By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

1. Introduction

The differential equations involving fractional derivatives in time have recently been studied extensively. One can see, for instance, the monographs [1–5] and the survey [6–8]. In particular, there has been a significant development in fractional evolution equations. Existence of solutions for fractional evolution equations has been studied by many authors during recent years. Many excellent results are obtained in this field; see [9–19] and the references therein. In [9, 10], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then the author investigated the existence, uniqueness, and regularity of solutions of fractional integrodifferential equations in [11, 12]. Recently, this theory was developed by Zhou et al. [13–16]. Particularly, they studied the existence and controllability of mild solution of fractional delay integrodifferential equations with a compact analytic semigroup in [16]. In [17–19], the authors studied the existence of mild solutions of fractional impulsive delay or impulsive evolution equations. But as far as we know, there are no results on the existence of positive solutions of fractional evolution equations.

In this paper, by using the fixed point theorems combined with the theory of analytic semigroup, we investigate the existence of positive mild solutions for the initial value problem (IVP) of fractional evolution equations in Banach space X as

(1)

where

denotes the Caputo fractional derivative of order q ∈ (0,1) with the lower limits zero, −A : D(A) ⊂ X → X is the infinitesimal generator of an analytic semigroup S(t) (t ≥ 0) of uniformly bounded linear operators, and f is the nonlinear term and will be specified later.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of the analytic semigroup and the definition of mild solutions of IVP(1). In Section 3, we study the existence of positive mild solutions for the IVP(1). In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section 3.

2. Preliminaries

In this section, we introduce some basic facts about the fractional power of the generator of analytic semigroup and the fractional calculus that are used throughout this paper.

Let X be a Banach space with norm ∥·∥. Throughout this paper, we assume that −A : D(A) ⊂ X → X is the infinitesimal generator of an analytic semigroup S(t) (t ≥ 0) of uniformly bounded linear operator in X; that is, there exists M ≥ 1 such that ∥S(t)∥≤M for all t ≥ 0. Without loss of generality, let 0 ∈ ρ(−A), where ρ(−A) is the resolvent set of −A. Then for any α > 0, we can define A^−α by

(2)

Then A^α can be defined by A^α : = (A^−α) ⁻¹ because A^−α is one to one. It can be shown that each A^α has dense domain and that D(A^β) ⊂ D(A^α) for 0 ≤ α ≤ β. Moreover, A^α+βx = A^αA^βx = A^βA^αx for every α, β ∈ ℝ and x ∈ D(A^μ) with μ : = max{α, β, α + β}, where A⁰ = I, I is the identity in X (for proofs of these facts we refer to the literature [20–22]).

We denote by X_α the Banach space of D(A^α) equipped with norm ∥x∥_α = ∥A^αx∥ for x ∈ D(A^α), which is equivalent to the graph norm of A^α. Then we have X_β↪X_α for 0 ≤ α ≤ β ≤ 1 (with X₀ = X), and the embedding is continuous. Moreover, A^α has the following basic properties.

Lemma 1 (see [23].) A^α has the following properties.

(i)
S(t) : X → X_α for each t > 0 and α ≥ 0.
(ii)
A^αS(t)x = S(t)A^αx for each x ∈ D(A^α) and t ≥ 0.
(iii)
For every t > 0, A^αS(t) is bounded in X and there exists M_α > 0 such that
(3)

Let J be a closed interval on ℝ⁺ = [0, ∞). In the following we denote by C(J, X_α) the Banach space of all continuous functions from J into X_α endowed with supnorm given by ∥u∥_C = sup_t∈J∥u(t)∥_α for u ∈ C(J, X_α). For any t ≥ 0, denote by S_α(t) the restriction of S(t) to X_α. From Lemma 1(i) and (ii), for any x ∈ X_α, we have

(4)

as t → 0. Therefore, S(t) (t ≥ 0) is a strongly continuous semigroup in X_α, and ∥S_α(t)∥_α ≤ ∥S(t)∥ for all t ≥ 0. To prove our main results, the following lemma is also needed.

Lemma 2 (see [24].)If S(t) (t ≥ 0) is a compact semigroup in X, then S_α(t) (t ≥ 0) is a compact semigroup in X_α, and hence it is norm continuous.

