 0 Thus,by applying Theorem 2.1, we get v1(t) ≤ w1(t). Assume that for some k > 1, vk−1(t) ≤ vk(t) ≤ wk(t) ≤ wk−1(t). We claim that 182 NANWARE vk(t) ≤ vk+1(t) ≤ wk+1(t) ≤ wk(t) on J. To prove this, set p(t) = vk(t) −vk+1(t). Since f(t,u) and g(t,u) are nondecreasing in u, we get [0D q t ]p(t) = [0D q t ]vk(t) − [0D q t ]vk+1(t) ≤ f(t,vk−1) −g(t,wk−1) −f(t,vk) + g(t,wk) [0D q t ]p(t) ≤ 0 p(0) = 0 By applying Theorem 2.1, we have vk ≤ vk+1 on J. By induction, it follows that vk ≤ vk+1 for all k ≥ 1, t ∈ J. Similarly we prove wk+1(t) ≤ wk(t) on J. Next we prove vk+1(t) ≤ wk+1(t). Con- sider p(t) = wk+1(t) −vk+1(t). Since f(t,u) and g(t,u) are nondecreasing in u, we have [0D q t ]p(t) = [0D q t ]wk+1(t) − [0D q t ]vk+1(t) ≥ f(t,wk) −g(t,vk) −f(t,vk) + g(t,wk) [0D q t ]p(t) ≥ 0 p(0) = 0 Hence, by applying Theorem 2.1, we get vk+1 ≤ wk+1 on J. By induction, we get vk+1 ≥ wk+1 for all k ≥ 1, t ∈ J. Thus we have sequences vn and wn on J such that v0 ≤ v1 ≤ v2 ≤ ... ≤ vn ≤ wn ≤ wn−1 ≤ ... ≤ w2 ≤ w1 ≤ w0. Clearly the sequences {vn} and {wn} are nondecreasing and bounded below and nondecreasing and bounded above respectively. By Lemma 2.2 it follows that the sequences {vn} and {wn} are equicontinuous and uniformly bounded. Applying Ascoli-Arzela theorem, there exist convergent subsequences {vnk} and {wnk} con- verging to v and w uniformly and monotonically on J respectively. Then we have {vn(t)}→ v(t) and {wn(t)}→ w(t) as n →∞. Using corresponding fractional Volterra integral equations (3.2) vn+1(t) = u0 + 1 Γ(q) ∫ T t0 (t−s)q−1 { f(s,vn(s)) −g(s,wn(s)) } ds wn+1(t) = u0 + 1 Γ(q) ∫ T t0 (t−s)q−1 { f(s,wn(s)) −g(s,vn(s)) } ds it follows that v(t) and w(t) are solutions of (3.1). Next we claim that v(t) and w(t) are the coupled minimal and maximal solutions of LIVP (3.1). For this, let u(t)) be any solution of nonlinear IVP (2.2) − (2.3) different from v(t) and w(t), so that there exists k such that vk(t) ≤ u(t) ≤ wk(t) on J and set p(t) = u(t) −vk+1(t) so that [0D q t ]p(t) = [0D q t ]u(t) − [0D q t ]vk+1(t) ≥ f(t,u) −g(t,u) −f(t,vk) + g(t,wk) [0D q t ]p(t) ≥ 0 p(0) = 0. By applying Theorem 2.1, we have vk+1(t) ≤ u(t) on J. Since v0(t) ≤ u(t) on J, by induction it follows that vk(t) ≤ u(t) for all k. NONLINEAR INITIAL VALUE PROBLEMS 183 Similarly we prove u(t) ≤ wk(t) for all k on J. Thus vk(t) ≤ uk(t) ≤ wk(t) on [0,T]. In limiting case, we have v(t) ≤ u(t) ≤ w(t) on [0,T]. This completes the proof. 4. REMARKS (1) If f(t,u) and g(t,u) in C(J×R) and if there exists positive constants M,N such that f(t,u) + Mu and g(t,u) + Nu are both nondecreasing, for t ∈ J and v0 ≤ u ≤ w0 then we may write G(t,u) = f(t,u) −g(t,u) = |f(t,u) + (M + N)u|− |g(t,u) + (M + N)u| = f1(t,u) −g1(t,u). Clearly f1 and g1 are both monotone nondecreasing functions and Theorem 3.1 can be applied. (2) If g = 0 and as in Theorem 2.1, f satisfies for some M > 0, f(t,u1) −f(t,u2) ≥−M(u1 −u2) whenever u1 ≥ u2. Define f1 = f(t,u) + Mu. Then f1 is nondecreasing in u and we may write [0D q t ]u(t) = f(t,u), u(0) = u0 as [0D q t ]u(t) = f1(t,u) −Mu, u(0) = u0 and with appropriate modifications we can apply Theorem 3.1 to [0D q t ]u(t) = f1(t,u) −Mu, u(0) = u0. Thus we obtain new result. (3) If f = 0 and g satisfies for some M > 0, g(t,u1) −g(t,u2) ≥−M(u1 −u2), whenever u1 ≥ u2, we define g1(t,u) = g(t,u) + Mu. Then g1 is nonde- creasing in u and we write [0D q t ]u(t) = −g(t,u), u(0) = u0 as [0D q t ]u(t) = Mu−g1(t,u), u(0) = u0. Theorem 2.1 may be applied to obtain the coupled minimal and maximal solutions of the original problem. (4) Theorem 3.1 can easily be modified to include the IVP of the form [0D q t ]u(t) + Ku(t) = f(t,u) −g(t,u), u(0) = 0. 184 NANWARE References [1] T.G.Bhaskar, F.A.McRae, Monotone Iterative Techniques for Nonlinear Problems Involving The Difference of Two Monotone Functions, Applied Mathematics and Computation 133 (2002), 187-192. [2] J.Vasundhara Devi, F.A.McRae, Z. 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