Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls

2020 ◽

Vol 21 (2) ◽

pp. 205-218

Author(s):

Haide Gou ◽

Yongxiang Li

Keyword(s):

Fractional Derivative ◽

Measure Of Noncompactness ◽

Evolution Equations ◽

Mild Solutions ◽

Nonlocal Conditions ◽

Sobolev Type ◽

Initial Value ◽

Liouville Fractional Derivative ◽

Characteristic Solution Operators ◽

Impulsive Evolution Equations

AbstractIn this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

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Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

Open Mathematics ◽

10.1515/math-2021-0027 ◽

2021 ◽

Vol 19 (1) ◽

pp. 111-120

Author(s):

Qinghua Zhang ◽

Zhizhong Tan

Keyword(s):

Cauchy Problem ◽

Asymptotic Behavior ◽

Evolution Equation ◽

Nonlinear Operator ◽

Evolution Equations ◽

Bounded Function ◽

Inhomogeneous Term ◽

Linear Evolution ◽

Abstract Evolution Equations ◽

Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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Fractional evolution equations with nonlocal conditions in partially ordered Banach space

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.03.13 ◽

2018 ◽

Vol 34 (3) ◽

pp. 379-390

Author(s):

HEMANT KUMAR NASHINE ◽

◽

HE YANG ◽

RAVI P. AGARWAL ◽

◽

...

Keyword(s):

Fixed Point ◽

Evolution Equations ◽

Mild Solutions ◽

Nonlocal Conditions ◽

Coupled Fixed Point ◽

Positive Cone ◽

Bounded Linear ◽

Partially Ordered ◽

C0 Semigroup ◽

Uniformly Bounded

In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form where C Dσ t denotes the Caputo fractional derivative of order σ ∈ (0, 1), −A : D(A) ⊂ X → X generates a positive C0-semigroup T(t)(t ≥ 0) of uniformly bounded linear operator in X, b > 0 is a constant, f is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone K is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.

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