Let us recall the following known definitions in fractional calculus. For more details, see [9, 13–16, 18, 19].

Definition 3. The fractional integral of order σ > 0 with the lower limits zero for a function f is defined by

(5)

where Γ is the gamma function.

The Riemann-Liouville fractional derivative of order n − 1 < σ < n with the lower limits zero for a function f can be written as

(6)

Also the Caputo fractional derivative of order n − 1 < σ < n with the lower limits zero for a function f ∈ Cⁿ[0, ∞) can be written as

(7)

Remark 4. (1) The Caputo derivative of a constant is equal to zero.

(2) If f is an abstract function with values in X, then integrals which appear in Definition 3 are taken in Bochner′s sense.

Lemma 5 (see [14].)A measurable function h : J → X is Bochner integrable if ∥h∥ is Lebesgue integrable.

For x ∈ X, we define two families {U(t)} _t≥0 and {V(t)} _t≥0 of operators by

(8)

where

(9)

where η_q is a probability density function defined on (0, ∞), which has properties η_q(θ) ≥ 0 for all θ ∈ (0, ∞) and

. It is not difficult to verify (see [14]) that for μ ∈ [0,1], we have

(10)

Clearly, if the semigroup S(t) (t ≥ 0) is positive, then, by the definitions, the operators U(t) and V(t) are also positive for all t ≥ 0.

The following lemma follows from the results in [14, Lemma 2.9] and [15, Lemmas 3.2–3.5].

Lemma 6. The operators U(t) and V(t) have the following properties.

(i)
For any fixed t ≥ 0 and any x ∈ X_α, one has
(11)
(ii)
The operators U(t) and V(t) are strongly continuous for all t ≥ 0.
(iii)
If the semigroup S(t) (t ≥ 0) is compact, then U(t) and V(t) are compact operators in X for t > 0.
(iv)
If the semigroup S_α(t) (t ≥ 0) is norm continuous, then the restriction of U(t) to X_α and the restriction of V(t) to X_α are uniformly continuous for t > 0.

Definition 7 (see [25], [26].)Let B be a bounded set of a real Banach space E. Set β(B) = inf{δ > 0: B can be expressed as the union of a finite number of sets such that the diameter of each set does not exceed δ; that is, with diam(B_i) ≤ δ, i = 1,2, …, m}. β(B) is called the Kuratowski measure of noncompactness of set B.

It is clear that 0 ≤ β(B) < ∞. For the Kuratowski measure of noncompactness, we have the following well-known results.

Lemma 8 (see [26].)If D ∈ C(J, E) is bounded and equicontinuous, then

(12)

where D(J) = {x(t) : x ∈ D, t ∈ J}.

Lemma 9 (see [27].)Let D be a countable set of strongly measurable function x : J → E such that there exists an M ∈ L(J, ℝ⁺) such that ∥x(t)∥≤M(t) a.e., t ∈ J for all x ∈ D. Then β(D(t)) ∈ L(J, ℝ⁺) and

(13)

Lemma 10 (see [25] Mönch fixed point theorem.)Let B be a closed and convex subset of E and y₀ ∈ B. Assume that the continuous operator A : B → B has the following property: D ⊂ B is countable, and is relatively compact. Then A has a fixed point in B.

Based on an overall observation of the previous related literature, in this paper we adopt the following definition of mild solution of IVP(1).

Definition 11. By a mild solution of the IVP(1), one means a function u ∈ C(J, X_α) satisfying

(14)

for all t ∈ J.

3. Existence of Positive Mild Solutions

In this section, we introduce the existence theorems of positive mild solutions of the IVP(1). The discussions are based on fractional calculus and fixed point theorems.

Let λ₁ be the smallest positive real eigenvalue of the linear operator A, and let e₁ ∈ D(A) be the positive eigenvector corresponding to λ₁. For any T > 0 and r > 0, we write

(15)

where σ > 0 is a constant. Our main results are as follows.

Theorem 12. Let −A : D(A) ⊂ X → X be the infinitesimal generator of a positive and compact analytic semigroup S(t) (t ≥ 0) of uniformly bounded linear operators. Assume that f ∈ C(ℝ⁺ × X_α, X) satisfies the following conditions.

(H₁ For any u ∈ Ω_r, one has
(16)
H₂ f maps bounded sets of ℝ⁺ × X_α into bounded sets of X.

If u₀ ∈ X_α with u₀ ≥ σe₁ and αq < 1/2 for some 1/2 < q < 1, then the IVP(1) has at least one positive mild solution u ∈ C([0, T), X_α). And if T < ∞, one has

Proof. For any t₀ ≥ 0 and x₀ ∈ X_α with x₀ ≥ σe₁, we first prove that the initial value problem (IVP) of fractional evolution equations

(17)

has at least one positive mild solution on

, where

is a positive constant and will be given later.

Let . Denote

(18)

Then

is a nonempty bounded convex closed set. The assumption (H₂) implies that there is a constant C = C(t₀) > 0 such that

(19)

for any t ∈ J and

Define an operator Q by

(20)

By the continuity of f, it is not difficult to prove that Q : C(J, X_α) → C(J, X_α) is continuous. By the positivity of the semigroup S(t) (t ≥ 0), the assumption (H₁), and (20), we easily see that (Qu)(t)≥(Qσe₁)(t). Clearly, the positive mild solution of the IVP(17) on J is equivalent to the fixed point of operator Q in

. We will use Schauder fixed point theorem to prove that Q has fixed points in

We first prove that is continuous. Let . For any and t ∈ J, by Lemma 6, (10), (19), and (20), we have

(21)

Let v₀ ≡ σe₁. Then v₀(t) = σe₁ for any t ∈ J and

(22)

By the positivity of the semigroup S(t) (t ≥ 0), assumption (H₁), and (20), for any t ∈ J, we have

(23)

Thus,

is continuous.

By using a similar argument as in the proof of Theorem 3.1 in [14], we can prove that is a compact operator. Hence by Schauder fixed point theorem, the operator Q has at least one fixed point u^* in , which satisfies u^*(t) ≥ σe₁ > 0 for all t ∈ J. Hence u^* is a positive mild solution of the IVP(1) on J.

Therefore, there exists [0, h₀] such that the IVP(1) has at least one positive mild solution u^* ∈ C([0, h₀], X_α). Now, by the standard proof method of extension theorem of initial value problem, u^* can be extended to a saturated solution u ∈ C([0, T), X_α) of the IVP(1), whose existence interval is [0, T), and if T < ∞, we have

For any T > 0 and r > 0, define Ω_r as in (15). If f(t, u) is increasing in Ω_r, that is, f(t, u) satisfies the condition

for any u₁, u₂ ∈ Ω_r with u₁(t) ≤ u₂(t) for all t ∈ [0, T], we have
(24)

then we have f(t, u(t)) ≥ f(t, σe₁) for any u ∈ Ω_r and t ∈ [0, T]. On the other hand, if f(t, u) satisfies linear growth condition, then it maps the bounded sets into the bounded sets. Hence by Theorem 12, we have the following existence result.

Corollary 13. Let −A : D(A) ⊂ X → X be the infinitesimal generator of a positive and compact analytic semigroup S(t) (t ≥ 0) of uniformly bounded linear operators. Assume that f ∈ C(ℝ⁺ × X_α, X) satisfies condition (H₁)* and

(H₂)* there exists a constant a_f > 0 such that
(25)

for all t ∈ [0, T] and x ∈ X_α.

If f(t, σe₁) ≥ λ₁σe₁ for all t ∈ [0, T], u₀ ∈ X_α with u₀ ≥ σe₁ and αq < 1/2 for some 1/2 < q < 1, then the IVP(1) has at least one positive mild solution u ∈ C([0, T), X_α). And if t < ∞, one has

Since the analytic semigroup is norm continuous, it follows that we can delete the compactness condition on the analytic semigroup S(t) (t ≥ 0) and obtain the following existence result.

Theorem 14. Assume that −A : D(A) ⊂ X → X is the infinitesimal generator of a positive analytic semigroup S(t) (t ≥ 0) of uniformly bounded linear operators, and that f ∈ C(ℝ⁺ × X_α, X) satisfies the condition (H₁) and

H₃ for any T > 0 and r > 0, f(t, Ω_r): = {f(t, u) : u ∈ Ω_r} is relatively compact in X_α for all t ∈ [0, T], where Ω_r is defined as in (15).

If u₀ ∈ X_α with u₀ ≥ σe₁ and αq < 1/2 for some 1/2 < q < 1, then the IVP(1) has at least one positive mild solution u ∈ C([0, T), X_α). And if t < ∞, one has

Proof. For any t₀ ≥ 0 and x₀ ∈ X_α with x₀ ≥ σe₁, we first prove that the IVP(17) has at least one positive mild solution on , where is a constant and will be specified later. Define an operator Q as in (20). Let . Write as in (18). The condition (H₃) implies that is bounded for any t ∈ J, that is, there is a positive constant C = C(t₀) such that

(26)

Let

. A similar argument as in the proof of Theorem 12 shows that

is continuous and

is equicontinuous.

Thus, for any , let D(t): = {u(t) : u ∈ D}, t ∈ J. Since is equicontinuous and bounded, by Lemma 8, we have

(27)

Now, let with for some . It is obvious that

(28)

Hence by Lemma 9 and (20), we have

(29)

It follows that β((QD)(t)) = 0 for all t ∈ J. By Lemma 8 and (27), we have β(QD) = max_t∈Jβ((QD)(t)) = 0. Thus, we have

(30)

This implies that D is relatively compact. Therefore, by Mönch fixed point theorem, the operator Q has at least one fixed point

, which satisfies u^*(t) ≥ σe₁ > 0 for all t ∈ J. Hence u^* is a positive mild solution of the IVP(17) on J.

Therefore, there exists [0, h₀] such that the IVP(1) has at least one positive mild solution u^* ∈ C([0, h₀], X_α). u^* can be extended to a saturated solution u ∈ C([0, T), X_α) of IVP(1), whose existence interval is [0, T) and when t ≤ ∞, we have .

4. Positive Mild Solutions of Parabolic Equations

Let Ω ⊂ ℝ^N be a bounded domain with a sufficiently smooth boundary ∂Ω. Let

(31)

be a uniformly elliptic differential operator of divergence form in

, where the coefficients

and

for some μ ∈ (0,1). We assume that [a_ij(x)] _N×N is a positive define symmetric matric for every

, and there exists a constant ν > 0 such that

(32)

Let a₀(x) ≥ 0 on

. We use (x, t, η) to denote a generic point of

, where ℝ = [0, +∞) and ℝ = (−∞, +∞). Let

be a continuous function. We discuss the existence of positive mild solutions for the parabolic initial boundary value problem (IBVP)

(33)

where 0 < q < 1 is a constant.

Let λ₁ be the smallest positive real eigenvalue of elliptic operator A(x, D) under the Dirichlet boundary condition u|_∂Ω = 0. It is well known (cf. Amann [22, 28]) that λ₁ > 0. Let e₁(x) be the positive eigenvector corresponding to λ₁. Assume that

is continuous and satisfies the following conditions.

F₁ For any T > 0 and r > 0, there exists a constant σ > 0 such that
(34)

where η ∈ ℝ with σe₁(x) ≤ η ≤ r.

F₂ For any T > 0, there exists a constant a_f > 0 such that
(35)

Let X = L²(Ω). Define an operator A : D(A) ⊂ X → X by

(36)

It is well known (cf. Li [29]) that −A generates a compact analytic semigroup S(t) (t ≥ 0) and

. By the maximum principle of the equation of the parabolic type, it is easy to prove that S(t) (t ≥ 0) is also a positive semigroup in X. The assumptions (F₁) and (F₂) imply that the mapping f : ℝ⁺ × X_1/2 → X defined by

(37)

is continuous and satisfies the conditions (H₁) and (H₂). Thus, the IBVP(33) can be rewritten into the abstract form of IVP(1). By Theorem 12, we have the following existence result for the IBVP(33).

Theorem 15. Assume that is continuous and satisfies conditions (F₁) and (F₂). If with φ(x) ≥ σe₁(x) for any and 1/2 < q < 1, then the IBVP(33) has at least one positive mild solution u that satisfies u(x, t) ≥ σe₁(x) for any and t ∈ [0, T]. And if T < +∞, one has

Acknowledgments

This research was supported by the NNSF of China (Grant no. 11261053), the Fundamental Research Funds for the Gansu Universities, and the Project of NWNU-LKQN-11-3.

References

1 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, New York, NY, USA, A Wiley-Interscience Publication, MR1219954, ZBL0943.82582.
Google Scholar
2 Podlubny I., Fractional Differential Equations, 1999, 198, Academic Press, San Diego, Calif, USA, Mathematics in Science and Engineering, MR1658022, ZBL1056.93542.
Web of Science® Google Scholar
3 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, 204, Elsevier Science B.V., Amsterdam, The Netherlands, North-Holland Mathematics Studies, MR2218073, https://doi.org/10.1016/S0304-0208(06)80001-0, ZBL1206.26007.
10.1016/S0304-0208(06)80001-0
Google Scholar
4 Lakshmikantham V., Leela S., and Devi J., Theory of Fractional Dynamic Systems, 2009, Cambridge Scientific Publishers, Cambridge, UK.
Google Scholar
5 Diethelm K., The Analysis of Fractional Differential Equations, 2010, 2004, Springer, Berlin, Germany, Lecture Notes in Mathematics, https://doi.org/10.1007/978-3-642-14574-2, MR2680847, ZBL1222.37107.
10.1007/978-3-642-14574-2
Web of Science® Google Scholar
6 Agarwal R. P., Belmekki M., and Benchohra M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations. (2009) 2009, 47, 981728, MR2505633, ZBL1182.34103.
10.1155/2009/981728
Web of Science® Google Scholar
7 Agarwal R. P., Benchohra M., and Hamani S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae. (2010) 109, no. 3, 973–1033, https://doi.org/10.1007/s10440-008-9356-6, MR2596185, ZBL1198.26004.
10.1007/s10440-008-9356-6
Web of Science® Google Scholar
8 Agarwal R. P., Lakshmikantham V., and Nieto J. J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis. Theory, Methods & Applications. (2010) 72, no. 6, 2859–2862, https://doi.org/10.1016/j.na.2009.11.029, MR2580143, ZBL1188.34005.
10.1016/j.na.2009.11.029
Web of Science® Google Scholar
9 El-Borai M. M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals. (2002) 14, no. 3, 433–440, https://doi.org/10.1016/S0960-0779(01)00208-9, MR1903295, ZBL1005.34051.
10.1016/S0960-0779(01)00208-9
Web of Science® Google Scholar
10 El-Borai M. M., The fundamental solutions for fractional evolution equations of parabolic type, Journal of Applied Mathematics and Stochastic Analysis. (2004) no. 3, 197–211, https://doi.org/10.1155/S1048953304311020, MR2102832, ZBL1081.34053.
10.1155/S1048953304311020
Google Scholar
11 El-Borai M. M., Semigroups and some nonlinear fractional differential equations, Applied Mathematics and Computation. (2004) 149, no. 3, 823–831, https://doi.org/10.1016/S0096-3003(03)00188-7, MR2033165, ZBL1046.34079.
10.1016/S0096-3003(03)00188-7
Web of Science® Google Scholar
12 El-Borai M., El-Nadi K., and El-Akabawy E., Fractional evolution equations with nonlocal conditions, International Journal of Applied Mathematics and Mechanics. (2008) 4, no. 6, 1–12.
Google Scholar
13 Zhou Y. and Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis. Real World Applications. (2010) 11, no. 5, 4465–4475, https://doi.org/10.1016/j.nonrwa.2010.05.029, MR2683890.
10.1016/j.nonrwa.2010.05.029
Web of Science® Google Scholar
14 Wang J. and Zhou Y., A class of fractional evolution equations and optimal controls, Nonlinear Analysis. Real World Applications. (2011) 12, no. 1, 262–272, https://doi.org/10.1016/j.nonrwa.2010.06.013, MR2728679, ZBL1214.34010.
10.1016/j.nonrwa.2010.06.013
Web of Science® Google Scholar
15 Zhou Y. and Jiao F., Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications. (2010) 59, no. 3, 1063–1077, https://doi.org/10.1016/j.camwa.2009.06.026, MR2579471, ZBL1189.34154.
10.1016/j.camwa.2009.06.026
Web of Science® Google Scholar
16 Wang J., Zhou Y., and Wei W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Communications in Nonlinear Science and Numerical Simulation. (2011) 16, no. 10, 4049–4059, https://doi.org/10.1016/j.cnsns.2011.02.003, MR2802711, ZBL1223.45007.
10.1016/j.cnsns.2011.02.003
Web of Science® Google Scholar
17 Tai Z. and Wang X., Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces, Applied Mathematics Letters. (2009) 22, no. 11, 1760–1765, https://doi.org/10.1016/j.aml.2009.06.017, MR2560992, ZBL1181.34078.
10.1016/j.aml.2009.06.017
Web of Science® Google Scholar
18 Debbouche A. and Baleanu D., Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Computers & Mathematics with Applications. (2011) 62, no. 3, 1442–1450, https://doi.org/10.1016/j.camwa.2011.03.075, MR2824731, ZBL1228.45013.
10.1016/j.camwa.2011.03.075
Web of Science® Google Scholar
19 Mophou G. M., Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications. (2010) 72, no. 3-4, 1604–1615, https://doi.org/10.1016/j.na.2009.08.046, MR2577561, ZBL1187.34108.
10.1016/j.na.2009.08.046
Web of Science® Google Scholar
20 Wang R.-N., Xiao T.-J., and Liang J., A note on the fractional Cauchy problems with nonlocal initial conditions, Applied Mathematics Letters. (2011) 24, no. 8, 1435–1442, https://doi.org/10.1016/j.aml.2011.03.026, MR2793647, ZBL1251.45008.
10.1016/j.aml.2011.03.026
Web of Science® Google Scholar
21 Sobolevskii P., Equations of parabolic type in a Banach space, American Mathematics Society Translations. Series 2. (1966) 49, 1–62.
10.1090/trans2/049/01
Google Scholar
22 Amann H., Periodic solutions of semilinear parabolic equations, Nonlinear Analysis, 1978, Academic Press, New York, NY, USA, 1–29, MR499089, ZBL0464.35050.
Google Scholar
23 Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 1983, 44, Springer, New York, NY, USA, Applied Mathematical Sciences, https://doi.org/10.1007/978-1-4612-5561-1, MR710486.
10.1007/978-1-4612-5561-1
Google Scholar
24 Liu H. and Chang J.-C., Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Analysis. Theory, Methods & Applications. (2009) 70, no. 9, 3076–3083, https://doi.org/10.1016/j.na.2008.04.009, MR2503051, ZBL1185.34112.
10.1016/j.na.2008.04.009
Web of Science® Google Scholar
25 Deimling K., Nonlinear Functional Analysis, 1985, Springer, Berlin, Germany, MR787404.
10.1007/978-3-662-00547-7
Google Scholar
26 Guo D., Lakshmikantham V., and Liu X., Nonlinear Integral Equations in Abstract Spaces, 1996, 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, Mathematics and Its Applications, MR1418859.
10.1007/978-1-4613-1281-9
Google Scholar
27 Liang J. and Xiao T.-J., Solvability of the Cauchy problem for infinite delay equations, Nonlinear Analysis. Theory, Methods & Applications. (2004) 58, no. 3-4, 271–297, https://doi.org/10.1016/j.na.2004.05.005, MR2073526, ZBL1058.34106.
10.1016/j.na.2004.05.005
Web of Science® Google Scholar
28 Amann H., Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, Nonlinear Operators and the Calculus of Variations, 1976, 543, Springer, Berlin, Germany, 1–55, Lecture Notes in Mathematics, MR0513051, ZBL0345.47045.
10.1007/BFb0079941
Google Scholar
29 Li Y., Existence and asymptotic stability of periodic solution for evolution equations with delays, Journal of Functional Analysis. (2011) 261, no. 5, 1309–1324, https://doi.org/10.1016/j.jfa.2011.05.001, MR2807101, ZBL1233.34028.
10.1016/j.jfa.2011.05.001
Web of Science® Google Scholar

Citing Literature

All articles

Positive Solutions for the Initial Value Problem of Fractional Evolution Equations

Abstract

1. Introduction

2. Preliminaries

3. Existence of Positive Mild Solutions

4. Positive Mild Solutions of Parabolic Equations

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Positive Solutions for the Initial Value Problem of Fractional Evolution Equations

Abstract

1. Introduction

2. Preliminaries

3. Existence of Positive Mild Solutions

4. Positive Mild Solutions of Parabolic Equations

Acknowledgments

References

Citing Literature

References

Related

Information